Introduction to Probability and Statistics 13th - Mendenhall, Beaver

Area

0 TABLE 3

Areas under the Normal Curve, pages 688–689

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

3.4 3.3 3.2 3.1 3.0

.0003 .0005 .0007 .0010 .0013

.0003 .0005 .0007 .0009 .0013

.0003 .0005 .0006 .0009 .0013

.0003 .0004 .0006 .0009 .0012

.0003 .0004 .0006 .0008 .0012

.0003 .0004 .0006 .0008 .0011

.0003 .0004 .0005 .0008 .0011

.0003 .0004 .0005 .0007 .0010

.0002 .0003 .0005 .0007 .0010

2.9 2.8 2.7 2.6 2.5

.0019 .0026 .0035 .0047 .0062

.0018 .0025 .0034 .0045 .0060

.0017 .0024 .0033 .0044 .0059

.0017 .0023 .0032 .0043 .0057

.0016 .0023 .0031 .0041 .0055

.0016 .0022 .0030 .0040 .0054

.0015 .0021 .0029 .0039 .0052

.0015 .0021 .0028 .0038 .0051

.0014 .0020 .0027 .0037 .0049

.0014 .0019 .0026 .0036 .0048

2.4 2.3 2.2 2.1 2.0

.0082 .0107 .0139 .0179 .0228

.0080 .0104 .0136 .0174 .0222

.0078 .0102 .0132 .0170 .0217

.0075 .0099 .0129 .0166 .0212

.0073 .0096 .0125 .0162 .0207

.0071 .0094 .0122 .0158 .0202

.0069 .0091 .0119 .0154 .0197

.0068 .0089 .0116 .0150 .0192

.0066 .0087 .0113 .0146 .0188

.0064 .0084 .0110 .0143 .0183

1.9 1.8 1.7 1.6 1.5

.0287 .0359 .0446 .0548 .0668

.0281 .0351 .0436 .0537 .0655

.0274 .0344 .0427 .0526 .0643

.0268 .0336 .0418 .0516 .0630

.0262 .0329 .0409 .0505 .0618

.0256 .0322 .0401 .0495 .0606

.0250 .0314 .0392 .0485 .0594

.0244 .0307 .0384 .0475 .0582

.0239 .0301 .0375 .0465 .0571

.0233 .0294 .0367 .0455 .0559

1.4 1.3 1.2 1.1 1.0

.0808 .0968 .1151 .1357 .1587

.0793 .0951 .1131 .1335 .1562

.0778 .0934 .1112 .1314 .1539

.0764 .0918 .1093 .1292 .1515

.0749 .0901 .1075 .1271 .1492

.0735 .0885 .1056 .1251 .1469

.0722 .0869 .1038 .1230 .1446

.0708 .0853 .1020 .1210 .1423

.0694 .0838 .1003 .1190 .1401

.0681 .0823 .0985 .1170 .1379

0.9 0.8 0.7 0.6 0.5

.1841 .2119 .2420 .2743 .3085

.1814 .2090 .2389 .2709 .3050

.1788 .2061 .2358 .2676 .3015

.1762 .2033 .2327 .2643 .2981

.1736 .2005 .2296 .2611 .2946

.1711 .1977 .2266 .2578 .2912

.1685 .1949 .2236 .2546 .2877

.1660 .1922 .2206 .2514 .2843

.1635 .1894 .2177 .2483 .2810

.1611 .1867 .2148 .2451 .2776

0.4 0.3 0.2 0.1 0.0

.3446 .3821 .4207 .4602 .5000

.3409 .3783 .4168 .4562 .4960

.3372 .3745 .4129 .4522 .4920

.3336 .3707 .4090 .4483 .4880

.3300 .3669 .4052 .4443 .4840

.3264 .3632 .4013 .4404 .4801

.3228 .3594 .3974 .4364 .4761

.3192 .3557 .3936 .4325 .4721

.3156 .3520 .3897 .4286 .4681

.3121 .3483 .3859 .4247 .4641

TABLE 3

(continued)

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.0 0.1 0.2 0.3 0.4

.5000 .5398 .5793 .6179 .6554

.5040 .5438 .5832 .6217 .6591

.5080 .5478 .5871 .6255 .6628

.5120 .5517 .5910 .6293 .6664

.5160 .5557 .5948 .6331 .6700

.5199 .5596 .5987 .6368 .6736

.5239 .5636 .6026 .6406 .6772

.5279 .5675 .6064 .6443 .6808

.5319 .5714 .6103 .6480 .6844

.5359 .5753 .6141 .6517 .6879

0.5 0.6 0.7 0.8 0.9

.6915 .7257 .7580 .7881 .8159

.6950 .7291 .7611 .7910 .8186

.6985 .7324 .7642 .7939 .8212

.7019 .7357 .7673 .7967 .8238

.7054 .7389 .7704 .7995 .8264

.7088 .7422 .7734 .8023 .8289

.7123 .7454 .7764 .8051 .8315

.7157 .7486 .7794 .8078 .8340

.7190 .7517 .7823 .8106 .8365

.7224 .7549 .7852 .8133 .8389

1.0 1.1 1.2 1.3 1.4

.8413 .8643 .8849 .9032 .9192

.8438 .8665 .8869 .9049 .9207

.8461 .8686 .8888 .9066 .9222

.8485 .8708 .8907 .9082 .9236

.8508 .8729 .8925 .9099 .9251

.8531 .8749 .8944 .9115 .9265

.8554 .8770 .8962 .9131 .9279

.8577 .8790 .8980 .9147 .9292

.8599 .8810 .8997 .9162 .9306

.8621 .8830 .9015 .9177 .9319

1.5 1.6 1.7 1.8 1.9

.9332 .9452 .9554 .9641 .9713

.9345 .9463 .9564 .9649 .9719

.9357 .9474 .9573 .9656 .9726

.9370 .9484 .9582 .9664 .9732

.9382 .9495 .9591 .9671 .9738

.9394 .9505 .9599 .9678 .9744

.9406 .9515 .9608 .9686 .9750

.9418 .9525 .9616 .9693 .9756

.9429 .9535 .9625 .9699 .9761

.9441 .9545 .9633 .9706 .9767

2.0 2.1 2.2 2.3 2.4

.9772 .9821 .9861 .9893 .9918

.9778 .9826 .9864 .9896 .9920

.9783 .9830 .9868 .9898 .9922

.9788 .9834 .9871 .9901 .9925

.9793 .9838 .9875 .9904 .9927

.9798 .9842 .9878 .9906 .9929

.9803 .9846 .9881 .9909 .9931

.9808 .9850 .9884 .9911 .9932

.9812 .9854 .9887 .9913 .9934

.9817 .9857 .9890 .9916 .9936

2.5 2.6 2.7 2.8 2.9

.9938 .9953 .9965 .9974 .9981

.9940 .9955 .9966 .9975 .9982

.9941 .9956 .9967 .9976 .9982

.9943 .9957 .9968 .9977 .9983

.9945 .9959 .9969 .9977 .9984

.9946 .9960 .9970 .9978 .9984

.9948 .9961 .9971 .9979 .9985

.9949 .9962 .9972 .9979 .9985

.9951 .9963 .9973 .9980 .9986

.9952 .9964 .9974 .9981 .9986

3.0 3.1 3.2 3.3 3.4

.9987 .9990 .9993 .9995 .9997

.9987 .9991 .9993 .9995 .9997

.9987 .9991 .9994 .9995 .9997

.9988 .9991 .9994 .9996 .9997

.9988 .9992 .9994 .9996 .9997

.9989 .9992 .9994 .9996 .9997

.9989 .9992 .9995 .9996 .9997

.9990 .9993 .9995 .9996 .9997

.9990 .9993 .9995 .9997 .9998

List of Applications Business and Economics Actuaries, 172 Advertising campaigns, 655 Airline occupancy rates, 361 America’s market basket, 415–416 Assembling electronic equipment, 460 Auto accidents, 328 Auto insurance, 58, 415, 477 Baseball bats, 286 Bidding on construction jobs, 476–477 Black jack, 286 Brass rivets, 286 Charitable contributions, 102 Coal burning power plant, 286 Coffee breaks, 172 College textbooks, 563–564 Color TVs, 638 Construction projects, 574–575 Consumer confidence, 306 Consumer Price Index, 101–102 Cordless phones, 124–125 Corporate profits, 565 Cost of flying, 520–521 Cost of lumber, 462, 466 Deli sales, 274 Does college pay off?, 362 Drilling oil wells, 171 Economic forecasts, 236 e-shopping, 317 Flextime, 362 Fortune 500 revenues, 58 Gas mileage, 475 Glare in rearview mirrors, 475 Grant funding, 156 Grocery costs, 113 Hamburger meat, 85, 234–235, 316–317, 361, 399 HDTVs, 59, 114, 526 Homeschool teachers, 623–624 Housing prices, 532–533 Inspection lines, 157 Internet on-the-go, 46–47 Interstate commerce, 176 Job security, 212 Legal immigration, 306, 334 Lexus, Inc., 113–114 Light bulbs, 424 Line length, 31–32 Loading grain, 236 Lumber specs, 286 Movie marketing, 376–377 MP3 players, 316 Multimedia kids, 306 Nuclear power plant, 286 Operating expenses, 334 Packaging hamburger meat, 72

Paper strength, 274 Particle board, 574 Product quality, 431 Property values, 642, 649 Raisins, 408–409 Rating tobacco leaves, 666 Real estate prices, 113 School workers, 339–340, 383–384 Service times, 32 Shipping charges, 172 Sports salaries, 59 Starbucks, 59 Strawberries, 514, 521, 533 Supermarket prices, 659–660 Tax assessors, 416–417 Tax audits, 236 Teaching credentials, 207–208 Telecommuting, 609–610 Telemarketers, 195 Timber tracts, 73 Tuna ﬁsh, 59, 73, 90, 397, 407–408, 431, 461–462 Utility bills in southern California, 66, 86 Vacation destinations, 217 Vehicle colors, 624 Warehouse shopping, 477–478 Water resistance in textiles, 475 Worker error, 162

General Interest “900” numbers, 307 100-meter run, 136, 143 9/11 conspiracy, 383 9-1-1, 322 Accident prone, 204 Airport safety, 204 Airport security, 162 Armspan and height, 513–514, 522 Art critics, 665–666 Barry Bonds, 93 Baseball and steroids, 327 Baseball fans, 327 Baseball stats, 539 Batting champions, 32–33 Birth order and college success, 327 Birthday problem, 156 Braking distances, 235 Brett Favre, 74, 122, 398 Car colors, 196 Cell phone etiquette, 251–252 Cheating on taxes, 162 Christmas trees, 235 Colored contacts, 372 Comparing NFL quarterbacks, 85, 409 Competitive running, 665 Cramming, 144

Creation, 136 Defective computer chips, 207 Defective equipment, 171 Dieting, 322 Different realities, 327 Dinner at Gerards, 143 Driving emergencies, 72 Elevator capacities, 235 Eyeglasses, 135 Fast food and gas stations, 197 Fear of terrorism, 46 Football strategies, 162 Free time, 101 Freestyle swimmers, 409 Going to the moon, 259–260 Golﬁng, 158 Gourmet cooking, 642, 649 GPAs, 335 GRE scores, 466 Hard hats, 424 Harry Potter, 196 Hockey, 538 Home security systems, 196 Hotel costs, 367–368 Human heights, 235 Hunting season, 335 In-home movies, 244 Instrument precision, 423–424 Insuring your diamonds, 171–172 Itineraries, 142–143 Jason and Shaq, 157–158 JFK assassination, 609 Length, 513 Letterman or Leno, 170–171 M&M’S, 101, 326–327, 377 Machine breakdowns, 649 Major world lakes, 43–44 Man’s best friend, 197, 373 Men on Mars, 307 Noise and stress, 368 Old Faithful, 73 PGA, 171 Phospate mine, 235 Playing poker, 143 Presidential vetoes, 85 President’s kids, 73–74 Professor Asimov, 512, 521, 525 Rating political candidates, 665 Red dye, 416 Roulette, 135, 171 Sandwich generation, 613 Smoke detectors, 157 Soccer injuries, 157 Starbucks or Peet’s, 156–157 Summer vacations, 306–307 SUVs, 317 (continued)

List of Applications (continued) Tennis, 171, 236 Tennis racquets, 665 Time on task, 59 Tom Brady, 533 Tomatoes, 274 Top 20 movies, 33 Traffic control, 649 Traffic problems, 143 Vacation plans, 143 Walking shoes, 549 What to wear, 142 WNBA, 143

Life Sciences Achilles tendon injuries, 274–275, 362 Acid rain, 316 Air pollution, 520, 525, 565 Alzheimer’s disease, 637 Archeological find, 47, 65, 74, 409 Baby’s sleeping position, 377 Back pain, 196–197 Bacteria in drinking water, 236 Bacteria in water, 274 Bacteria in water samples, 204–205 Biomass, 306 Birth order and personality, 58 Blood thinner, 259 Blood types, 196 Body temperature and heart rate, 539 Breathing rates, 72, 235 Bulimia, 398 Calcium, 461, 465–466 Calcium content, 32 Cancer in rats, 259 Cerebral blood flow, 235 Cheese, 539 Chemical experiment, 512 Chemotherapy, 638 Chicago weather, 195 Childhood obesity, 371–372 Cholesterol, 399 Clopidogrel and aspirin, 377 Color preferences in mice, 196 Cotton versus cucumber, 573 Cure for insomnia, 372–373 Cure for the common cold, 366–367 Deep-sea research, 614 Digitalis and calcium uptake, 476 Diseased chickens, 613 Disinfectants, 408 Dissolved O2 content, 397–398, 409, 461, 638 Drug potency, 424 E. coli outbreak, 205 Early detection of breast cancer, 372 Excedrin or Tylenol, 328 FDA testing, 172 Fruit flies, 136 Geothermal power, 538–539 Glucose tolerance, 466

Good tasting medicine, 660 Ground or air, 416 Hazardous waste, 33 Healthy eating, 367 Healthy teeth, 407, 416 Heart rate and exercise, 655 Hormone therapy and Alzheimer’s disease, 377 HRT, 377 Hungry rats, 307 Impurities, 431–432 Invasive species, 361–362 Jigsaw puzzles, 649–650 Lead levels in blood, 642–643 Lead levels in drinking water, 367 Legal abortions, 291, 317 Less red meat, 335, 572–573 Lobsters, 398, 538 Long-term care, 613–614 Losing weight, 280 Mandatory health care, 608 Measurement error, 273–274 Medical diagnostics, 162 Mercury concentration in dolphins, 84–85 MMT in gasoline, 368 Monkey business, 144 Normal temperatures, 274 Ore samples, 72 pH in rainfall, 335 pH levels in water, 655 Physical fitness, 499 Plant genetics, 157, 372 Polluted rain, 335 Potassium levels, 274 Potency of an antibiotic, 362 Prescription costs, 280 Pulse rates, 236 Purifying organic compounds, 398 Rain and snow, 124 Recovery rates, 643 Recurring illness, 31 Red blood cell count, 32, 399 Runners and cyclists, 408, 415, 431 San Andreas Fault, 306 Screening tests, 162–163 Seed treatments, 208 Selenium, 322, 335 Slash pine seedlings, 475–476 Sleep deprivation, 512 Smoking and lung capacity, 398 Sunflowers, 235 Survival times, 50, 73, 85–86 Swampy sites, 460–461, 465, 655 Sweet potato whitefly, 372 Taste test for PTC, 197 Titanium, 408 Toxic chemicals, 660 Treatment versus control, 376 Vegi-burgers, 564–565 Waiting for a prescription, 609

Weights of turtles, 638 What’s normal?, 49, 86, 317, 323, 362, 368 Whiteﬂy infestation, 196

Social Sciences A female president?, 338–339 Achievement scores, 573–574 Achievement tests, 512–513, 545 Adolescents and social stress, 381 American presidents, 32 Anxious infants, 608–609 Back to work, 17 Catching a cold, 327 Choosing a mate, 157 Churchgoing and age, 614 Disabled students, 113 Discovery-based teaching, 621 Drug offenders, 156 Drug testing, 156 Election 2008, 16 Eye movement, 638 Faculty salaries, 273 Gender bias, 144, 171, 207 Generation Next, 327–328, 380 Hospital survey, 143 Household size, 102, 614 Images and word recall, 650 Intensive care, 204 Jury duty, 135–136 Laptops and learning, 522, 526 Medical bills, 196 Memory experiments, 417 Midterm scores, 125 Music in the workplace, 417 Native American youth, 259 No pass, no play rule for athletics, 162 Organized religion, 31 Political corruption, 334–335 Preschool, 31 Race distributions in the Armed Forces, 16–17 Racial bias, 259 Reducing hostility, 460 Rocking the vote, 317 SAT scores, 195–196, 431, 445 Smoking and cancer, 157 Social Security numbers, 72–73 Social skills training, 538, 666 Spending patterns, 609 Starting salaries, 322–323, 367 Student ratings, 665 Teaching biology, 322 Teen magazines, 212 Test interviews, 513 Union, yes!, 327 Violent crime, 161–162 Want to be president?, 16 Who votes?, 373 YouTube, 566

How Do I Construct a Stem and Leaf Plot? 20 How Do I Construct a Relative Frequency Histogram? How Do I Calculate Sample Quartiles?

How Do I Calculate the Correlation Coefficient? How Do I Calculate the Regression Line? 111

111

What’s the Difference between Mutually Exclusive and Independent Events? 153 How Do I Use Table 1 to Calculate Binomial Probabilities? 190 How Do I Calculate Poisson Probabilities Using the Formula? 198 How Do I Use Table 2 to Calculate Poisson Probabilities? 199 How Do I Use Table 3 to Calculate Probabilities under the Standard Normal Curve? 228 How Do I Calculate Binomial Probabilities Using the Normal Approximation? 240

How Do I Calculate Probabilities for the Sample Mean x苶? 268 How Do I Calculate Probabilities for the Sample Proportion pˆ ? 277 How Do I Estimate a Population Mean or Proportion? 303 How Do I Choose the Sample Size? 331 Rejection Regions, p-Values, and Conclusions How Do I Calculate b? 360 How Do I Decide Which Test to Use?

355

432

How Do I Know Whether My Calculations Are Accurate? 459 How Do I Make Sure That My Calculations Are Correct? 508 How Do I Determine the Appropriate Number of Degrees of Freedom? 606, 611

Index of Applet Figures CHAPTER 1 Figure 1.17 Building a Dotplot applet Figure 1.18 Building a Histogram applet Figure 1.19 Flipping Fair Coins applet Figure 1.20 Flipping Fair Coins applet CHAPTER 2 Figure 2.4 How Extreme Values Affect the Mean and Median applet Figure 2.9 Why Divide n 1? Figure 2.19 Building a Box Plot applet CHAPTER 3 Figure 3.6 Building a Scatterplot applet Figure 3.9 Exploring Correlation applet Figure 3.12 How a Line Works applet CHAPTER 4 Figure 4.6 Tossing Dice applet Figure 4.16 Flipping Fair Coins applet Figure 4.17 Flipping Weighted Coins applet

CHAPTER 8 Figure 8.10 Interpreting Conﬁdence Intervals applet CHAPTER 9 Figure 9.7 Large Sample Test of a Population Mean applet Figure 9.9 Power of a z-Test applet CHAPTER 10 Figure 10.3 Student’s t Probabilities applet Figure 10.5 Comparing t and z applet Figure 10.9 Small Sample Test of a Population Mean applet Figure 10.12 Two-Sample t Test: Independent Samples applet Figure 10.17 Chi-Square Probabilities applet Figure 10.21 F Probabilities applet CHAPTER 11 Figure 11.6 F Probabilities applet

CHAPTER 5 Figure 5.2 Calculating Binomial Probabilities applet Figure 5.3 Java Applet for Example 5.6

CHAPTER 12 Figure 12.4 Method of Least Squares applet Figure 12.7 t Test for the Slope applet Figure 12.17 Exploring Correlation applet

CHAPTER 6 Figure 6.7 Visualizing Normal Curves applet Figure 6.14 Normal Distribution Probabilities applet Figure 6.17 Normal Probabilities and z-Scores applet Figure 6.21 Normal Approximation to Binomial Probabilities applet

CHAPTER 14 Figure 14.1 Goodness-of-Fit applet Figure 14.2 Chi-Square Test of Independence applet Figure 14.4 Chi-Square Test of Independence applet

CHAPTER 7 Figure 7.7 Central Limit Theorem applet Figure 7.10 Normal Probabilities for Means applet

a ta

TABLE 4

Critical Values of t page 691

t.100

t.050

t.025

t.010

t.005

1 2 3 4 5

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2.015

12.706 4.303 3.182 2.776 2.571

31.821 6.965 4.541 3.747 3.365

63.657 9.925 5.841 4.604 4.032

1 2 3 4 5

6 7 8 9 10

1.440 1.415 1.397 1.383 1.372

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.143 2.998 2.896 2.821 2.764

3.707 3.499 3.355 3.250 3.169

6 7 8 9 10

11 12 13 14 15

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 2.624 2.602

3.106 3.055 3.012 2.977 2.947

11 12 13 14 15

16 17 18 19 20

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

2.921 2.898 2.878 2.861 2.845

16 17 18 19 20

21 22 23 24 25

1.323 1.321 1.319 1.318 1.316

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.518 2.508 2.500 2.492 2.485

2.831 2.819 2.807 2.797 2.787

21 22 23 24 25

26 27 28 29

1.315 1.314 1.313 1.311 1.282

1.706 1.703 1.701 1.699 1.645

2.056 2.052 2.048 2.045 1.960

2.479 2.473 2.467 2.462 2.326

2.779 2.771 2.763 2.756 2.576

26 27 28 29

SOURCE: From “Table of Percentage Points of the t-Distribution,” Biometrika 32 (1941):300. Reproduced by permission of the Biometrika Trustees.

Introduction to Probability and Statistics 13th

EDITION

William Mendenhall University of Florida, Emeritus

Robert J. Beaver University of California, Riverside, Emeritus

Barbara M. Beaver University of California, Riverside

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

Introduction to Probability and Statistics, Thirteenth Edition William Mendenhall, Robert J. Beaver, Barbara M. Beaver Acquisitions Editor: Carolyn Crockett Development Editor: Kristin Marrs Assistant Editor: Catie Ronquillo Editorial Assistant: Rebecca Dashiell

© 2009, 2006 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher.

Technology Project Manager: Sam Subity Marketing Manager: Amanda Jellerichs Marketing Assistant: Ashley Pickering Marketing Communications Manager: Talia Wise Project Manager, Editorial Production: Jennifer Risden Creative Director: Rob Hugel Art Director: Vernon Boes Print Buyer: Linda Hsu Permissions Editor: Mardell Glinski Schultz Production Service: ICC Macmillan Inc. Text Designer: John Walker Photo Researcher: Rose Alcorn Copy Editor: Richard Camp Cover Designer: Cheryl Carrington Cover Image: R. Creation/Getty Images Compositor: ICC Macmillan Inc.

For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at cengage.com/permissions. Further permissions questions can be e-mailed to [emailprotected].

MINITAB is a trademark of Minitab, Inc., and is used herein with the owner’s permission. Portions of MINITAB Statistical Software input and output contained in this book are printed with permission of Minitab, Inc. The applets in this book are from Seeing Statistics™, an online, interactive statistics textbook. Seeing Statistics is a registered service mark used herein under license. The applets in this book were designed to be used exclusively with Introduction to Probability and Statistics, Thirteenth Edition, by Mendenhall, Beaver & Beaver, and they may not be copied, duplicated, or reproduced for any reason. Library of Congress Control Number: 2007931223 ISBN-13: 978-0-495-38953-8 ISBN-10: 0-495-38953-6 Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at international.cengage.com/region. Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit academic.cengage.com.

Printed in Canada 1 2 3 4 5 6 7 12 11 10 09 08

Purchase any of our products at your local college store or at our preferred online store www.ichapters.com.

Preface Every time you pick up a newspaper or a magazine, watch TV, or surf the Internet, you encounter statistics. Every time you ﬁll out a questionnaire, register at an online website, or pass your grocery rewards card through an electronic scanner, your personal information becomes part of a database containing your personal statistical information. You cannot avoid the fact that in this information age, data collection and analysis are an integral part of our day-to-day activities. In order to be an educated consumer and citizen, you need to understand how statistics are used and misused in our daily lives. To that end we need to “train your brain” for statistical thinking—a theme we emphasize throughout the thirteenth edition by providing you with a “personal trainer.”

THE SECRET TO OUR SUCCESS The ﬁrst college course in introductory statistics that we ever took used Introduction to Probability and Statistics by William Mendenhall. Since that time, this text—currently in the thirteenth edition—has helped several generations of students understand what statistics is all about and how it can be used as a tool in their particular area of application. The secret to the success of Introduction to Probability and Statistics is its ability to blend the old with the new. With each revision we try to build on the strong points of previous editions, while always looking for new ways to motivate, encourage, and interest students using new technological tools.

HALLMARK FEATURES OF THE THIRTEENTH EDITION The thirteenth edition retains the traditional outline for the coverage of descriptive and inferential statistics. This revision maintains the straightforward presentation of the twelfth edition. In this spirit, we have continued to simplify and clarify the language and to make the language and style more readable and “user friendly”—without sacriﬁcing the statistical integrity of the presentation. Great effort has been taken to “train your brain” to explain not only how to apply statistical procedures, but also to explain • • • •

how to meaningfully describe real sets of data what the results of statistical tests mean in terms of their practical applications how to evaluate the validity of the assumptions behind statistical tests what to do when statistical assumptions have been violated

iv ❍

PREFACE

Exercises In the tradition of all previous editions, the variety and number of real applications in the exercise sets is a major strength of this edition. We have revised the exercise sets to provide new and interesting real-world situations and real data sets, many of which are drawn from current periodicals and journals. The thirteenth edition contains over 1300 problems, many of which are new to this edition. Any exercises from previous editions that have been deleted will be available to the instructor as Classic Exercises on the Instructor’s Companion Website (academic.cengage.com/statistics/mendenhall). Exercises are graduated in level of difficulty; some, involving only basic techniques, can be solved by almost all students, while others, involving practical applications and interpretation of results, will challenge students to use more sophisticated statistical reasoning and understanding.

Organization and Coverage Chapters 1–3 present descriptive data analysis for both one and two variables, using state-of-the-art MINITAB graphics. We believe that Chapters 1 through 10—with the possible exception of Chapter 3—should be covered in the order presented. The remaining chapters can be covered in any order. The analysis of variance chapter precedes the regression chapter, so that the instructor can present the analysis of variance as part of a regression analysis. Thus, the most effective presentation would order these three chapters as well. Chapter 4 includes a full presentation of probability and probability distributions. Three optional sections—Counting Rules, the Total Law of Probability, and Bayes’ Rule—are placed into the general flow of text, and instructors will have the option of complete or partial coverage. The sections that present event relations, independence, conditional probability, and the Multiplication Rule have been rewritten in an attempt to clarify concepts that often are difficult for students to grasp. As in the twelfth edition, the chapters on analysis of variance and linear regression include both calculational formulas and computer printouts in the basic text presentation. These chapters can be used with equal ease by instructors who wish to use the “hands-on” computational approach to linear regression and ANOVA and by those who choose to focus on the interpretation of computer-generated statistical printouts. One important change implemented in this and the last two editions involves the emphasis on p-values and their use in judging statistical significance. With the advent of computer-generated p-values, these probabilities have become essential components in reporting the results of a statistical analysis. As such, the observed value of the test statistic and its p-value are presented together at the outset of our discussion of statistical hypothesis testing as equivalent tools for decision-making. Statistical significance is defined in terms of preassigned values of a, and the p-value approach is presented as an alternative to the critical value approach for testing a statistical hypothesis. Examples are presented using both the p-value and critical value approaches to hypothesis testing. Discussion of the practical interpretation of statistical results, along with the difference between statistical significance and practical significance, is emphasized in the practical examples in the text.

Special Feature of the Thirteenth Edition— MyPersonal Trainer A special feature of this edition are the MyPersonal Trainer sections, consisting of deﬁnitions and/or step-by-step hints on problem solving. These sections are followed by Exercise Reps, a set of exercises involving repetitive problems concerning a speciﬁc

PREFACE

❍

topic or concept. These Exercise Reps can be compared to sets of exercises speciﬁed by a trainer for an athlete in training. The more “reps” the athlete does, the more he acquires strength or agility in muscle sets or an increase in stamina under stress conditions.

How Do I Calculate Sample Quartiles? 1. Arrange the data set in order of magnitude from smallest to largest. 2. Calculate the quartile positions: •

Position of Q1: .25(n 1)

•

Position of Q3: .75(n 1)

3. If the positions are integers, then Q1 and Q3 are the values in the ordered data set found in those positions. 4. If the positions in step 2 are not integers, find the two measurements in positions just above and just below the calculated position. Calculate the quartile by finding a value either one-fourth, one-half, or three-fourths of the way between these two measurements. Exercise Reps A. Below you will find two practice data sets. Fill in the blanks to find the necessary quartiles. The first data set is done for you. Data Set

Sorted

Position of Q1

Position of Q3

Lower Quartile, Q1

Upper Quartile, Q3

2, 5, 7, 1, 1, 2, 8

1, 1, 2, 2, 5, 7, 8

2nd

6th

5, 0, 1, 3, 1, 5, 5, 2, 4, 4, 1

B. Below you will find three data sets that have already been sorted. The positions of the upper and lower quartiles are shown in the table. Find the measurements just above and just below the quartile position. Then find the upper and lower quartiles. The first data set is done for you. Sorted Data Set

Position of Q1

Measurements Above and Below

0, 1, 4, 4, 5, 9

1.75

0 and 1

Q1 0 .75(1) .75

Position of Q3

Measurements Above and Below

5.25

5 and 9

Q3 5 .25(4) 6

0, 1, 3, 3, 4, 7, 7, 8

2.25

and

6.75

and

1, 1, 2, 5, 6, 6, 7, 9, 9

2.5

and

7.5

and

The MyPersonal Trainer sections with Exercise Reps are used frequently in early chapters where it is important to establish basic concepts and statistical thinking, coupled up with straightforward calculations. The answers to the “Exercise Reps,” when needed, are found on a perforated card in the back of the text. The MyPersonal Trainer sections appear in all but two chapters—Chapters 13 and 15. However, the Exercise Reps problem sets appear only in the ﬁrst 10 chapters where problems can be solved using pencil and paper, or a calculator. We expect that by the time a student has completed the ﬁrst 10 chapters, statistical concepts and approaches will have been mastered. Further, the computer intensive nature of the remaining chapters is not amenable to a series of simple repetitive and easily calculated exercises, but rather is amenable to a holistic approach—that is, a synthesis of the results of a complete analysis into a set of conclusions and recommendations for the experimenter.

Other Features of the Thirteenth Edition •

MyApplet: Easy access to the Internet has made it possible for students to visualize statistical concepts using an interactive webtool called an applet. Applets written by Gary McClelland, author of Seeing Statistics™, have been customized speciﬁcally to match the presentation and notation used in this edition. Found on the Premium Website that accompanies the text, they

vi ❍

PREFACE

provide visual reinforcement of the concepts presented in the text. Applets allow the user to perform a statistical experiment, to interact with a statistical graph to change its form, or to access an interactive “statistical table.” At appropriate points in the text, a screen capture of each applet is displayed and explained, and each student is encouraged to learn interactively by using the “MyApplet” exercises at the end of each chapter. We are excited to see these applets integrated into statistical pedagogy and hope that you will take advantage of their visual appeal to your students.

You can compare the accuracy of estimators of the population variance s 2 using the Why Divide by n 1? applet. The applet selects samples from a population with standard deviation s 29.2. It then calculates the standard deviation s using (n 1) in the denominator as well as a standard deviation calculated using n in the denominator. You can choose to compare the estimators for a single new sample, for 10 samples, or for 100 samples. Notice that each of the 10 samples shown in Figure 2.9 has a different sample standard deviation. However, when the 10 standard deviations are averaged at the bottom of the applet, one of the two estimators is closer to the population standard deviation, s 29.2. Which one is it? We will use this applet again for the MyApplet Exercises at the end of the chapter. FIGURE 2.9

Why Divide by n 1? applet

●

Exercises 2.86 Refer to Data Set #1 in the How Extreme Val-

ues Affect the Mean and Median applet. This applet loads with a dotplot for the following n 5 observations: 2, 5, 6, 9, 11. a. What are the mean and median for this data set? b. Use your mouse to change the value x 11 (the moveable green dot) to x 13. What are the mean and median for the new data set? c. Use your mouse to move the green dot to x 33. When the largest value is extremely large compared to the other observations, which is larger, the mean or the median? d. What effect does an extremely large value have on the mean? What effect does it have on the median? 2.87 Refer to Data Set #2 in the How Extreme Val-

n 3 from a population in which the standard deviation is s 29.2. a. Click . A sample consisting of n 3 observations will appear. Use your calculator to verify the values of the standard deviation when dividing by n 1 and n as shown in the applet. b. Click again. Calculate the average of the two standard deviations (dividing by n 1) from parts a and b. Repeat the process for the two standard deviations (dividing by n). Compare your results to those shown in red on the applet. c. You can look at how the two estimators in part a behave “in the long run” by clicking or a number of times, until the average of all the standard deviations begins to stabilize. Which of the two methods gives a standard deviation closer to s 29.2? d. In the long run, how far off is the standard deviation when dividing by n? 2.90 Refer to Why Divide by n 1 applet. The

second applet on the page randomly selects sample of n 10 from the same population in which the standard deviation is s 29.2.

PREFACE

•

MINITAB histogram for Example 2.8

vii

Graphical and numerical data description includes both traditional and EDA methods, using computer graphics generated by MINITAB 15 for Windows.

● 6/25

Relative Frequency

F I G URE 2 . 1 2

❍

4/25

2/25

0 8.5

14.5

20.5 Scores

26.5

FIGURE 2.16

MINITAB output for the data in Example 2.13

•

32.5

● Descriptive Statistics: x Variable X

N N* Mean SE Mean 10 0 13.50 1.98

StDev Minimum 6.28 4.00

Q1 Median Q3 Maximum 8.75 12.00 18.50 25.00

The presentation in Chapter 4 has been rewritten to clarify the presentation of simple events and the sample space as well as the presentation of conditional probability, independence, and the Multiplication Rule. All examples and exercises in the text contain printouts based on MINITAB 15 and consistent with MINITAB 14. MINITAB printouts are provided for some exercises, while other exercises require the student to obtain solutions without using the computer. y p graphs? c. Use a line chart to describe the predicted number of wired households for the years 2002 to 2008. d. Use a bar chart to describe the predicted number of wireless households for the years 2002 to 2008. 1.51 Election Results The 2004 election

was a race in which the incumbent, George W. Bush, defeated John Kerry, Ralph Nader, and other candidates, receiving 50.7% of the popular vote. The popular vote (in thousands) for George W. Bush in each of the 50 states is listed below:8

EX0151

AL AK AZ AR CA CO CT DE FL GA

1176 191 1104 573 5510 1101 694 172 3965 1914

HI ID IL IN IA KS KY LA ME MD

194 409 2346 1479 572 736 1069 1102 330 1025

MA MI MN MS MO MT NE NV NH NJ

1071 2314 1347 685 1456 266 513 419 331 1670

NM NY NC ND OH OK OR PA RI SC

377 2962 1961 197 2860 960 867 2794 169 938

SD TN TX UT VT VA WA WV WI WY

233 1384 4527 664 121 1717 1305 424 1478 168

a. By just looking at the table, what shape do you think the data distribution for the popular vote by state will have? b. Draw a relative frequency histogram to describe the distribution of the popular vote for President Bush in the 50 states. c. Did the histogram in part b conﬁrm your guess in part a? Are there any outliers? How can you explain them?

1.53 Election Results, continued Refer to

Exercises 1.51 and 1.52. The accompanying stem and leaf plots were generated using MINITAB for the variables named “Popular Vote” and “Percent Vote.” Stem-and-Leaf Display: Popular Vote, Percent Vote Stem-and-leaf of Popular Vote N = 50 Leaf Unit = 100

Stem-and-leaf of Percent Vote N = 50 Leaf Unit = 1.0

7 12 18 22 25 25 18 15 12 10 8 8 6 6 5

3 8 19 (9) 22 13 5 1

0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 HI

1111111 22333 444555 6667 899 0001111 333 444 67 99

3 4 4 5 5 6 6 7

799 03444 55666788899 001122344 566778899 00011223 6689 3

33 7 89 39, 45, 55

a. Describe the shapes of the two distributions. Are there any outliers? b. Do the stem and leaf plots resemble the relative frequency histograms constructed in Exercises 1.51 and 1.52? c. Explain why the distribution of the popular vote for President Bush by state is skewed while the

viii ❍

PREFACE

The Role of the Computer in the Thirteenth Edition—My MINITAB Computers are now a common tool for college students in all disciplines. Most students are accomplished users of word processors, spreadsheets, and databases, and they have no trouble navigating through software packages in the Windows environment. We believe, however, that advances in computer technology should not turn statistical analyses into a “black box.” Rather, we choose to use the computational shortcuts and interactive visual tools that modern technology provides to give us more time to emphasize statistical reasoning as well as the understanding and interpretation of statistical results. In this edition, students will be able to use the computer for both standard statistical analyses and as a tool for reinforcing and visualizing statistical concepts. MINITAB 15 (consistent with MINITAB 14 ) is used exclusively as the computer package for statistical analysis. Almost all graphs and ﬁgures, as well as all computer printouts, are generated using this version of MINITAB. However, we have chosen to isolate the instructions for generating this output into individual sections called “My MINITAB ” at the end of each chapter. Each discussion uses numerical examples to guide the student through the MINITAB commands and options necessary for the procedures presented in that chapter. We have included references to visual screen captures from MINITAB 15, so that the student can actually work through these sections as “mini-labs.”

Numerical Descriptive Measures MINITAB provides most of the basic descriptive statistics presented in Chapter 2 using a single command in the drop-down menus. Once you are on the Windows desktop, double-click on the MINITAB icon or use the Start button to start MINITAB. Practice entering some data into the Data window, naming the columns appropriately in the gray cell just below the column number. When you have ﬁnished entering your data, you will have created a MINITAB worksheet, which can be saved either singly or as a MINITAB project for future use. Click on File 씮 Save Current Worksheet or File 씮 Save Project. You will need to name the worksheet (or project)—perhaps “test data”—so that you can retrieve it later. The following data are the ﬂoor lengths (in inches) behind the second and third seats in nine different minivans:12 Second seat: Third seat:

62.0, 62.0, 64.5, 48.5, 57.5, 61.0, 45.5, 47.0, 33.0 27.0, 27.0, 24.0, 16.5, 25.0, 27.5, 14.0, 18.5, 17.0

Since the data involve two variables, we enter the two rows of numbers into columns C1 and C2 in the MINITAB worksheet and name them “2nd Seat” and “3rd Seat,” respectively. Using the drop-down menus, click on Stat 씮 Basic Statistics 씮 Display Descriptive Statistics. The Dialog box is shown in Figure 2.21. F I G URE 2 . 2 1

●

provides printing options for multiple box plots. Labels will let you annotate the graph with titles and footnotes. If you have entered data into the worksheet as a frequency distribution (values in one column, frequencies in another), the Data Options will allow the data to be read in that format. The box plot for the third seat lengths is shown in Figure 2.24. You can use the MINITAB commands from Chapter 1 to display stem and leaf plots or histograms for the two variables. How would you describe the similarities and differences in the two data sets? Save this worksheet in a ﬁle called “Minivans” before exiting MINITAB. We will use it again in Chapter 3. FIGURE 2.22

FIGURE 2 23

●

PREFACE

❍

If you do not need “hands-on” knowledge of MINITAB, or if you are using another software package, you may choose to skip these sections and simply use the MINITAB printouts as guides for the basic understanding of computer printouts. Any student who has Internet access can use the applets found on the Student Premium Website to visualize a variety of statistical concepts (access instructions for the Student Premium Website are listed on the Printed Access Card that is an optional bundle with this text). In addition, some of the applets can be used instead of computer software to perform simple statistical analyses. Exercises written speciﬁcally for use with these applets appear in a section at the end of each chapter. Students can use the applets at home or in a computer lab. They can use them as they read through the text material, once they have ﬁnished reading the entire chapter, or as a tool for exam review. Instructors can assign applet exercises to the students, use the applets as a tool in a lab setting, or use them for visual demonstrations during lectures. We believe that these applets will be a powerful tool that will increase student enthusiasm for, and understanding of, statistical concepts and procedures.

STUDY AIDS The many and varied exercises in the text provide the best learning tool for students embarking on a ﬁrst course in statistics. An exercise number printed in color indicates that a detailed solution appears in the Student Solutions Manual, which is available as a supplement for students. Each application exercise now has a title, making it easier for students and instructors to immediately identify both the context of the problem and the area of application.

y 5.46 Accident Prone, continued Refer to Exer-

APPLICATIONS 5.43 Airport Safety The increased number of small

commuter planes in major airports has heightened concern over air safety. An eastern airport has recorded a monthly average of ﬁve near-misses on landings and takeoffs in the past 5 years. a. Find the probability that during a given month there are no near-misses on landings and takeoffs at the airport.

cise 5.45. a. Calculate the mean and standard deviation for x, the number of injuries per year sustained by a schoolage child. b. Within what limits would you expect the number of injuries per year to fall? 5.47 Bacteria in Water Samples If a drop of

water is placed on a slide and examined under a microscope, the number x of a particular type of bacteria

Students should be encouraged to use the MyPersonal Trainer sections and the Exercise Reps whenever they appear in the text. Students can “ﬁll in the blanks” by writing directly in the text and can get immediate feedback by checking the answers on the perforated card in the back of the text. In addition, there are numerous hints called MyTip, which appear in the margins of the text.

Empirical Rule ⇔ mound-shaped data Tchebysheff ⇔ any shaped data

Is Tchebysheff’s Theorem applicable? Yes, because it can be used for any set of data. According to Tchebysheff’s Theorem, • •

at least 3/4 of the measurements will fall between 10.6 and 32.6. at least 8/9 of the measurements will fall between 5.1 and 38.1.

❍

PREFACE

The MyApplet sections appear within the body of the text, explaining the use of a particular Java applet. Finally, sections called Key Concepts and Formulas appear in each chapter as a review in outline form of the material covered in that chapter. CHAPTER REVIEW Key Concepts and Formulas I.

Measures of the Center of a Data Distribution

1. Arithmetic mean (mean) or average a. Population: m

Sx b. Sample of n measurements: x苶 i n 2. Median; position of the median .5(n 1) 3. Mode 4. The median may be preferred to the mean if the data are highly skewed. II. Measures of Variability

1. Range: R largest smallest 2. Variance a. Population of N measurements: S(xi m)2 s2 N

68%, 95%, and 99.7% of the measurements are within one, two, and three standard deviations of the mean, respectively. IV. Measures of Relative Standing

x 苶x 1. Sample z-score: z s 2. pth percentile; p% of the measurements are smaller, and (100 p)% are larger. 3. Lower quartile, Q1; position of Q1 .25 (n 1) 4. Upper quartile, Q3; position of Q3 .75 (n 1) 5. Interquartile range: IQR Q3 Q1 V. The Five-Number Summary and Box Plots

1. The ﬁve-number summary: Min

b. Sample of n measurements: (Sxi)2 Sx 2i n S(xi x苶 )2 s2 n1 n1

Median Q3

Max

One-fourth of the measurements in the data set lie between each of the four adjacent pairs of numbers. 2. Box plots are used for detecting outliers and h f di ib i

The Student Premium Website, a password-protected resource that can be accessed with a Printed Access Card (optional bundle item), provides students with an array of study resources, including the complete set of Java applets used for the MyApplet sections, PowerPoint® slides for each chapter, and a Graphing Calculator Manual, which includes instructions for performing many of the techniques in the text using the popular TI-83 graphing calculator. In addition, sets of Practice (or Self-Correcting) Exercises are included for each chapter. These exercise sets are followed by the complete solutions to each of the exercises. These solutions can be used pedagogically to allow students to pinpoint any errors made at each of the calculational steps leading to ﬁnal answers. Data sets (saved in a variety of formats) for many of the text exercises can be found on the book’s website (academic.cengage.com/statistics/mendenhall).

PREFACE

❍

INSTRUCTOR RESOURCES The Instructor’s Companion Website (academic.cengage.com/statistics/mendenhall), available to adopters of the thirteenth edition, provides a variety of teaching aids, including •

• • • •

All the material from the Student Companion Website, including exercises using the Large Data Sets, which is accompanied by three large data sets that can be used throughout the course. A file named “Fortune” contains the revenues (in millions) for the Fortune 500 largest U.S. industrial corporations in a recent year; a file named “Batting” contains the batting averages for the National and American baseball league batting champions from 1876 to 2006; and a file named “Blood Pressure” contains the age and diastolic and systolic blood pressures for 965 men and 945 women compiled by the National Institutes of Health. Classic exercises with data sets and solutions PowerPoints created by Barbara Beaver Applets by Gary McClelland (the complete set of Java applets used for the MyApplet sections) Graphing Calculator manual, which includes instructions for performing many of the techniques in the text using the TI-83 graphing calculator

Also available for instructors: WebAssign WebAssign, the most widely used homework system in higher education, allows you to assign, collect, grade, and record homework assignments via the web. Through a partnership between WebAssign and Brooks/Cole Cengage Learning, this proven homework system has been enhanced to include links to textbook sections, video examples, and problem-speciﬁc tutorials. PowerLecture™ PowerLecture with ExamView® for Introduction to Probability and Statistics contains the Instructor’s Solutions Manual, PowerPoint lectures prepared by Barbara Beaver, ExamView Computerized Testing, Classic Exercises, and TI-83 Manual prepared by James Davis.

ACKNOWLEDGMENTS The authors are grateful to Carolyn Crockett and the editorial staff of Brooks/Cole for their patience, assistance, and cooperation in the preparation of this edition. A special thanks to Gary McClelland for his careful customization of the Java applets used in the text, and for his patient and even enthusiastic responses to our constant emails! Thanks are also due to thirteenth edition reviewers Bob Denton, Timothy Husband, Ron LaBorde, Craig McBride, Marc Sylvester, Kanapathi Thiru, and Vitaly Voloshin and twelfth edition reviewers David Laws, Dustin Paisley, Krishnamurthi Ravishankar, and Maria Rizzo. We wish to thank authors and organizations for allowing us to reprint selected material; acknowledgments are made wherever such material appears in the text. Robert J. Beaver Barbara M. Beaver William Mendenhall

Brief Contents INTRODUCTION 1 1

DESCRIBING DATA WITH GRAPHS 7

DESCRIBING DATA WITH NUMERICAL MEASURES 52

DESCRIBING BIVARIATE DATA 97

PROBABILITY AND PROBABILITY DISTRIBUTIONS 127

SEVERAL USEFUL DISCRETE DISTRIBUTIONS 183

THE NORMAL PROBABILITY DISTRIBUTION 219

SAMPLING DISTRIBUTIONS 254

LARGE-SAMPLE ESTIMATION 297

LARGE-SAMPLE TESTS OF HYPOTHESES 343

INFERENCE FROM SMALL SAMPLES 386

THE ANALYSIS OF VARIANCE 447

LINEAR REGRESSION AND CORRELATION 502

MULTIPLE REGRESSION ANALYSIS 551

ANALYSIS OF CATEGORICAL DATA 594

NONPARAMETRIC STATISTICS 629 APPENDIX I 679 DATA SOURCES 712 ANSWERS TO SELECTED EXERCISES 722 INDEX 737 CREDITS 744

Contents Introduction: Train Your Brain for Statistics

The Population and the Sample 3 Descriptive and Inferential Statistics 4 Achieving the Objective of Inferential Statistics: The Necessary Steps 4 Training Your Brain for Statistics 5 1

DESCRIBING DATA WITH GRAPHS

1.1 Variables and Data 8 1.2 Types of Variables 10 1.3 Graphs for Categorical Data 11 Exercises 14

1.4 Graphs for Quantitative Data 17 Pie Charts and Bar Charts 17 Line Charts 19 Dotplots 20 Stem and Leaf Plots 20 Interpreting Graphs with a Critical Eye 22

1.5 Relative Frequency Histograms 24 Exercises 29 Chapter Review 34 CASE STUDY: How Is Your Blood Pressure? 50 2

DESCRIBING DATA WITH NUMERICAL MEASURES

2.1 Describing a Set of Data with Numerical Measures 53 2.2 Measures of Center 53 Exercises 57

2.3 Measures of Variability 60 Exercises 65

2.4 On the Practical Signiﬁcance of the Standard Deviation 66

xiv

❍

CONTENTS

2.5 A Check on the Calculation of s 70 Exercises 71

2.6 Measures of Relative Standing 75 2.7 The Five-Number Summary and the Box Plot 80 Exercises 84 Chapter Review 87 CASE STUDY: The Boys of Summer 96 3

DESCRIBING BIVARIATE DATA

3.1 Bivariate Data 98 3.2 Graphs for Qualitative Variables 98 Exercises 101

3.3 Scatterplots for Two Quantitative Variables 102 3.4 Numerical Measures for Quantitative Bivariate Data 105 Exercises 112 Chapter Review 114 CASE STUDY: Are Your Dishes Really Clean? 126 4

PROBABILITY AND PROBABILITY DISTRIBUTIONS

127

4.1 The Role of Probability in Statistics 128 4.2 Events and the Sample Space 128 4.3 Calculating Probabilities Using Simple Events 131 Exercises 134

4.4 Useful Counting Rules (Optional) 137 Exercises 142

4.5 Event Relations and Probability Rules 144 Calculating Probabilities for Unions and Complements 146

4.6 Independence, Conditional Probability, and the Multiplication Rule 149 Exercises 154

4.7 Bayes’ Rule (Optional) 158 Exercises 161

4.8 Discrete Random Variables and Their Probability Distributions 163 Random Variables 163 Probability Distributions 163 The Mean and Standard Deviation for a Discrete Random Variable 166 Exercises 170 Chapter Review 172 CASE STUDY: Probability and Decision Making in the Congo 181

CONTENTS

SEVERAL USEFUL DISCRETE DISTRIBUTIONS

❍

183

5.1 Introduction 184 5.2 The Binomial Probability Distribution 184 Exercises 193

5.3 The Poisson Probability Distribution 197 Exercises 202

5.4 The Hypergeometric Probability Distribution 205 Exercises 207 Chapter Review 208 CASE STUDY: A Mystery: Cancers Near a Reactor 218 6

THE NORMAL PROBABILITY DISTRIBUTION

219

6.1 Probability Distributions for Continuous Random Variables 220 6.2 The Normal Probability Distribution 223 6.3 Tabulated Areas of the Normal Probability Distribution 225 The Standard Normal Random Variable 225 Calculating Probabilities for a General Normal Random Variable 229 Exercises 233

6.4 The Normal Approximation to the Binomial Probability Distribution (Optional) 237 Exercises 243 Chapter Review 246 CASE STUDY: The Long and Short of It 252 7

SAMPLING DISTRIBUTIONS

254

7.1 Introduction 255 7.2 Sampling Plans and Experimental Designs 255 Exercises 258

7.3 Statistics and Sampling Distributions 260 7.4 The Central Limit Theorem 263 7.5 The Sampling Distribution of the Sample Mean 266 Standard Error 267 Exercises 272

7.6 The Sampling Distribution of the Sample Proportion 275 Exercises 279

7.7 A Sampling Application: Statistical Process Control (Optional) 281 A Control Chart for the Process Mean: The x苶 Chart 281 A Control Chart for the Proportion Defective: The p Chart 283 Exercises 285

xvi

❍

CONTENTS

Chapter Review 287 CASE STUDY: Sampling the Roulette at Monte Carlo 295 8

LARGE-SAMPLE ESTIMATION

297

8.1 Where We’ve Been 298 8.2 Where We’re Going—Statistical Inference 298 8.3 Types of Estimators 299 8.4 Point Estimation 300 Exercises 305

8.5 Interval Estimation 307 Constructing a Confidence Interval 308 Large-Sample Confidence Interval for a Population Mean m 310 Interpreting the Confidence Interval 311 Large-Sample Confidence Interval for a Population Proportion p 314 Exercises 316

8.6 Estimating the Difference between Two Population Means 318 Exercises 321 8.7 Estimating the Difference between Two Binomial Proportions 324 Exercises 326 8.8 One-Sided Conﬁdence Bounds 328 8.9 Choosing the Sample Size 329 Exercises 333 Chapter Review 336 CASE STUDY: How Reliable Is That Poll? CBS News: How and Where America Eats 341 9

LARGE-SAMPLE TESTS OF HYPOTHESES

343

9.1 Testing Hypotheses about Population Parameters 344 9.2 A Statistical Test of Hypothesis 344 9.3 A Large-Sample Test about a Population Mean 347 The Essentials of the Test 348 Calculating the p-Value 351 Two Types of Errors 356 The Power of a Statistical Test 356 Exercises 360

9.4 A Large-Sample Test of Hypothesis for the Difference between Two Population Means 363 Hypothesis Testing and Conﬁdence Intervals 365 Exercises 366

CONTENTS

❍

xvii

9.5 A Large-Sample Test of Hypothesis for a Binomial Proportion 368 Statistical Signiﬁcance and Practical Importance 370 Exercises 371

9.6 A Large-Sample Test of Hypothesis for the Difference between Two Binomial Proportions 373 Exercises 376

9.7 Some Comments on Testing Hypotheses 378 Chapter Review 379 CASE STUDY: An Aspirin a Day . . . ? 384 10

INFERENCE FROM SMALL SAMPLES

386

10.1 Introduction 387 10.2 Student’s t Distribution 387 Assumptions behind Student’s t Distribution 391

10.3 Small-Sample Inferences Concerning a Population Mean 391 Exercises 397

10.4 Small-Sample Inferences for the Difference between Two Population Means: Independent Random Samples 399 Exercises 406

10.5 Small-Sample Inferences for the Difference between Two Means: A Paired-Difference Test 410 Exercises 414

10.6 Inferences Concerning a Population Variance 417 Exercises 423

10.7 Comparing Two Population Variances 424 Exercises 430

10.8 Revisiting the Small-Sample Assumptions 432 Chapter Review 433 CASE STUDY: How Would You Like a Four-Day Workweek? 445 11

THE ANALYSIS OF VARIANCE

447

11.1 The Design of an Experiment 448 11.2 What Is an Analysis of Variance? 449 11.3 The Assumptions for an Analysis of Variance 449 11.4 The Completely Randomized Design: A One-Way Classiﬁcation 450 11.5 The Analysis of Variance for a Completely Randomized Design 451 Partitioning the Total Variation in an Experiment 451 Testing the Equality of the Treatment Means 454 Estimating Differences in the Treatment Means 456 Exercises 459

xviii

❍

CONTENTS

11.6 Ranking Population Means 462 Exercises 465

11.7 The Randomized Block Design: A Two-Way Classiﬁcation 466 11.8 The Analysis of Variance for a Randomized Block Design 467 Partitioning the Total Variation in the Experiment 467 Testing the Equality of the Treatment and Block Means 470 Identifying Differences in the Treatment and Block Means 472 Some Cautionary Comments on Blocking 473 Exercises 474

11.9 The a b Factorial Experiment: A Two-Way Classiﬁcation 478 11.10 The Analysis of Variance for an a b Factorial Experiment 480 Exercises 484

11.11 Revisiting the Analysis of Variance Assumptions 487 Residual Plots 488

11.12 A Brief Summary 490 Chapter Review 491 CASE STUDY: “A Fine Mess” 501 12

LINEAR REGRESSION AND CORRELATION

502

12.1 Introduction 503 12.2 A Simple Linear Probabilistic Model 503 12.3 The Method of Least Squares 506 12.4 An Analysis of Variance for Linear Regression 509 Exercises 511

12.5 Testing the Usefulness of the Linear Regression Model 514 Inferences Concerning b, the Slope of the Line of Means 514 The Analysis of Variance F-Test 518 Measuring the Strength of the Relationship: The Coefficient of Determination 518 Interpreting the Results of a Signiﬁcant Regression 519 Exercises 520

12.6 Diagnostic Tools for Checking the Regression Assumptions 522 Dependent Error Terms 523 Residual Plots 523 Exercises 524

12.7 Estimation and Prediction Using the Fitted Line 527 Exercises 531

12.8 Correlation Analysis 533 Exercises 537

CONTENTS

❍

xix

Chapter Review 540 CASE STUDY: Is Your Car “Made in the U.S.A.”? 550 13

MULTIPLE REGRESSION ANALYSIS

551

13.1 Introduction 552 13.2 The Multiple Regression Model 552 13.3 A Multiple Regression Analysis 553 The Method of Least Squares 554 The Analysis of Variance for Multiple Regression 555 Testing the Usefulness of the Regression Model 556 Interpreting the Results of a Signiﬁcant Regression 557 Checking the Regression Assumptions 558 Using the Regression Model for Estimation and Prediction 559

13.4 A Polynomial Regression Model 559 Exercises 562

13.5 Using Quantitative and Qualitative Predictor Variables in a Regression Model 566 Exercises 572

13.6 Testing Sets of Regression Coefficients 575 13.7 Interpreting Residual Plots 578 13.8 Stepwise Regression Analysis 579 13.9 Misinterpreting a Regression Analysis 580 Causality 580 Multicollinearity 580

13.10 Steps to Follow When Building a Multiple Regression Model 582 Chapter Review 582 CASE STUDY: “Made in the U.S.A.”—Another Look 592 14

ANALYSIS OF CATEGORICAL DATA

594

14.1 A Description of the Experiment 595 14.2 Pearson’s Chi-Square Statistic 596 14.3 Testing Speciﬁed Cell Probabilities: The Goodness-of-Fit Test 597 Exercises 599

14.4 Contingency Tables: A Two-Way Classiﬁcation 602 The Chi-Square Test of Independence 602 Exercises 608

14.5 Comparing Several Multinomial Populations: A Two-Way Classiﬁcation with Fixed Row or Column Totals 610 Exercises 613

❍

CONTENTS

14.6 The Equivalence of Statistical Tests 614 14.7 Other Applications of the Chi-Square Test 615 Chapter Review 616 CASE STUDY: Can a Marketing Approach Improve Library Services? 628 15

NONPARAMETRIC STATISTICS

629

15.1 Introduction 630 15.2 The Wilcoxon Rank Sum Test: Independent Random Samples 630 Normal Approximation for the Wilcoxon Rank Sum Test 634 Exercises 637

15.3 The Sign Test for a Paired Experiment 639 Normal Approximation for the Sign Test 640 Exercises 642

15.4 A Comparison of Statistical Tests 643 15.5 The Wilcoxon Signed-Rank Test for a Paired Experiment 644 Normal Approximation for the Wilcoxon Signed-Rank Test 647 Exercises 648

15.6 The Kruskal–Wallis H-Test for Completely Randomized Designs 650 Exercises 654

15.7 The Friedman Fr-Test for Randomized Block Designs 656 Exercises 659

15.8 Rank Correlation Coefficient 660 Exercises 664

15.9 Summary 666 Chapter Review 667 CASE STUDY: How’s Your Cholesterol Level? 677

APPENDIX I

679

Table 1

Cumulative Binomial Probabilities 680

Table 2

Cumulative Poisson Probabilities 686

Table 3

Areas under the Normal Curve 688

Table 4

Critical Values of t 691

Table 5

Critical Values of Chi-Square 692

Table 6

Percentage Points of the F Distribution 694

Table 7

Critical Values of T for the Wilcoxon Rank Sum Test, n1 n2 702

Table 8

Critical Values of T for the Wilcoxon Signed-Rank Test, n 5(1)50 704

CONTENTS

Table 9

❍

Critical Values of Spearman’s Rank Correlation Coefficient for a One-Tailed Test 705

Table 10 Random Numbers 706 Table 11 Percentage Points of the Studentized Range, qa(k, df ) 708

DATA SOURCES

712

ANSWERS TO SELECTED EXERCISES INDEX CREDITS

737 744

722

xxi

This page intentionally left blank

Introduction Train Your Brain for Statistics

What is statistics? Have you ever met a statistician? Do you know what a statistician does? Perhaps you are thinking of the person who sits in the broadcast booth at the Rose Bowl, recording the number of pass completions, yards rushing, or interceptions thrown on New Year’s Day. Or perhaps the mere mention of the word statistics sends a shiver of fear through you. You may think you know nothing about statistics; however, it is almost inevitable that you encounter statistics in one form or another every time you pick up a daily newspaper. Here is an example:

Polls See Republicans Keeping Senate Control NEW YORK–Just days from the midterm elections, the ﬁnal round of MSNBC/McClatchy polls shows a tightening race to the ﬁnish in the battle for control of the U.S. Senate. Democrats are leading in several races that could result in party pickups, but Republicans have narrowed the gap in other close races, according to Mason-Dixon polls in 12 states. In all, these key Senate races show the following: •

Two Republican incumbents in serious trouble: Santorum and DeWine. Democrats could gain two seats.

•

Four Republican incumbents essentially tied with their challengers: Allen, Burns, Chafee, and Talent. Four toss-ups that could turn into Democratic gains.

•

Three Democratic incumbents with leads: Cantwell, Menendez, and Stabenow.

•

One Republican incumbent ahead of his challenger: Kyl.

•

One Republican open seat with the Republican leading: Tennessee.

•

One open Democratic seat virtually tied: Maryland.

❍

INTRODUCTION TRAIN YOUR BRAIN FOR STATISTICS

The results show that the Democrats have a good chance of gaining at least two seats in the Senate. As of now, they must win four of the toss-up seats, while holding on to Maryland in order to gain control of the Senate. A total of 625 likely voters in each state were interviewed by telephone. The margin for error, according to standards customarily used by statisticians, is no more than plus or minus 4 percentage points in each poll. —www.msnbc.com1

Articles similar to this one are commonplace in our newspapers and magazines, and in the period just prior to a presidential election, a new poll is reported almost every day. In fact, in the national election on November 7th, the Democrats were able to take control of both the House of Representatives and the Senate of the United States. The language of this article is very familiar to us; however, it leaves the inquisitive reader with some unanswered questions. How were the people in the poll selected? Will these people give the same response tomorrow? Will they give the same response on election day? Will they even vote? Are these people representative of all those who will vote on election day? It is the job of a statistician to ask these questions and to ﬁnd answers for them in the language of the poll. Most Believe “Cover-Up” of JFK Assassination Facts A majority of the public believes the assassination of President John F. Kennedy was part of a larger conspiracy, not the act of one individual. In addition, most Americans think there was a cover-up of facts about the 1963 shooting. More than 40 years after JFK’s assassination, a FOX News poll shows most Americans disagree with the government’s conclusions about the killing. The Warren Commission found that Lee Harvey Oswald acted alone when he shot Kennedy, but 66 percent of the public today think the assassination was “part of a larger conspiracy” while only 25 percent think it was the “act of one individual.” “For older Americans, the Kennedy assassination was a traumatic experience that began a loss of conﬁdence in government,” commented Opinion Dynamics President John Gorman. “Younger people have grown up with movies and documentaries that have pretty much pushed the ‘conspiracy’ line. Therefore, it isn’t surprising there is a fairly solid national consensus that we still don’t know the truth.” (The poll asked): “Do you think that we know all the facts about the assassination of President John F. Kennedy or do you think there was a cover-up?”

All Democrats Republicans Independents

We Know All the Facts

There Was a Cover-Up

(Not Sure)

14% 11% 18% 12%

74 81 69 71

12 8 13 17

—www.foxnews.com2

When you see an article like this one in a magazine, do you simply read the title and the ﬁrst paragraph, or do you read further and try to understand the meaning of the numbers? How did the authors get these numbers? Did they really interview every American with each political affiliation? It is the job of the statistician to interpret the language of this study. Hot News: 98.6 Not Normal After believing for more than a century that 98.6 was the normal body temperature for humans, researchers now say normal is not normal anymore. For some people at some hours of the day, 99.9 degrees could be ﬁne. And readings as low as 96 turn out to be highly human. The 98.6 standard was derived by a German doctor in 1868. Some physicians have always been suspicious of the good doctor’s research. His claim: 1 million readings—in an epoch without computers.

THE POPULATION AND THE SAMPLE

❍

So Mackowiak & Co. took temperature readings from 148 healthy people over a three-day period and found that the mean temperature was 98.2 degrees. Only 8 percent of the readings were 98.6. —The Press-Enterprise3

What questions come to your mind when you read this article? How did the researcher select the 148 people, and how can we be sure that the results based on these 148 people are accurate when applied to the general population? How did the researcher arrive at the normal “high” and “low” temperatures given in the article? How did the German doctor record 1 million temperatures in 1868? Again, we encounter a statistical problem with an application to everyday life. Statistics is a branch of mathematics that has applications in almost every facet of our daily life. It is a new and unfamiliar language for most people, however, and, like any new language, statistics can seem overwhelming at ﬁrst glance. We want you to “train your brain” to understand this new language one step at a time. Once the language of statistics is learned and understood, it provides a powerful tool for data analysis in many different ﬁelds of application.

THE POPULATION AND THE SAMPLE In the language of statistics, one of the most basic concepts is sampling. In most statistical problems, a speciﬁed number of measurements or data—a sample—is drawn from a much larger body of measurements, called the population. Sample

Population

For the body-temperature experiment, the sample is the set of body-temperature measurements for the 148 healthy people chosen by the experimenter. We hope that the sample is representative of a much larger body of measurements—the population— the body temperatures of all healthy people in the world! Which is of primary interest, the sample or the population? In most cases, we are interested primarily in the population, but the population may be difficult or impossible to enumerate. Imagine trying to record the body temperature of every healthy person on earth or the presidential preference of every registered voter in the United States! Instead, we try to describe or predict the behavior of the population on the basis of information obtained from a representative sample from that population. The words sample and population have two meanings for most people. For example, you read in the newspapers that a Gallup poll conducted in the United States was based on a sample of 1823 people. Presumably, each person interviewed is asked a particular question, and that person’s response represents a single measurement in the sample. Is the sample the set of 1823 people, or is it the 1823 responses that they give? When we use statistical language, we distinguish between the set of objects on which the measurements are taken and the measurements themselves. To experimenters, the objects on which measurements are taken are called experimental units. The sample survey statistician calls them elements of the sample.

❍

INTRODUCTION TRAIN YOUR BRAIN FOR STATISTICS

DESCRIPTIVE AND INFERENTIAL STATISTICS When ﬁrst presented with a set of measurements—whether a sample or a population— you need to ﬁnd a way to organize and summarize it. The branch of statistics that presents techniques for describing sets of measurements is called descriptive statistics. You have seen descriptive statistics in many forms: bar charts, pie charts, and line charts presented by a political candidate; numerical tables in the newspaper; or the average rainfall amounts reported by the local television weather forecaster. Computer-generated graphics and numerical summaries are commonplace in our everyday communication. Descriptive statistics consists of procedures used to summarize and describe the important characteristics of a set of measurements. Definition

If the set of measurements is the entire population, you need only to draw conclusions based on the descriptive statistics. However, it might be too expensive or too time consuming to enumerate the entire population. Perhaps enumerating the population would destroy it, as in the case of “time to failure” testing. For these or other reasons, you may have only a sample from the population. By looking at the sample, you want to answer questions about the population as a whole. The branch of statistics that deals with this problem is called inferential statistics. Inferential statistics consists of procedures used to make inferences about population characteristics from information contained in a sample drawn from this population. Definition

The objective of inferential statistics is to make inferences (that is, draw conclusions, make predictions, make decisions) about the characteristics of a population from information contained in a sample.

ACHIEVING THE OBJECTIVE OF INFERENTIAL STATISTICS: THE NECESSARY STEPS How can you make inferences about a population using information contained in a sample? The task becomes simpler if you train yourself to organize the problem into a series of logical steps. 1. Specify the questions to be answered and identify the population of interest. In the presidential election poll, the objective is to determine who will get the most votes on election day. Hence, the population of interest is the set of all votes in the presidential election. When you select a sample, it is important that the sample be representative of this population, not the population of voter preferences on July 5 or on some other day prior to the election. 2. Decide how to select the sample. This is called the design of the experiment or the sampling procedure. Is the sample representative of the population of interest? For example, if a sample of registered voters is selected from the state of Arkansas, will this sample be representative of all voters in the United States?

TRAINING YOUR BRAIN FOR STATISTICS

❍

Will it be the same as a sample of “likely voters”—those who are likely to actually vote in the election? Is the sample large enough to answer the questions posed in step 1 without wasting time and money on additional information? A good sampling design will answer the questions posed with minimal cost to the experimenter. 3. Select the sample and analyze the sample information. No matter how much information the sample contains, you must use an appropriate method of analysis to extract it. Many of these methods, which depend on the sampling procedure in step 2, are explained in the text. 4. Use the information from step 3 to make an inference about the population. Many different procedures can be used to make this inference, and some are better than others. For example, 10 different methods might be available to estimate human response to an experimental drug, but one procedure might be more accurate than others. You should use the best inference-making procedure available (many of these are explained in the text). 5. Determine the reliability of the inference. Since you are using only a fraction of the population in drawing the conclusions described in step 4, you might be wrong! How can this be? If an agency conducts a statistical survey for you and estimates that your company’s product will gain 34% of the market this year, how much confidence can you place in this estimate? Is this estimate accurate to within 1, 5, or 20 percentage points? Is it reliable enough to be used in setting production goals? Every statistical inference should include a measure of reliability that tells you how much confidence you have in the inference. Now that you have learned some of the basic terms and concepts in the language of statistics, we again pose the question asked at the beginning of this discussion: Do you know what a statistician does? It is the job of the statistician to implement all of the preceding steps. This may involve questioning the experimenter to make sure that the population of interest is clearly defined, developing an appropriate sampling plan or experimental design to provide maximum information at minimum cost, correctly analyzing and drawing conclusions using the sample information, and finally, measuring the reliability of the conclusions based on the experimental results.

TRAINING YOUR BRAIN FOR STATISTICS As you proceed through the book, you will learn more and more words, phrases, and concepts from this new language of statistics. Statistical procedures, for the most part, consist of commonsense steps that, given enough time, you would most likely have discovered for yourself. Since statistics is an applied branch of mathematics, many of these basic concepts are mathematical—developed and based on results from calculus or higher mathematics. However, you do not have to be able to derive results in order to apply them in a logical way. In this text, we use numerical examples and intuitive arguments to explain statistical concepts, rather than more complicated mathematical arguments. To help you in your statistical training, we have included a section called “MyPersonal Trainer” at appropriate points in the text. This is your “personal trainer,” which will take you step-by-step through some of the procedures that tend to be confusing to students. Once you read the step-by-step explanation, try doing the “Exercise Reps,”

❍

INTRODUCTION TRAIN YOUR BRAIN FOR STATISTICS

which usually appear in table form. Write the answers—right in your book—and then check your answers against the answers on the perforated card at the back of the book. If you’re still having trouble, you will find more “Exercise Reps” in the exercise set for that section. You should also watch for quick study tips—named “My Tip”—found in the margin of the text as you read through the chapter. In recent years, computers have become readily available to many students and provide them with an invaluable tool. In the study of statistics, even the beginning student can use packaged programs to perform statistical analyses with a high degree of speed and accuracy. Some of the more common statistical packages available at computer facilities are MINITABTM, SAS (Statistical Analysis System), and SPSS (Statistical Package for the Social Sciences); personal computers will support packages such as MINITAB, MS Excel, and others. There are even online statistical programs and interactive “applets” on the Internet. These programs, called statistical software, differ in the types of analyses available, the options within the programs, and the forms of printed results (called output). However, they are all similar. In this book, we primarily use MINITAB as a statistical tool; understanding the basic output of this package will help you interpret the output from other software systems. At the end of most chapters, you will find a section called “My MINITAB.” These sections present numerical examples to guide you through the MINITAB commands and options that are used for the procedures in that chapter. If you are using MINITAB in a lab or home setting, you may want to work through this section at your own computer so that you become familiar with the hands-on methods in MINITAB analysis. If you do not need hands-on knowledge of MINITAB, you may choose to skip this section and simply use the MINITAB printouts for analysis as they appear in the text. You will also find a section called “MyApplet” in many of the chapters. These sections provide a useful introduction to the statistical applets available on the Premium Website. You can use these applets to visualize many of the chapter concepts and to find solutions to exercises in a new section called “MyApplet Exercises.” Most important, using statistics successfully requires common sense and logical thinking. For example, if we want to find the average height of all students at a particular university, would we select our entire sample from the members of the basketball team? In the body-temperature example, the logical thinker would question an 1868 average based on 1 million measurements—when computers had not yet been invented. As you learn new statistical terms, concepts, and techniques, remember to view every problem with a critical eye and be sure that the rule of common sense applies. Throughout the text, we will remind you of the pitfalls and dangers in the use or misuse of statistics. Benjamin Disraeli once said that there are three kinds of lies: lies, damn lies, and statistics! Our purpose is to dispel this claim—to show you how to make statistics work for you and not lie for you! As you continue through the book, refer back to this “training manual” periodically. Each chapter will increase your knowledge of the language of statistics and should, in some way, help you achieve one of the steps described here. Each of these steps is essential in attaining the overall objective of inferential statistics: to make inferences about a population using information contained in a sample drawn from that population.

Describing Data with Graphs

GENERAL OBJECTIVES Many sets of measurements are samples selected from larger populations. Other sets constitute the entire population, as in a national census. In this chapter, you will learn what a variable is, how to classify variables into several types, and how measurements or data are generated. You will then learn how to use graphs to describe data sets.

CHAPTER INDEX ● Data distributions and their shapes (1.1, 1.4) ● Dotplots (1.4) ● Pie charts, bar charts, line charts (1.3, 1.4) ● Qualitative and quantitative variables—discrete and continuous (1.2)

How Is Your Blood Pressure? Is your blood pressure normal, or is it too high or too low? The case study at the end of this chapter examines a large set of blood pressure data. You will use graphs to describe these data and compare your blood pressure with that of others of your same age and gender.

● Relative frequency histograms (1.5) ● Stem and leaf plots (1.4) ● Univariate and bivariate data (1.1) ● Variables, experimental units, samples and populations, data (1.1)

How Do I Construct a Stem and Leaf Plot? How Do I Construct a Relative Frequency Histogram?

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

1.1

VARIABLES AND DATA In Chapters 1 and 2, we will present some basic techniques in descriptive statistics— the branch of statistics concerned with describing sets of measurements, both samples and populations. Once you have collected a set of measurements, how can you display this set in a clear, understandable, and readable form? First, you must be able to deﬁne what is meant by measurements or “data” and to categorize the types of data that you are likely to encounter in real life. We begin by introducing some deﬁnitions—new terms in the statistical language that you need to know. A variable is a characteristic that changes or varies over time and/or for different individuals or objects under consideration. Definition

For example, body temperature is a variable that changes over time within a single individual; it also varies from person to person. Religious affiliation, ethnic origin, income, height, age, and number of offspring are all variables—characteristics that vary depending on the individual chosen. In the Introduction, we deﬁned an experimental unit or an element of the sample as the object on which a measurement is taken. Equivalently, we could deﬁne an experimental unit as the object on which a variable is measured. When a variable is actually measured on a set of experimental units, a set of measurements or data result. Definition An experimental unit is the individual or object on which a variable is measured. A single measurement or data value results when a variable is actually measured on an experimental unit.

If a measurement is generated for every experimental unit in the entire collection, the resulting data set constitutes the population of interest. Any smaller subset of measurements is a sample. Definition

A population is the set of all measurements of interest to the investi-

gator.

Definition

A sample is a subset of measurements selected from the population of

interest.

EXAMPLE

1.1

A set of five students is selected from all undergraduates at a large university, and measurements are entered into a spreadsheet as shown in Figure 1.1. Identify the various elements involved in generating this set of measurements. Solution There are several variables in this example. The experimental unit on which the variables are measured is a particular undergraduate student on the campus, identified in column C1. Five variables are measured for each student: grade point average (GPA), gender, year in college, major, and current number of units enrolled. Each of these characteristics varies from student to student. If we consider the GPAs of all students at this university to be the population of interest, the five GPAs in column C2 represent a sample from this population. If the GPA of each undergraduate student at the university had been measured, we would have generated the entire population of measurements for this variable.

1.1 VARIABLES AND DATA

F I GU R E 1 .1

Measurements on ﬁve undergraduate students

❍

●

The second variable measured on the students is gender, in column C3-T. This variable can take only one of two values—male (M) or female (F). It is not a numerically valued variable and hence is somewhat different from GPA. The population, if it could be enumerated, would consist of a set of Ms and Fs, one for each student at the university. Similarly, the third and fourth variables, year and major, generate nonnumerical data. Year has four categories (Fr, So, Jr, Sr), and major has one category for each undergraduate major on campus. The last variable, current number of units enrolled, is numerically valued, generating a set of numbers rather than a set of qualities or characteristics. Although we have discussed each variable individually, remember that we have measured each of these five variables on a single experimental unit: the student. Therefore, in this example, a “measurement” really consists of five observations, one for each of the five measured variables. For example, the measurement taken on student 2 produces this observation: (2.3, F, So, Mathematics, 15) You can see that there is a difference between a single variable measured on a single experimental unit and multiple variables measured on a single experimental unit as in Example 1.1. Univariate data result when a single variable is measured on a single experimental unit. Definition

Bivariate data result when two variables are measured on a single experimental unit. Multivariate data result when more than two variables are measured. Definition

If you measure the body temperatures of 148 people, the resulting data are univariate. In Example 1.1, ﬁve variables were measured on each student, resulting in multivariate data.

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

1.2

TYPES OF VARIABLES Variables can be classiﬁed into one of two categories: qualitative or quantitative. Definition Qualitative variables measure a quality or characteristic on each experimental unit. Quantitative variables measure a numerical quantity or amount on each experimental unit.

Qualitative ⇔ “quality” or characteristic Quantitative ⇔ “quantity” or number

Qualitative variables produce data that can be categorized according to similarities or differences in kind; hence, they are often called categorical data. The variables gender, year, and major in Example 1.1 are qualitative variables that produce categorical data. Here are some other examples: • • •

Political affiliation: Republican, Democrat, Independent Taste ranking: excellent, good, fair, poor Color of an M&M’S® candy: brown, yellow, red, orange, green, blue

Quantitative variables, often represented by the letter x, produce numerical data, such as those listed here: • • • •

x Prime interest rate x Number of passengers on a ﬂight from Los Angeles to New York City x Weight of a package ready to be shipped x Volume of orange juice in a glass

Notice that there is a difference in the types of numerical values that these quantitative variables can assume. The number of passengers, for example, can take on only the values x 0, 1, 2, . . . , whereas the weight of a package can take on any value greater than zero, or 0 x . To describe this difference, we define two types of quantitative variables: discrete and continuous. Definition A discrete variable can assume only a finite or countable number of values. A continuous variable can assume the infinitely many values corresponding to the points on a line interval.

Discrete ⇔ “listable” Continuous ⇔ “unlistable”

EXAMPLE

1.2

The name discrete relates to the discrete gaps between the possible values that the variable can assume. Variables such as number of family members, number of new car sales, and number of defective tires returned for replacement are all examples of discrete variables. On the other hand, variables such as height, weight, time, distance, and volume are continuous because they can assume values at any point along a line interval. For any two values you pick, a third value can always be found between them! Identify each of the following variables as qualitative or quantitative: 1. The most frequent use of your microwave oven (reheating, defrosting, warming, other) 2. The number of consumers who refuse to answer a telephone survey 3. The door chosen by a mouse in a maze experiment (A, B, or C) 4. The winning time for a horse running in the Kentucky Derby 5. The number of children in a ﬁfth-grade class who are reading at or above grade level

1.3 GRAPHS FOR CATEGORICAL DATA

❍

Solution Variables 1 and 3 are both qualitative because only a quality or characteristic is measured for each individual. The categories for these two variables are shown in parentheses. The other three variables are quantitative. Variable 2, the number of consumers, is a discrete variable that can take on any of the values x 0, 1, 2, . . . , with a maximum value depending on the number of consumers called. Similarly, variable 5, the number of children reading at or above grade level, can take on any of the values x 0, 1, 2, . . . , with a maximum value depending on the number of children in the class. Variable 4, the winning time for a Kentucky Derby horse, is the only continuous variable in the list. The winning time, if it could be measured with sufﬁcient accuracy, could be 121 seconds, 121.5 seconds, 121.25 seconds, or any values between any two times we have listed.

Discrete variables often involve the “number of” items in a set.

Figure 1.2 depicts the types of data we have deﬁned. Why should you be concerned about different kinds of variables and the data that they generate? The reason is that the methods used to describe data sets depend on the type of data you have collected. For each set of data that you collect, the key will be to determine what type of data you have and how you can present them most clearly and understandably to your audience!

FIGU R E 1 .2

●

Types of data

Data

Qualitative

Quantitative

Discrete

1.3

Continuous

GRAPHS FOR CATEGORICAL DATA After the data have been collected, they can be consolidated and summarized to show the following information: • •

What values of the variable have been measured How often each value has occurred

For this purpose, you can construct a statistical table that can be used to display the data graphically as a data distribution. The type of graph you choose depends on the type of variable you have measured. When the variable of interest is qualitative, the statistical table is a list of the categories being considered along with a measure of how often each value occurred. You can measure “how often” in three different ways: • • •

The frequency, or number of measurements in each category The relative frequency, or proportion of measurements in each category The percentage of measurements in each category

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

For example, if you let n be the total number of measurements in the set, you can ﬁnd the relative frequency and percentage using these relationships: Frequency Relative frequency n Percent 100 Relative frequency You will ﬁnd that the sum of the frequencies is always n, the sum of the relative frequencies is 1, and the sum of the percentages is 100%. The categories for a qualitative variable should be chosen so that • •

For example, if you categorize meat products according to the type of meat used, you might use these categories: beef, chicken, seafood, pork, turkey, other. To categorize ranks of college faculty, you might use these categories: professor, associate professor, assistant professor, instructor, lecturer, other. The “other” category is included in both cases to allow for the possibility that a measurement cannot be assigned to one of the earlier categories. Once the measurements have been categorized and summarized in a statistical table, you can use either a pie chart or a bar chart to display the distribution of the data. A pie chart is the familiar circular graph that shows how the measurements are distributed among the categories. A bar chart shows the same distribution of measurements in categories, with the height of the bar measuring how often a particular category was observed.

Three steps to a data distribution: (1) raw data ⇒ (2) statistical table ⇒ (3) graph

EXAMPLE

a measurement will belong to one and only one category each measurement has a category to which it can be assigned

1.3

In a survey concerning public education, 400 school administrators were asked to rate the quality of education in the United States. Their responses are summarized in Table 1.1. Construct a pie chart and a bar chart for this set of data. To construct a pie chart, assign one sector of a circle to each category. The angle of each sector should be proportional to the proportion of measurements (or relative frequency) in that category. Since a circle contains 360°, you can use this equation to ﬁnd the angle:

Solution

Angle Relative frequency 360° TABLE 1.1

●

U.S. Education Rating by 400 Educators Rating

Proportions add to 1. Percents add to 100. Sector angles add to 360°.

Frequency

A B C D

35 260 93 12

Total

400

Table 1.2 shows the ratings along with the frequencies, relative frequencies, percentages, and sector angles necessary to construct the pie chart. Figure 1.3 shows the pie chart constructed from the values in the table. While pie charts use percentages to determine the relative sizes of the “pie slices,” bar charts usually plot frequency against the categories. A bar chart for these data is shown in Figure 1.4.

1.3 GRAPHS FOR CATEGORICAL DATA

TABLE 1.2

●

❍

Calculations for the Pie Chart in Example 1.3 Rating

Frequency

Relative Frequency

A B C D

35 260 93 12

35/400 .09 260/400 .65 93/400 .23 12/400 .03

400

1.00

Total

Percent 9% 65% 23% 3% 100%

Angle .09 360 32.4º 234.0º 82.8º 10.8º 360º

The visual impact of these two graphs is somewhat different. The pie chart is used to display the relationship of the parts to the whole; the bar chart is used to emphasize the actual quantity or frequency for each category. Since the categories in this example are ordered “grades” (A, B, C, D), we would not want to rearrange the bars in the chart to change its shape. In a pie chart, the order of presentation is irrelevant. FIGU R E 1 .3

Pie chart for Example 1.3

● D 3.0%

A 8.8%

C 23.3%

B 65.0%

FIGU R E 1 .4

Bar chart for Example 1.3

● 250

Frequency

200

150

100

0 A

Rating

EXAMPLE

1.4

A snack size bag of peanut M&M’S candies contains 21 candies with the colors listed in Table 1.3. The variable “color” is qualitative, so Table 1.4 lists the six categories along with a tally of the number of candies of each color. The last three columns of Table 1.4 give the three different measures of how often each category occurred. Since the categories are colors and have no particular order, you could construct bar charts with many different shapes just by reordering the bars. To emphasize that brown is the most frequent color, followed by blue, green, and orange, we order the bars from largest to smallest and generate the bar chart using MINITAB in Figure 1.5. A bar chart in which the bars are ordered from largest to smallest is called a Pareto chart.

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

TABLE 1.3

●

Raw Data: Colors of 21 Candies Brown Red Yellow Brown Orange Yellow

TABLE 1.4

●

Green Red Orange Blue Blue

Brown Green Green Blue Brown

Statistical Table: M&M’S Data for Example 1.4 Category

Tally

Frequency

Relative Frequency

Brown Green Orange Yellow Red Blue

6 3 3 2 2 5

6/21 3/21 3/21 2/21 2/21 5/21

Total

FI GU R E 1 .5

MINITAB bar chart for Example 1.4

Blue Brown Blue Brown Orange

Percent 28% 14 14 10 10 24

100%

● 6

Frequency

5 4 3 2 1 0 Brown

Blue

Green

Orange

Yellow

Red

Color

1.3

EXERCISES

UNDERSTANDING THE CONCEPTS

1.2 Qualitative or Quantitative? Identify each

1.1 Experimental Units Identify the experimental

variable as quantitative or qualitative: a. Amount of time it takes to assemble a simple puzzle b. Number of students in a ﬁrst-grade classroom c. Rating of a newly elected politician (excellent, good, fair, poor) d. State in which a person lives

units on which the following variables are measured: a. b. c. d. e.

Gender of a student Number of errors on a midterm exam Age of a cancer patient Number of ﬂowers on an azalea plant Color of a car entering a parking lot

1.3 GRAPHS FOR CATEGORICAL DATA

1.3 Discrete or Continuous? Identify the following quantitative variables as discrete or continuous: a. Population in a particular area of the United States b. Weight of newspapers recovered for recycling on a single day c. Time to complete a sociology exam d. Number of consumers in a poll of 1000 who consider nutritional labeling on food products to be important 1.4 Discrete or Continuous? Identify each quantitative variable as discrete or continuous. a. Number of boating accidents along a 50-mile stretch of the Colorado River b. Time required to complete a questionnaire c. Cost of a head of lettuce d. Number of brothers and sisters you have e. Yield in kilograms of wheat from a 1-hectare plot in a wheat ﬁeld 1.5 Parking on Campus Six vehicles are selected

from the vehicles that are issued campus parking permits, and the following data are recorded:

Vehicle

Type

Make

1 2 3 4 5

Car Car Truck Van Motorcycle Car

Honda Toyota Toyota Dodge HarleyDavidson Chevrolet

Carpool?

One-way Commute Distance (miles)

Age of Vehicle (years)

No No No Yes No

23.6 17.2 10.1 31.7 25.5

6 3 4 2 1

5.4

a. What are the experimental units? b. What are the variables being measured? What types of variables are they? c. Is this univariate, bivariate, or multivariate data? 1.6 Past U.S. Presidents A data set consists of the

ages at death for each of the 38 past presidents of the United States now deceased. a. Is this set of measurements a population or a sample? b. What is the variable being measured? c. Is the variable in part b quantitative or qualitative? 1.7 Voter Attitudes You are a candidate for your

state legislature, and you want to survey voter attitudes regarding your chances of winning. Identify the population that is of interest to you and from which you would like to select your sample. How is this population dependent on time?

❍

1.8 Cancer Survival Times A medical researcher

wants to estimate the survival time of a patient after the onset of a particular type of cancer and after a particular regimen of radiotherapy. a. What is the variable of interest to the medical researcher? b. Is the variable in part a qualitative, quantitative discrete, or quantitative continuous? c. Identify the population of interest to the medical researcher. d. Describe how the researcher could select a sample from the population. e. What problems might arise in sampling from this population? 1.9 New Teaching Methods An educational

researcher wants to evaluate the effectiveness of a new method for teaching reading to deaf students. Achievement at the end of a period of teaching is measured by a student’s score on a reading test. a. What is the variable to be measured? What type of variable is it? b. What is the experimental unit? c. Identify the population of interest to the experimenter. BASIC TECHNIQUES 1.10 Fifty people are grouped into four categories—

A, B, C, and D—and the number of people who fall into each category is shown in the table: Category

Frequency

A B C D

11 14 20 5

a. What is the experimental unit? b. What is the variable being measured? Is it qualitative or quantitative? c. Construct a pie chart to describe the data. d. Construct a bar chart to describe the data. e. Does the shape of the bar chart in part d change depending on the order of presentation of the four categories? Is the order of presentation important? f. What proportion of the people are in category B, C, or D? g. What percentage of the people are not in category B?

16 ❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

1.11 Jeans A manufacturer of jeans has plants in

California, Arizona, and Texas. A group of 25 pairs of jeans is randomly selected from the computerized database, and the state in which each is produced is recorded: CA CA AZ CA CA

AZ CA AZ AZ AZ

AZ TX CA TX AZ

TX TX AZ TX CA

CA TX TX TX CA

a. What is the experimental unit? b. What is the variable being measured? Is it qualitative or quantitative? c. Construct a pie chart to describe the data. d. Construct a bar chart to describe the data. e. What proportion of the jeans are made in Texas? f. What state produced the most jeans in the group? g. If you want to ﬁnd out whether the three plants produced equal numbers of jeans, or whether one produced more jeans than the others, how can you use the charts from parts c and d to help you? What conclusions can you draw from these data? APPLICATIONS 1.12 Election 2008 During the spring of 2006 the

news media were already conducting opinion polls that tracked the fortunes of the major candidates hoping to become the president of the United States. One such poll conducted by Financial Dynamics showed the following results:1 “Thinking ahead to the next presidential election, if the 2008 election were held today and the candidates were Democrat [see below] and Republican [see below], for whom would you vote?”

The results were based on a sample taken May 16–18, 2006, of 900 registered voters nationwide. a. If the pollsters were planning to use these results to predict the outcome of the 2008 presidential election, describe the population of interest to them. b. Describe the actual population from which the sample was drawn. c. Some pollsters prefer to select a sample of “likely” voters. What is the difference between “registered voters” and “likely voters”? Why is this important? d. Is the sample selected by the pollsters representative of the population described in part a? Explain. 1.13 Want to Be President? Would you want to be the president of the United States? Although many teenagers think that they could grow up to be the president, most don’t want the job. In an opinion poll conducted by ABC News, nearly 80% of the teens were not interested in the job.2 When asked “What’s the main reason you would not want to be president?” they gave these responses: Other career plans/no interest Too much pressure Too much work Wouldn’t be good at it Too much arguing

a. Are all of the reasons accounted for in this table? Add another category if necessary. b. Would you use a pie chart or a bar chart to graphically describe the data? Why? c. Draw the chart you chose in part b. d. If you were the person conducting the opinion poll, what other types of questions might you want to investigate?

John McCain (R) % 46

Hillary Clinton (D) % 42

Unsure % 13

John McCain % 51

Al Gore % 33

Unsure % 15

Rudy Giuliani % 49

Hillary Clinton % 40

Unsure % 12

Rudy Giuliani % 50

White Black Hispanic Other

Al Gore % 37

Unsure % 13

Source: Time magazine

Source: www.pollingreport.com

40% 20% 15% 14% 5%

1.14 Race Distributions in the Armed Forces

The four branches of the armed forces in the United States are quite different in their makeup with regard to gender, race, and age distributions. The table below shows the racial breakdown of the members of the United States Army and the United States Air Force.3 Army

Air Force

58.4% 26.3% 8.9% 6.4%

75.5% 16.2% 5.0% 3.3%

a. Deﬁne the variable that has been measured in this table.

1.4 GRAPHS FOR QUANTITATIVE DATA

b. Is the variable quantitative or qualitative? c. What do the numbers represent? d. Construct a pie chart to describe the racial breakdown in the U.S. Army. e. Construct a bar chart to describe the racial breakdown in the U.S. Air Force. f. What percentage of the members of the U.S. Army are minorities—that is, not white? What is this percentage in the U.S. Air Force?

❍

from vacation? A bar graph with data from the Snapshots section of USA Today is shown below:4 a. Are all of the opinions accounted for in the table? Add another category if necessary. b. Is the bar chart drawn accurately? That is, are the three bars in the correct proportion to each other? c. Use a pie chart to describe the opinions. Which graph is more interesting to look at?

1.15 Back to Work How long does it take you to adjust to your normal work routine after coming back

Adjustment from Vacation One day A few days No time 0%

10% 20% 30% 40%

1.4

GRAPHS FOR QUANTITATIVE DATA Quantitative variables measure an amount or quantity on each experimental unit. If the variable can take only a ﬁnite or countable number of values, it is a discrete variable. A variable that can assume an inﬁnite number of values corresponding to points on a line interval is called continuous.

Pie Charts and Bar Charts Sometimes information is collected for a quantitative variable measured on different segments of the population, or for different categories of classiﬁcation. For example, you might measure the average incomes for people of different age groups, different genders, or living in different geographic areas of the country. In such cases, you can use pie charts or bar charts to describe the data, using the amount measured in each category rather than the frequency of occurrence of each category. The pie chart displays how the total quantity is distributed among the categories, and the bar chart uses the height of the bar to display the amount in a particular category.

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

EXAMPLE

1.5

TABLE 1.5

The amount of money expended in ﬁscal year 2005 by the U.S. Department of Defense in various categories is shown in Table 1.5.5 Construct both a pie chart and a bar chart to describe the data. Compare the two forms of presentation. ●

Expenses by Category Category

Amount (in billions)

Military personnel Operation and maintenance Procurement Research and development Military construction Other

$127.5 188.1 82.3 65.7 5.3 5.5

Total

$474.4

Source: The World Almanac and Book of Facts 2007

Solution Two variables are being measured: the category of expenditure (qualitative) and the amount of the expenditure (quantitative). The bar chart in Figure 1.6 displays the categories on the horizontal axis and the amounts on the vertical axis. For FI GU R E 1 .6

● 200 Amount ($ Billions)

Bar chart for Example 1.5

150 100 50

tru

t pm

de M

ili

tio

ili M

t en em ur oc Pr

the pie chart in Figure 1.7, each “pie slice” represents the proportion of the total expenditures ($474.4 billion) corresponding to its particular category. For example, for the research and development category, the angle of the sector is 65.7 360° 49.9° 474.4 FI GU R E 1 .7

Pie chart for Example 1.5

●

Military construction 5.3 Research and development 65.7

Military personnel 127.5

Procurement 82.3 Other 5.5

Operation and maintenance 188.1

1.4 GRAPHS FOR QUANTITATIVE DATA

❍

Both graphs show that the largest amounts of money were spent on personnel and operations. Since there is no inherent order to the categories, you are free to rearrange the bars or sectors of the graphs in any way you like. The shape of the bar chart has no bearing on its interpretation.

Line Charts When a quantitative variable is recorded over time at equally spaced intervals (such as daily, weekly, monthly, quarterly, or yearly), the data set forms a time series. Time series data are most effectively presented on a line chart with time as the horizontal axis. The idea is to try to discern a pattern or trend that will likely continue into the future, and then to use that pattern to make accurate predictions for the immediate future. EXAMPLE

1.6

TABLE 1.6

In the year 2025, the oldest “baby boomers” (born in 1946) will be 79 years old, and the oldest “Gen-Xers” (born in 1965) will be two years from Social Security eligibility. How will this affect the consumer trends in the next 15 years? Will there be sufficient funds for “baby boomers” to collect Social Security beneﬁts? The United States Bureau of the Census gives projections for the portion of the U.S. population that will be 85 and over in the coming years, as shown below.5 Construct a line chart to illustrate the data. What is the effect of stretching and shrinking the vertical axis on the line chart? ●

Population Growth Projections Year

2010

2020

2030

2040

2050

6.1

7.3

9.6

15.4

20.9

85 and over (millions)

The quantitative variable “85 and over” is measured over ﬁve time intervals, creating a time series that you can graph with a line chart. The time intervals are marked on the horizontal axis and the projections on the vertical axis. The data points are then connected by line segments to form the line charts in Figure 1.8. Notice the marked difference in the vertical scales of the two graphs. Shrinking the scale on the vertical axis causes large changes to appear small, and vice versa. To avoid misleading conclusions, you must look carefully at the scales of the vertical and horizontal axes. However, from both graphs you get a clear picture of the steadily increasing number of those 85 and older in the early years of the new millennium.

Solution Beware of stretching or shrinking axes when you look at a graph!

● 100

22.5 20.0

85 and Older (Millions)

Line charts for Example 1.6

85 and Older (Millions)

FIGU R E 1 .8

17.5 15.0 12.5 10.0 7.5 5.0

80 60 40 20 0

2010

2020

2030 Year

2040

2050

2010

2020

2030 Year

2040

2050

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

Dotplots Many sets of quantitative data consist of numbers that cannot easily be separated into categories or intervals of time. You need a different way to graph this type of data! The simplest graph for quantitative data is the dotplot. For a small set of measurements—for example, the set 2, 6, 9, 3, 7, 6—you can simply plot the measurements as points on a horizontal axis. This dotplot, generated by MINITAB, is shown in Figure 1.9(a). For a large data set, however, such as the one in Figure 1.9(b), the dotplot can be uninformative and tedious to interpret. (a)

FI GU R E 1 .9

Dotplots for small and large data sets

●

6 Small Set

(b)

0.98

1.05

1.12

1.19 1.26 Large Set

1.33

1.40

1.47

Stem and Leaf Plots Another simple way to display the distribution of a quantitative data set is the stem and leaf plot. This plot presents a graphical display of the data using the actual numerical values of each data point.

How Do I Construct a Stem and Leaf Plot? 1. Divide each measurement into two parts: the stem and the leaf. 2. List the stems in a column, with a vertical line to their right. 3. For each measurement, record the leaf portion in the same row as its corresponding stem. 4. Order the leaves from lowest to highest in each stem. 5. Provide a key to your stem and leaf coding so that the reader can re-create the actual measurements if necessary.

EXAMPLE

1.7

Table 1.7 lists the prices (in dollars) of 19 different brands of walking shoes. Construct a stem and leaf plot to display the distribution of the data.

1.4 GRAPHS FOR QUANTITATIVE DATA

TABLE 1.7

●

❍

Prices of Walking Shoes 90 65 75 70

70 68 70

70 60 68

70 74 65

75 70 40

70 95 65

Solution To create the stem and leaf, you could divide each observation between the ones and the tens place. The number to the left is the stem; the number to the right is the leaf. Thus, for the shoes that cost $65, the stem is 6 and the leaf is 5. The stems, ranging from 4 to 9, are listed in Figure 1.10, along with the leaves for each of the 19 measurements. If you indicate that the leaf unit is 1, the reader will realize that the stem and leaf 6 and 8, for example, represent the number 68, recorded to the nearest dollar.

FIGU R E 1 .1 0

Stem and leaf plot for the data in Table 1.7

●

4 5 6 7 8 9

580855 0005040500 05

• •

TABLE 1.8

4 5 Reordering ⎯→ 6 7 8 9

0 055588 0000000455 05

Sometimes the available stem choices result in a plot that contains too few stems and a large number of leaves within each stem. In this situation, you can stretch the stems by dividing each one into several lines, depending on the leaf values assigned to them. Stems are usually divided in one of two ways:

stem | leaf

EXAMPLE

Leaf unit 1

1.8

Into two lines, with leaves 0–4 in the first line and leaves 5–9 in the second line Into five lines, with leaves 0–1, 2–3, 4–5, 6–7, and 8–9 in the five lines, respectively

The data in Table 1.8 are the weights at birth of 30 full-term babies, born at a metropolitan hospital and recorded to the nearest tenth of a pound.6 Construct a stem and leaf plot to display the distribution of the data. ●

Birth Weights of 30 Full-Term Newborn Babies 7.2 8.0 8.2 5.8 6.1 8.5

7.8 8.2 7.7 6.8 7.9 9.0

6.8 5.6 7.5 6.8 9.4 7.7

6.2 8.6 7.2 8.5 9.0 6.7

8.2 7.1 7.7 7.5 7.8 7.7

The data, though recorded to an accuracy of only one decimal place, are measurements of the continuous variable x weight, which can take on any positive value. By examining Table 1.8, you can quickly see that the highest and lowest weights are 9.4 and 5.6, respectively. But how are the remaining weights distributed? Solution

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

If you use the decimal point as the dividing line between the stem and the leaf, you have only ﬁve stems, which does not produce a very good picture. When you divide each stem into two lines, there are eight stems, since the ﬁrst line of stem 5 and the second line of stem 9 are empty! This produces a more descriptive plot, as shown in Figure 1.11. For these data, the leaf unit is .1, and the reader can infer that the stem and leaf 8 and 2, for example, represent the measurement x 8.2.

FI GU R E 1 .1 1

Stem and leaf plot for the data in Table 1.8

● 5

86 12 8887 221 879577587 0222 565 040

6 6 7 7 8 8 9

Reordering →

Leaf unit .1

5 6 6 7 7 8 8 9

68 12 7888 122 557777889 0222 556 004

If you turn the stem and leaf plot sideways, so that the vertical line is now a horizontal axis, you can see that the data have “piled up” or been “distributed” along the axis in a pattern that can be described as “mound-shaped”—much like a pile of sand on the beach. This plot again shows that the weights of these 30 newborns range between 5.6 and 9.4; many weights are between 7.5 and 8.0 pounds.

Interpreting Graphs with a Critical Eye Once you have created a graph or graphs for a set of data, what should you look for as you attempt to describe the data? • •

•

First, check the horizontal and vertical scales, so that you are clear about what is being measured. Examine the location of the data distribution. Where on the horizontal axis is the center of the distribution? If you are comparing two distributions, are they both centered in the same place? Examine the shape of the distribution. Does the distribution have one “peak,” a point that is higher than any other? If so, this is the most frequently occurring measurement or category. Is there more than one peak? Are there an approximately equal number of measurements to the left and right of the peak? Look for any unusual measurements or outliers. That is, are any measurements much bigger or smaller than all of the others? These outliers may not be representative of the other values in the set.

Distributions are often described according to their shapes.

Definition A distribution is symmetric if the left and right sides of the distribution, when divided at the middle value, form mirror images. A distribution is skewed to the right if a greater proportion of the measurements lie to the right of the peak value. Distributions that are skewed right contain a few unusually large measurements.

1.4 GRAPHS FOR QUANTITATIVE DATA

❍

A distribution is skewed to the left if a greater proportion of the measurements lie to the left of the peak value. Distributions that are skewed left contain a few unusually small measurements. A distribution is unimodal if it has one peak; a bimodal distribution has two peaks. Bimodal distributions often represent a mixture of two different populations in the data set.

EXAMPLE

1.9

FIGU R E 1 .1 2

Shapes of data distributions for Example 1.9

Examine the three dotplots generated by MINITAB and shown in Figure 1.12. Describe these distributions in terms of their locations and shapes. ●

The ﬁrst dotplot shows a relatively symmetric distribution with a single peak located at x 4. If you were to fold the page at this peak, the left and right halves would almost be mirror images. The second dotplot, however, is far from symmetric. It has a long “right tail,” meaning that there are a few unusually large observations. If you were to fold the page at the peak, a larger proportion of measurements would be on the right side than on the left. This distribution is skewed to the right. Similarly, the third dotplot with the long “left tail” is skewed to the left. Solution

Symmetric ⇔ mirror images Skewed right ⇔ long right tail Skewed left ⇔ long left tail

EXAMPLE

1.10

An administrative assistant for the athletics department at a local university is monitoring the grade point averages for eight members of the women’s volleyball team. He enters the GPAs into the database but accidentally misplaces the decimal point in the last entry. 2.8

3.0

3.3

2.4

3.4

3.0

.21

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

Use a dotplot to describe the data and uncover the assistant’s mistake. The dotplot of this small data set is shown in Figure 1.13(a). You can clearly see the outlier or unusual observation caused by the assistant’s data entry error. Once the error has been corrected, as in Figure 1.13(b), you can see the correct distribution of the data set. Since this is a very small set, it is difficult to describe the shape of the distribution, although it seems to have a peak value around 3.0 and it appears to be relatively symmetric.

Solution

FI GU R E 1 .1 3

Distributions of GPAs for Example 1.10

●

(a)

0.5

1.0

1.5

2.0

2.5

3.0

3.5

GPAs

(b)

2.2

Outliers lie out, away from the main body of data.

1.5

2.4

2.6

2.8 GPAs

3.0

3.2

3.4

When comparing graphs created for two data sets, you should compare their scales of measurement, locations, and shapes, and look for unusual measurements or outliers. Remember that outliers are not always caused by errors or incorrect data entry. Sometimes they provide very valuable information that should not be ignored. You may need additional information to decide whether an outlier is a valid measurement that is simply unusually large or small, or whether there has been some sort of mistake in the data collection. If the scales differ widely, be careful about making comparisons or drawing conclusions that might be inaccurate!

RELATIVE FREQUENCY HISTOGRAMS A relative frequency histogram resembles a bar chart, but it is used to graph quantitative rather than qualitative data. The data in Table 1.9 are the birth weights of 30 fullterm newborn babies, reproduced from Example 1.8 and shown as a dotplot in Figure 1.14(a). First, divide the interval from the smallest to the largest measurements into subintervals or classes of equal length. If you stack up the dots in each subinterval (Figure 1.14(b)), and draw a bar over each stack, you will have created a frequency histogram or a relative frequency histogram, depending on the scale of the vertical axis.

1.5 RELATIVE FREQUENCY HISTOGRAMS

TABLE 1.9

●

How to construct a histogram

Birth Weights of 30 Full-Term Newborn Babies 7.2 8.0 8.2 5.8 6.1 8.5

FIGU R E 1 .1 4

❍

7.8 8.2 7.7 6.8 7.9 9.0

6.8 5.6 7.5 6.8 9.4 7.7

6.2 8.6 7.2 8.5 9.0 6.7

8.2 7.1 7.7 7.5 7.8 7.7

● (a) 6.0

6.6

7.2

7.8 Birth Weights

8.4

9.0

(b) 6.0

6.5

7.0

7.5 8.0 Birth Weights

8.5

9.0

9.5

Definition A relative frequency histogram for a quantitative data set is a bar graph in which the height of the bar shows “how often” (measured as a proportion or relative frequency) measurements fall in a particular class or subinterval. The classes or subintervals are plotted along the horizontal axis.

As a rule of thumb, the number of classes should range from 5 to 12; the more data available, the more classes you need.† The classes must be chosen so that each measurement falls into one and only one class. For the birth weights in Table 1.9, we decided to use eight intervals of equal length. Since the total span of the birth weights is 9.4 5.6 3.8 the minimum class width necessary to cover the range of the data is (3.8 8) .475. For convenience, we round this approximate width up to .5. Beginning the ﬁrst interval at the lowest value, 5.6, we form subintervals from 5.6 up to but not including 6.1, 6.1 up to but not including 6.6, and so on. By using the method of left inclusion, and including the left class boundary point but not the right boundary point in the class, we eliminate any confusion about where to place a measurement that happens to fall on a class boundary point. Table 1.10 shows the eight classes, labeled from 1 to 8 for identiﬁcation. The boundaries for the eight classes, along with a tally of the number of measurements that fall in each class, are also listed in the table. As with the charts in Section 1.3, you can now measure how often each class occurs using frequency or relative frequency.

†

You can use this table as a guide for selecting an appropriate number of classes. Remember that this is only a guide; you may use more or fewer classes than the table recommends if it makes the graph more descriptive.

Sample Size Number of Classes

100

200

500

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

To construct the relative frequency histogram, plot the class boundaries along the horizontal axis. Draw a bar over each class interval, with height equal to the relative frequency for that class. The relative frequency histogram for the birth weight data, Figure 1.15, shows at a glance how birth weights are distributed over the interval 5.6 to 9.4. TABLE 1.10

●

Relative Frequencies for the Data of Table 1.9

Class

Class Boundaries

1 2 3 4 5 6 7 8

5.6 to 6.1 6.1 to 6.6 6.6 to 7.1 7.1 to 7.6 7.6 to 8.1 8.1 to 8.6 8.6 to 9.1 9.1 to 9.6

Relative frequencies add to 1; frequencies add to n.

FI GU R E 1 .1 5

Tally

Class Frequency

Class Relative Frequency

II II IIII IIII IIII III IIII III I

2 2 4 5 8 5 3 1

2/30 2/30 4/30 5/30 8/30 5/30 3/30 1/30

●

Relative frequency histogram

8/30

Relative Frequency

7/30 6/30 5/30 4/30 3/30 2/30 1/30 0 5.6

EXAMPLE

TABLE 1.11

6.1

6.6

7.1

7.6 8.1 Birth Weights

8.6

9.1

9.6

Twenty-ﬁve Starbucks® customers are polled in a marketing survey and asked, “How often do you visit Starbucks in a typical week?” Table 1.11 lists the responses for these 25 customers. Construct a relative frequency histogram to describe the data.

1.11

●

Number of Visits in a Typical Week for 25 Customers 6 4 6 5 3

7 6 5 5 5

1 4 6 5 7

5 6 3 7 5

6 8 4 6 5

The variable being measured is “number of visits to Starbucks,” which is a discrete variable that takes on only integer values. In this case, it is simplest to choose the classes or subintervals as the integer values over the range of observed values: 1, 2, 3, 4, 5, 6, and 7. Table 1.12 shows the classes and their corresponding frequencies and relative frequencies. The relative frequency histogram, generated using MINITAB, is shown in Figure 1.16.

Solution

1.5 RELATIVE FREQUENCY HISTOGRAMS

FIGU R E 1 .1 6

MINITAB histogram for Example 1.11

●

Frequency Table for Example 1.11 Number of Visits to Starbucks

Frequency

Relative Frequency

1 2 3 4 5 6 7 8

1 — 2 3 8 7 3 1

.04 — .08 .12 .32 .28 .12 .04

● 8/25

Relative frequency

TABLE 1.12

❍

6/25

4/25

2/25

0 1

Visits

Notice that the distribution is skewed to the left and that there is a gap between 1 and 3.

How Do I Construct a Relative Frequency Histogram? 1. Choose the number of classes, usually between 5 and 12. The more data you have, the more classes you should use. 2. Calculate the approximate class width by dividing the difference between the largest and smallest values by the number of classes. 3. Round the approximate class width up to a convenient number. 4. If the data are discrete, you might assign one class for each integer value taken on by the data. For a large number of integer values, you may need to group them into classes. 5. Locate the class boundaries. The lowest class must include the smallest measurement. Then add the remaining classes using the left inclusion method. 6. Construct a statistical table containing the classes, their frequencies, and their relative frequencies. 7. Construct the histogram like a bar graph, plotting class intervals on the horizontal axis and relative frequencies as the heights of the bars. (continued)

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

Exercise Reps A. For the following data sets, ﬁnd the range, the minimum class width, and a convenient class width. The ﬁrst data set is done for you. Number of Measurements

Smallest and Largest Values

Number of Classes

Range

Minimum Class Width

Convenient Class Width

10 to 100

12.86

0.1 to 6.0

100

500 to 700

B. For the same data sets, select a convenient starting point, and list the class boundaries for the ﬁrst two classes. The ﬁrst data set is done for you. Number of Measurements

Smallest and Largest Values

Convenient Starting Point

10 to 100

First Two Classes 0 to 15 15 to 30

0.1 to 6.0

100

500 to 700

Progress Report •

Still having trouble? Try again using the Exercise Reps at the end of this section.

•

Mastered relative frequency histograms? You can skip the Exercise Reps and go straight to the Basic Techniques Exercises at the end of this section.

Answers are located on the perforated card at the back of this book.

A relative frequency histogram can be used to describe the distribution of a set of data in terms of its location and shape, and to check for outliers as you did with other graphs. For example, the birth weight data were relatively symmetric, with no unusual measurements, while the Starbucks data were skewed left. Since the bar constructed above each class represents the relative frequency or proportion of the measurements in that class, these heights can be used to give us further information: • •

The proportion of the measurements that fall in a particular class or group of classes The probability that a measurement drawn at random from the set will fall in a particular class or group of classes

Consider the relative frequency histogram for the birth weight data in Figure 1.15. What proportion of the newborns have birth weights of 7.6 or higher? This involves all

1.5 RELATIVE FREQUENCY HISTOGRAMS

❍

classes beyond 7.6 in Table 1.10. Because there are 17 newborns in those classes, the proportion who have birth weights of 7.6 or higher is 17/30, or approximately 57%. This is also the percentage of the total area under the histogram in Figure 1.15 that lies to the right of 7.6. Suppose you wrote each of the 30 birth weights on a piece of paper, put them in a hat, and drew one at random. What is the chance that this piece of paper contains a birth weight of 7.6 or higher? Since 17 of the 30 pieces of paper fall in this category, you have 17 chances out of 30; that is, the probability is 17/30. The word probability is not unfamiliar to you; we will discuss it in more detail in Chapter 4. Although we are interested in describing a set of n 30 measurements, we might also be interested in the population from which the sample was drawn, which is the set of birth weights of all babies born at this hospital. Or, if we are interested in the weights of newborns in general, we might consider our sample as representative of the population of birth weights for newborns at similar metropolitan hospitals. A sample histogram provides valuable information about the population histogram—the graph that describes the distribution of the entire population. Remember, though, that different samples from the same population will produce different histograms, even if you use the same class boundaries. However, you can expect that the sample and population histograms will be similar. As you add more and more data to the sample, the two histograms become more and more alike. If you enlarge the sample to include the entire population, the two histograms are identical!

1.5

EXERCISES EXERCISE REPS These exercises refer back to the MyPersonal Trainer section on page 27. 1.16 For the following data sets, ﬁnd the range, the minimum class width, and a conve-

nient class width. Number of Measurements

Smallest and Largest Values

Number of Classes

0.5 to 1.0

0 to 100

200

1200 to 1500

Range

Minimum Class Width

Convenient Class Width

1.17 Refer to Exercise 1.16. For the same data sets, select a convenient starting point,

and list the class boundaries for the ﬁrst two classes. Number of Measurements

Smallest and Largest Values

0.5 to 1.0

0 to 100

200

1200 to 1500

Convenient Starting Point

First Two Classes

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

BASIC TECHNIQUES 1.18 Construct a stem and leaf plot for these

50 measurements:

EX0118

3.1 2.9 3.8 2.5 4.3

4.9 2.1 6.2 3.6 5.7

2.8 3.5 2.5 5.1 3.7

3.6 4.0 2.9 4.8 4.6

2.5 3.7 2.8 1.6 4.0

4.5 2.7 5.1 3.6 5.6

3.5 4.0 1.8 6.1 4.9

3.7 4.4 5.6 4.7 4.2

4.1 3.7 2.2 3.9 3.1

4.9 4.2 3.4 3.9 3.9

a. Describe the shape of the data distribution. Do you see any outliers? b. Use the stem and leaf plot to ﬁnd the smallest observation. c. Find the eighth and ninth largest observations. 1.19 Refer to Exercise 1.18. Construct a relative fre-

quency histogram for the data. a. Approximately how many class intervals should you use? b. Suppose you decide to use classes starting at 1.6 with a class width of .5 (i.e., 1.6 to 2.1, 2.1 to 2.6). Construct the relative frequency histogram for the data. c. What fraction of the measurements are less than 5.1? d. What fraction of the measurements are larger than 3.6? e. Compare the relative frequency histogram with the stem and leaf plot in Exercise 1.18. Are the shapes similar? 1.20 Consider this set of data: EX0120

4.5 4.3 3.9 4.4

3.2 4.8 3.7 4.0

3.5 3.6 4.3 3.6

3.9 3.3 4.4 3.5

3.5 4.3 3.4 3.9

3.9 4.2 4.2 4.0

a. Construct a stem and leaf plot by using the leading digit as the stem. b. Construct a stem and leaf plot by using each leading digit twice. Does this technique improve the presentation of the data? Explain. 1.21 A discrete variable can take on only the values

0, 1, or 2. A set of 20 measurements on this variable is shown here: 1 2 2 0

2 1 2 1

1 1 1 2

0 0 1 1

2 0 0 1

a. Construct a relative frequency histogram for the data.

b. What proportion of the measurements are greater than 1? c. What proportion of the measurements are less than 2? d. If a measurement is selected at random from the 20 measurements shown, what is the probability that it is a 2? e. Describe the shape of the distribution. Do you see any outliers? 1.22 Refer to Exercise 1.21.

a. Draw a dotplot to describe the data. b. How could you deﬁne the stem and the leaf for this data set? c. Draw the stem and leaf plot using your decision from part b. d. Compare the dotplot, the stem and leaf plot, and the relative frequency histogram (Exercise 1.21). Do they all convey roughly the same information? 1.23 Navigating a Maze An experimental psychologist measured the length of time it took for a rat to successfully navigate a maze on each of ﬁve days. The results are shown in the table. Create a line chart to describe the data. Do you think that any learning is taking place? Day

Time (sec.)

1.24 Measuring over Time The value of a

quantitative variable is measured once a year for a 10-year period. Here are the data:

EX0124

Year

Measurement

Year

Measurement

1 2 3 4 5

61.5 62.3 60.7 59.8 58.0

6 7 8 9 10

58.2 57.5 57.5 56.1 56.0

a. Create a line chart to describe the variable as it changes over time. b. Describe the measurements using the chart constructed in part a. 1.25 Test Scores The test scores on a 100-point test were recorded for 20 students:

EX0125

61 94

93 89

91 67

86 62

55 72

63 87

86 68

82 65

76 75

57 84

a. Use an appropriate graph to describe the data. b. Describe the shape and location of the scores.

1.5 RELATIVE FREQUENCY HISTOGRAMS

c. Is the shape of the distribution unusual? Can you think of any reason the distribution of the scores would have such a shape?

APPLICATIONS 1.26 A Recurring Illness The length of time (in months) between the onset of a particular illness and its recurrence was recorded for n 50 patients:

EX0126

2.1 14.7 4.1 14.1 1.6

4.4 9.6 18.4 1.0 3.5

2.7 16.7 .2 2.4 11.4

32.3 7.4 6.1 2.4 18.0

9.9 8.2 13.5 18.0 26.7

9.0 19.2 7.4 8.7 3.7

2.0 6.9 .2 24.0 12.6

6.6 4.3 8.3 1.4 23.1

3.9 3.3 .3 8.2 5.6

1.6 1.2 1.3 5.8 .4

a. Construct a relative frequency histogram for the data. b. Would you describe the shape as roughly symmetric, skewed right, or skewed left? c. Give the fraction of recurrence times less than or equal to 10 months.

❍

a. Construct a stem and leaf display for the data. b. Construct a relative frequency histogram for these data. Start the lower boundary of the first class at 30 and use a class width of 5 months. c. Compare the graphs in parts a and b. Are there any significant differences that would cause you to choose one as the better method for displaying the data? d. What proportion of the children were 35 months (2 years, 11 months) or older, but less than 45 months (3 years, 9 months) of age when first enrolled in preschool? e. If one child were selected at random from this group of children, what is the probability that the child was less than 50 months old (4 years, 2 months) when first enrolled in preschool?

1.27 Education Pays Off! Education pays off,

according to a snapshot provided in a report to the city of Riverside by the Riverside County Office of Education.7 The average annual incomes for six different levels of education are shown in the table: Educational Level

Text not available due to copyright restrictions

Average Annual Income

High school graduate Some college, no degree Bachelor’s degree Master’s degree Doctorate Professional (Doctor, Lawyer)

$26,795 29,095 50,623 63,592 85,675 101,375

Source: U.S. Census Bureau

a. What graphical methods could you use to describe the data? b. Select the method from part a that you think best describes the data. c. How would you summarize the information that you see in the graph regarding educational levels and salary? 1.28 Preschool The ages (in months) at

which 50 children were ﬁrst enrolled in a preschool are listed below.

EX0128

38 47 32 55 42

40 35 34 39 50

30 34 41 33 37

35 43 30 32 39

39 41 46 32 33

40 36 35 45 45

48 41 40 42 38

36 43 30 41 46

31 48 46 36 36

36 40 37 50 31

1.30 How Long Is the Line? To decide on the number of service counters needed for stores to be built in the future, a supermarket chain wanted to obtain information on the length of time (in minutes) required to service customers. To ﬁnd the distribution of customer service times, a sample of 1000 customers’ service times was recorded. Sixty of these are shown here:

EX0130

3.6 1.1 1.4 .6 1.1 1.6

1.9 1.8 .2 2.8 1.2 1.9

2.1 .3 1.3 2.5 .8 5.2

.3 1.1 3.1 1.1 1.0 .5

.8 .5 .4 .4 .9 1.8

.2 1.2 2.3 1.2 .7 .3

1.0 .6 1.8 .4 3.1 1.1

1.4 1.1 4.5 1.3 1.7 .6

1.8 .8 .9 .8 1.1 .7

1.6 1.7 .7 1.3 2.2 .6

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

a. Construct a stem and leaf plot for the data. b. What fraction of the service times are less than or equal to 1 minute? c. What is the smallest of the 60 measurements? 1.31 Service Times, continued Refer to Exercise 1.30. Construct a relative frequency histogram for the supermarket service times.

a. Describe the shape of the distribution. Do you see any outliers? b. Assuming that the outliers in this data set are valid observations, how would you explain them to the management of the supermarket chain? c. Compare the relative frequency histogram with the stem and leaf plot in Exercise 1.30. Do the two graphs convey the same information? 1.32 Calcium Content The calcium (Ca)

content of a powdered mineral substance was analyzed ten times with the following percent compositions recorded:

EX0132

.0271 .0271

.0282 .0281

.0279 .0269

.0281 .0275

.0268 .0276

a. Draw a dotplot to describe the data. (HINT: The scale of the horizontal axis should range from .0260 to .0290.) b. Draw a stem and leaf plot for the data. Use the numbers in the hundredths and thousandths places as the stem. c. Are any of the measurements inconsistent with the other measurements, indicating that the technician may have made an error in the analysis? 1.33 American Presidents Listed below are the ages at the time of death for the 38 deceased American presidents from George Washington to Ronald Reagan:5

EX0133

Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren W. H. Harrison Tyler Polk Taylor Fillmore Pierce Buchanan Lincoln A. Johnson Grant Hayes

67 90 83 85 73 80 78 79 68 71 53 65 74 64 77 56 66 63 70

Garﬁeld Arthur Cleveland B. Harrison Cleveland McKinley T. Roosevelt Taft Wilson Harding Coolidge Hoover F. D. Roosevelt Truman Eisenhower Kennedy L. Johnson Nixon Reagan

49 56 71 67 71 58 60 72 67 57 60 90 63 88 78 46 64 81 93

a. Before you graph the data, try to visualize the distribution of the ages at death for the presidents. What shape do you think it will have? b. Construct a stem and leaf plot for the data. Describe the shape. Does it surprise you? c. The five youngest presidents at the time of death appear in the lower “tail” of the distribution. Three of the five youngest have one common trait. Identify the five youngest presidents at death. What common trait explains these measurements? 1.34 RBC Counts The red blood cell count

of a healthy person was measured on each of 15 days. The number recorded is measured in 106 cells per microliter ( L).

EX0134

5.4 5.3 5.3

5.2 5.4 4.9

5.0 5.2 5.4

5.2 5.1 5.2

5.5 5.3 5.2

a. Use an appropriate graph to describe the data. b. Describe the shape and location of the red blood cell counts. c. If the person’s red blood cell count is measured today as 5.7 106/ L, would you consider this unusual? What conclusions might you draw? 1.35 Batting Champions The officials of

major league baseball have crowned a batting champion in the National League each year since 1876. A sample of winning batting averages is listed in the table:5

EX0135

Year

Name

2005 2000 1915 1917 1934 1911 1898 1924 1963 1992 1954 1975 1958 1942 1948 1971 1996 1961 1968 1885

Derreck Lee Todd Helton Larry Doyle Edd Roush Paul Waner Honus Wagner Willie Keeler Roger Hornsby Tommy Davis Gary Sheffield Willie Mays Bill Madlock Richie Ashburn Ernie Lombardi Stan Musial Joe Torre Tony Gwynn Roberto Clemente Pete Rose Roger Connor

Average .335 .372 .320 .341 .362 .334 .379 .424 .326 .330 .345 .354 .350 .330 .376 .363 .353 .351 .335 .371

1.5 RELATIVE FREQUENCY HISTOGRAMS

a. Construct a relative frequency histogram to describe the batting averages for these 20 champions. b. If you were to randomly choose one of the 20 names, what is the chance that you would choose a player whose average was above .400 for his championship year? 1.36 Top 20 Movies The table that follows shows the weekend gross ticket sales for the top 20 movies during the week of August 4, 2006:9

EX0136

Weekend Gross ($ millions)

Movie 1.Talladega Nights: The Ballad of Ricky Bobby 2. Barnyard 3. Pirates of the Caribbean: Dead Man’s Chest 4. Miami Vice 5. The Descent 6. John Tucker Must Die 7. Monster House 8. The Ant Bully 9. You, Me and Dupree 10. The Night Listener 11. The Devil Wears Prada 12. Lady in the Water 13. Little Man 14. Superman Returns 15. Scoop 16. Little Miss Sunshine 17. Clerks II 18. My Super Ex-Girlfriend 19. Cars 20. Click

$47.0 15.8 11.0 10.2 8.9 6.2 6.1 3.9 3.6 3.6 3.0 2.7 2.5 2.2 1.8 1.5 1.3 1.2 1.1 0.8

Source: www.radiofree.com/mov-tops.shtml

a. Draw a stem and leaf plot for the data. Describe the shape of the distribution. Are there any outliers? b. Construct a dotplot for the data. Which of the two graphs is more informative? Explain. 1.37 Hazardous Waste How safe is your EX0137

neighborhood? Are there any hazardous waste

❍

sites nearby? The table shows the number of hazardous waste sites in each of the 50 states and the District of Columbia in the year 2006:5 AL AK AZ AR CA CO CT DE DC FL GA

15 6 9 10 95 19 16 15 1 50 17

HI ID IL IN IA KS KY LA ME MD

3 9 48 30 12 11 14 14 12 18

MA MI MN MS MO MT NE NV NH NJ

33 68 24 5 26 15 14 1 21 117

NM NY NC ND OH OK OR PA RI SC

13 87 31 0 37 11 11 96 12 26

SD TN TX UT VT VA WA WV WI WY

2 14 44 18 11 28 47 9 38 2

a. What variable is being measured? Is the variable discrete or continuous? b. A stem and leaf plot generated by MINITAB is shown here. Describe the shape of the data distribution. Identify the unusually large measurements marked “HI” by state. Stem-and-Leaf Display: Hazardous Waste Stem-and-leaf of Sites N = 51 Leaf Unit = 1.0 6 11 24 (8) 19 17 14 11 9 8 6

0 0 1 1 2 2 3 3 4 4 5

011223 56999 0111122234444 55567889 14 668 013 78 4 78 0

HI 68, 87, 95, 96, 117

c. Can you think of any reason these ﬁve states would have a large number of hazardous waste sites? What other variable might you measure to help explain why the data behave as they do?

As you continue to work through the exercises in this chapter, you will become more experienced in recognizing different types of data and in determining the most appropriate graphical method to use. Remember that the type of graphic you use is not as important as the interpretation that accompanies the picture. Look for these important characteristics: • • •

Location of the center of the data Shape of the distribution of data Unusual observations in the data set

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

Using these characteristics as a guide, you can interpret and compare sets of data using graphical methods, which are only the ﬁrst of many statistical tools that you will soon have at your disposal.

CHAPTER REVIEW Key Concepts I.

2. Quantitative data

How Data Are Generated

1. Experimental units, variables, measurements 2. Samples and populations 3. Univariate, bivariate, and multivariate data II. Types of Variables

1. Qualitative or categorical 2. Quantitative a. Discrete b. Continuous III. Graphs for Univariate Data Distributions

a. Pie and bar charts b. Line charts c. Dotplots d. Stem and leaf plots e. Relative frequency histograms 3. Describing data distributions a. Shapes—symmetric, skewed left, skewed right, unimodal, bimodal b. Proportion of measurements in certain intervals c. Outliers

1. Qualitative or categorical data a. Pie charts b. Bar charts

Easy access to the web has made it possible for you to understand statistical concepts using an interactive web tool called an applet. These applets provide visual reinforcement for the concepts that have been presented in the chapter. Sometimes you will be able to perform statistical experiments, sometimes you will be able to interact with a statistical graph to change its form, and sometimes you will be able to use the applet as an interactive “statistical table.” At the end of each chapter, you will find exercises designed specifically for use with a particular applet. The applets have been customized specifically to match the presentation and notation used in your text. They can be found on the Premium Website. If necessary, follow the instructions to download the latest web browser and/or Java plug-in, or just click the appropriate link to load the applets. Your web browser will open the index of applets, organized by chapter and name. When you click a particular applet title, the applet will appear in your browser. To return to the index of applets, simply click the link at the bottom of the page.

CHAPTER REVIEW

❍

Dotplots Click the Chapter 1 applet called Building a Dotplot. If you move your cursor over the applet marked Dotplot Demo you will see a green line with a value that changes as you move along the horizontal axis. When you left-click your mouse, a dot will appear at that point on the dotplot. If two measurements are identical, the dots will pile up on top of each other (Figure 1.17). Follow the directions in the Dotplot Demo, using the sample data given there. If you make a mistake, the applet will tell you. The second applet will not correct your mistakes; you can add as many dots as you want!

FIGU R E 1 .1 7

Building a Dotplot applet

●

Histograms Click the Chapter 1 applet called Building a Histogram. If you scroll down to the applet marked Histogram Demo, you will see the interval boundaries (or interval midpoints) for the histogram along the horizontal axis. As you move the mouse across the graph, a light gray box will show you where the measurement will be added at your next mouse click. When you release the mouse, the box turns dark blue (dark blue in Figure 1.18). The partially completed histogram in Figure 1.18 contains one 3, one 4, one 5, three 6s, and one 7. Follow the directions in the Histogram Demo using the sample data given there. Click the link to compare your results to the correct histogram. The second applet will be used for some of the MyApplet Exercises. Click the applet called Flipping Fair Coins, and scroll down to the applet marked sample size 3. The computer will collect some data by “virtually” tossing 3 coins and recording the quantitative discrete variable x number of heads observed Click on “New Coin Flip.” You will see the result of your three tosses in the upperleft-hand corner, along with the value of x. For the experiment in Figure 1.19 we observed x 2. The applet begins to build a relative frequency histogram to describe the data set, which at this point contains only one observation. Click “New Coin Flip” a few more times. Watch the coins appear, along with the value of x, and watch the

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

FI GU R E 1 .1 8

Building a Histogram applet

FI GU R E 1 .1 9

Flipping Fair Coins applet

FI GU R E 1 .2 0

Flipping Fair Coins applet

●

MY MINITAB

❍

relative frequency histogram grow. The red area (light blue in Figures 1.19 and 1.20) represents the current data added to the histogram, and the dark blue area in Figure 1.20 is contributed from the previous coin ﬂips. You can ﬂip the three coins 10 at a time or 100 at a time to generate data more quickly. Figure 1.20 shows the relative frequency histogram for 500 observations in our data set. Your data set will look a little different. However, it should have the same approximate shape—it should be relatively symmetric. For our histogram, we can say that the values x 0 and x 3 occurred about 12–13% of the time, while the values x 1 and x 2 occurred between 38% and 40% of the time. Does your histogram produce similar results?

Introduction to MINITABTM MINITAB is a computer software package that is available in many forms for different computer environments. The current version of MINITAB at the time of this printing is MINITAB 15, which is used in the Windows environment. We will assume that you are familiar with Windows. If not, perhaps a lab or teaching assistant can help you to master the basics. Once you have started Windows, there are two ways to start MINITAB: • •

If there is a MINITAB shortcut icon on the desktop, double-click on the icon. Click the Start button on the taskbar. Follow the menus, highlighting All Programs 씮 MINITAB Solutions 씮 MINITAB 15 Statistical Software English. Click on MINITAB 15 Statistical Software English to start the program.

When MINITAB is opened, the main MINITAB screen will be displayed (see Figure 1.21). It contains two windows: the Data window and the Session window. Clicking FIGU R E 1 .2 1

●

38 ❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

anywhere on the window will make that window active so that you can either enter data or type commands. Although it is possible to manually type MINITAB commands in the Session window, we choose to use the Windows approach, which will be familiar to most of you. If you prefer to use the typed commands, consult the MINITAB manual for detailed instructions. At the top of the Session window, you will see a Menu bar. Highlighting and clicking on any command on the Menu bar will cause a menu to drop down, from which you may then select the necessary command. We will use the standard notation to indicate a sequence of commands from the drop-down menus. For example, File 씮 Open Worksheet will allow you to retrieve a “worksheet”—a set of data from the Data window—which you have previously saved. To close the program, the command sequence is File 씮 Exit. MINITAB 15 allows multiple worksheets to be saved as “projects.” When you are working on a project, you can add new worksheets or open worksheets from other projects to add to your current project. As you become more familiar with MINITAB, you will be able to organize your information into either “worksheets” or “projects,” depending on the complexity of your task.

Graphing with MINITAB The ﬁrst data set to be graphed consists of qualitative data whose frequencies have already been recorded. The class status of 105 students in an introductory statistics class are listed in Table 1.13. Before you enter the data into the Minitab Data window, start a project called “Chapter 1” by highlighting File 씮 New. A Dialog box called “New” will appear. Highlight Minitab Project and click OK. Before you continue, let’s save this project as “Chapter 1” using the series of commands File 씮 Save Project. Type Chapter 1 in the File Name box, and select a location using the white box marked “Save in:” at the top of the Dialog box. Click Save. In the Data window at the top of the screen, you will see your new project name, “Chapter 1.MPJ.”

TABLE 1.13

●

Status of Students in Statistics Class Status Frequency

Freshman

Sophomore

Junior

Senior

Grad Student

To enter the data into the worksheet, click on the gray cell just below the name C1 in the Data window. You can enter your own descriptive name for the categories— possibly “Status.” Now use the down arrow앗 or your mouse to continue down column C1, entering the ﬁve status descriptions. Notice that the name C1 has changed to C1-T because you are entering text rather than numbers. Continue by naming column 2 (C2) “Frequency,” and enter the ﬁve numerical frequencies into C2. The Data window will appear as in Figure 1.22. To construct a pie chart for these data, click on Graph 씮 Pie Chart, and a Dialog box will appear (see Figure 1.23). In this box, you must specify how you want to create the chart. Click the radio button marked Chart values from a table. Then place your cursor in the box marked “Categorical variable.” Either (1) highlight C1 in the list at the left and choose Select, (2) double-click on C1 in the list at the left, or (3) type C1 in the “Categorical variable” box. Similarly, place the cursor in the box marked

MY MINITAB

FIGU R E 1 .2 2

●

FIGU R E 1 .2 3

●

❍

“Summary variables” and select C2. Click Labels and select the tab marked Slice Labels. Check the boxes marked “Category names” and “Percent.” When you click OK, MINITAB will create the pie chart in Figure 1.24. We have removed the legend by selecting and deleting it. As you become more proﬁcient at using the pie chart command, you may want to take advantage of some of the options available. Once the chart is created, right-click on the pie chart and select Edit Pie. You can change the colors and format of the chart, “explode” important sectors of the pie, and change the order of the categories. If you right-click on the pie chart and select Update Graph Automatically, the pie chart will automatically update when you change the data in columns C1 and C2 of the MINITAB worksheet. If you would rather construct a bar chart, use the command Graph 씮 Bar Chart. In the Dialog box that appears, choose Simple. Choose an option in the “Bars represent” drop-down list, depending on the way that the data has been entered into the

40 ❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

FI GU R E 1 .2 4

●

worksheet. For the data in Table 1.13, we choose “Values from a table” and click OK. When the Dialog box appears, place your cursor in the “Graph variables” box and select C2. Place your cursor in the “Categorical variable” box, and select C1. Click OK to ﬁnish the bar chart, shown in Figure 1.25. Once the chart is created, right-click on various parts of the bar chart and choose Edit to change the look of the chart. MINITAB can create dotplots, stem and leaf plots, and histograms for quantitative data. The top 40 stocks on the over-the-counter (OTC) market, ranked by percentage of outstanding shares traded on a particular day, are listed in Table 1.14. Although we could simply enter these data into the third column (C3) of Worksheet 1 in the “Chapter 1” project, let’s start a new worksheet within “Chapter 1” using File 씮 New, highlighting Minitab Worksheet, and clicking OK. Worksheet 2 will appear on the screen. Enter the data into column C1 and name them “Stocks” in the gray cell just below the C1.

TABLE 1.14

●

Percentage of OTC Stocks Traded 11.88 7.99 7.15 7.13

6.27 6.07 5.98 5.91

5.49 5.26 5.07 4.94

4.81 4.79 4.55 4.43

4.40 4.05 3.94 3.93

3.78 3.69 3.62 3.48

3.44 3.36 3.26 3.20

3.11 3.03 2.99 2.89

2.88 2.74 2.74 2.69

2.68 2.63 2.62 2.61

To create a dotplot, use Graph 씮 Dotplot. In the Dialog box that appears, choose One Y 씮 Simple and click OK. To create a stem and leaf plot, use Graph 씮 Stemand-Leaf. For either graph, place your cursor in the “Graph variables” box, and select “Stocks” from the list to the left (see Figure 1.26).

MY MINITAB

F IGU R E 1 .2 5

●

F IGU R E 1 .2 6

●

❍

You can choose from a variety of formatting options before clicking OK. The dotplot appears as a graph, while the stem and leaf plot appears in the Session window. To print either a Graph window or the Session window, click on the window to make it active and use File 씮 Print Graph (or Print Session Window). To create a histogram, use Graph 씮 Histogram. In the Dialog box that appears, choose Simple and click OK, selecting “Stocks” for the “Graph variables” box.

42 ❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

Select Scale 씮 Y-Scale Type and click the radio button marked “Frequency.” (You can edit the histogram later to show relative frequencies.) Click OK twice. Once the histogram has been created, right-click on the Y-axis and choose Edit Y Scale. Under the tab marked “Scale,” you can click the radio button marked “Position of ticks” and type in 0 5 10 15. Then click the tab marked “Labels,” the radio button marked “Speciﬁed” and type 0 5/40 10/40 15/40. Click OK. This will reduce the number of ticks on the y-axis and change them to relative frequencies. Finally, double-click on the word “Frequency” along the y-axis. Change the box marked “Text” to read “Relative frequency” and click OK. To adjust the type of boundaries for the histogram, right-click on the bars of the histogram and choose Edit Bars. Use the tab marked “Binning” to choose either “Cutpoints” or “Midpoints” for the histogram; you can specify the cutpoint or midpoint positions if you want. In this same Edit box, you can change the colors, ﬁll type, and font style of the histogram. If you right-click on the bars and select Update Graph Automatically, the histogram will automatically update when you change the data in the “Stocks” column. As you become more familiar with MINITAB for Windows, you can explore the various options available for each type of graph. It is possible to plot more than one variable at a time, to change the axes, to choose the colors, and to modify graphs in many ways. However, even with the basic default commands, it is clear that the distribution of OTC stocks is highly skewed to the right. Make sure to save your work using the File 씮 Save Project command before you exit MINITAB!

FI GU R E 1 .2 7

●

SUPPLEMENTARY EXERCISES

❍

Supplementary Exercises 1.38 Quantitative or Qualitative? Identify each variable as quantitative or qualitative: a. Ethnic origin of a candidate for public office

a. Number of people in line at a supermarket checkout counter b. Depth of a snowfall

b. Score (0–100) on a placement examination

c. Length of time for a driver to respond when faced with an impending collision

c. Fast-food establishment preferred by a student (McDonald’s, Burger King, or Carl’s Jr.) d. Mercury concentration in a sample of tuna 1.39 Symmetric or Skewed? Do you expect the distributions of the following variables to be symmetric or skewed? Explain. a. Size in dollars of nonsecured loans b. Size in dollars of secured loans c. Price of an 8-ounce can of peas d. Height in inches of freshman women at your university e. Number of broken taco shells in a package of 100 shells f. Number of ticks found on each of 50 trapped cottontail rabbits 1.40 Continuous or Discrete? Identify each variable as continuous or discrete: a. Number of homicides in Detroit during a one-month period b. Length of time between arrivals at an outpatient clinic c. Number of typing errors on a page of manuscript d. Number of defective lightbulbs in a package containing four bulbs e. Time required to ﬁnish an examination

d. Number of aircraft arriving at the Atlanta airport in a given hour 1.43 Aqua Running Aqua running has been suggested as a method of cardiovascular conditioning for injured athletes and others who want a low-impact aerobics program. A study reported in the Journal of Sports Medicine investigated the relationship between exercise cadence and heart rate by measuring the heart rates of 20 healthy volunteers at a cadence of 48 cycles per minute (a cycle consisted of two steps).10 The data are listed here:

EX0143

87 109 79 80 96 95 90 92 96 98 101 91 78 112 94 98 94 107 81 96

Construct a stem and leaf plot to describe the data. Discuss the characteristics of the data distribution. 1.44 Major World Lakes A lake is a body

of water surrounded by land. Hence, some bodies of water named “seas,” like the Caspian Sea, are actual salt lakes. In the table that follows, the length in miles is listed for the major natural lakes of the world, excluding the Caspian Sea, which has an area of 143,244 square miles, a length of 760 miles, and a maximum depth of 3,363 feet.5

EX0144

Name

Length (mi)

1.42 Continuous or Discrete, again Identify each

Superior Victoria Huron Michigan Aral Sea Tanganyika Baykal Great Bear Nyasa Great Slave Erie Winnipeg Ontario Balkhash Ladoga Maracaibo Onega

variable as continuous or discrete:

Source: The World Almanac and Book of Facts 2007

1.41 Continuous or Discrete, again Identify each

variable as continuous or discrete: a. Weight of two dozen shrimp b. A person’s body temperature c. Number of people waiting for treatment at a hospital emergency room d. Number of properties for sale by a real estate agency e. Number of claims received by an insurance company during one day

350 250 206 307 260 420 395 192 360 298 241 266 193 376 124 133 145

Name Eyre Titicaca Nicaragua Athabasca Reindeer Turkana Issyk Kul Torrens Vänern Nettilling Winnipegosis Albert Nipigon Gairdner Urmia Manitoba Chad

Length (mi) 90 122 102 208 143 154 115 130 91 67 141 100 72 90 90 140 175

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

a. Use a stem and leaf plot to describe the lengths of the world’s major lakes. b. Use a histogram to display these same data. How does this compare to the stem and leaf plot in part a? c. Are these data symmetric or skewed? If skewed, what is the direction of the skewing? 1.45 Ages of Pennies We collected 50 pennies and recorded their ages, by calculaEX0145 ting AGE CURRENT YEAR YEAR ON PENNY. 5 1 5 0 19

1 4 21 1 36

9 4 19 19 23

1 3 9 0 0

2 0 0 2 1

20 25 5 0 17

0 3 0 20 6

25 3 2 16 0

0 8 1 22 5

17 28 0 10 0

a. Before drawing any graphs, try to visualize what the distribution of penny ages will look like. Will it be mound-shaped, symmetric, skewed right, or skewed left? b. Draw a relative frequency histogram to describe the distribution of penny ages. How would you describe the shape of the distribution? 1.46 Ages of Pennies, continued The data below represent the ages of a different set of 50 pennies, again calculated using AGE CURRENT YEAR YEAR ON PENNY.

EX0146

41 2 3 14 0

9 10 1 9 7

0 4 14 3 3

4 0 7 5 5

3 14 2 3 23

0 0 4 0 7

3 25 4 8 28

8 12 5 17 17

21 24 1 16 9

3 19 20 0 2

a. Draw a relative frequency histogram to describe the distribution of penny ages. Is the shape similar to the shape of the relative frequency histogram in Exercise 1.41? b. Draw a stem and leaf plot to describe the penny ages. Are there any unusually large or small measurements in the set? 1.47 Presidential Vetoes Here is a list of

the 43 presidents of the United States along with the number of regular vetoes used by each:5

EX0147

Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren W. H. Harrison Tyler Polk Taylor Fillmore

2 0 0 5 1 0 5 0 0 6 2 0 0

B. Harrison Cleveland McKinley T. Roosevelt Taft Wilson Harding Coolidge Hoover F. D. Roosevelt Truman Eisenhower Kennedy

19 42 6 42 30 33 5 20 21 372 180 73 12

Pierce Buchanan Lincoln A. Johnson Grant Hayes Garﬁeld Arthur Cleveland

9 4 2 21 45 12 0 4 304

L. Johnson Nixon Ford Carter Reagan G. H. W. Bush Clinton G. W. Bush

16 26 48 13 39 29 36 1

Source: The World Almanac and Book of Facts 2007

Use an appropriate graph to describe the number of vetoes cast by the 43 presidents. Write a summary paragraph describing this set of data. 1.48 Windy Cities Are some cities more

windy than others? Does Chicago deserve to be nicknamed “The Windy City”? These data are the average wind speeds (in miles per hour) for 55 selected cities in the United States:5

EX0148

8.9 7.1 9.1 8.8 10.2 8.7

12.4 11.8 9.0 10.8 8.6 5.8

12.9 10.3 10.5 8.7 10.7 10.2

8.4 7.7 11.3 7.6 9.6 6.9

7.8 9.2 7.8 5.5 8.3 9.2

11.5 10.5 8.8 35.1 8.0 10.2

8.2 9.3 12.2 10.5 9.5 6.2

9.0 8.7 7.9 10.4 7.7 9.6

8.8 8.7 8.8 11.0 9.4 12.2

9.0

Source: The World Almanac and Book of Facts 2007

a. Construct a relative frequency histogram for the data. (HINT: Choose the class boundaries without including the value x 35.1 in the range of values.) b. The value x 35.1 was recorded at Mt. Washington, New Hampshire. Does the geography of that city explain the observation? c. The average wind speed in Chicago is recorded as 10.3 miles per hour. Do you consider this unusually windy? 1.49 Kentucky Derby The following data

set shows the winning times (in seconds) for the Kentucky Derby races from 1950 to 2007.11

EX0149

(1950) (1960) (1970) (1980) (1990) (2000)

121.3 122.2 123.2 122.0 122.0 121.0

122.3 124.0 123.1 122.0 123.0 119.97

121.3 120.2 121.4 122.2 123.0 121.13

122.0 121.4 119.2† 122.1 122.2 121.19

123.0 120.0 124.0 122.2 123.3 124.06

121.4 121.1 122.0 120.1 121.1 122.75

123.2 122.0 121.3 122.4 121.0 121.36

122.1 120.3 122.1 123.2 122.4 122.17

125.0 122.1 121.1 122.2 122.2

122.1 121.4 122.2 125.0 123.2

† Record time set by Secretariat in 1973 Source: www.kentuckyderby.com

a. Do you think there will be a trend in the winning times over the years? Draw a line chart to verify your answer. b. Describe the distribution of winning times using an appropriate graph. Comment on the shape of the distribution and look for any unusual observations.

SUPPLEMENTARY EXERCISES

1.50 Computer Networks at Home As Americans become more knowledgeable about computer hardware and software, as prices drop and installation becomes easier, home networking of PCs is expected to penetrate 27 percent of U.S. households by 2008, with wireless technology leading the way.12

EX0150

U.S. Home Networks (in millions) Year Wired Wireless 2002 2003 2004 2005 2006 2007 2008

6.1 6.5 6.2 5.7 4.9 4.1 3.4

1.7 4.5 8.7 13.7 19.1 24.0 28.2

Source: Jupiter Research

a. What graphical methods could you use to describe the data? b. Before you draw a graph, look at the predicted number of wired and wireless households in the table. What trends do you expect to see in the graphs? c. Use a line chart to describe the predicted number of wired households for the years 2002 to 2008. d. Use a bar chart to describe the predicted number of wireless households for the years 2002 to 2008. 1.51 Election Results The 2004 election

EX0151

AL AK AZ AR CA CO CT DE FL GA

1176 191 1104 573 5510 1101 694 172 3965 1914

HI ID IL IN IA KS KY LA ME MD

194 409 2346 1479 572 736 1069 1102 330 1025

MA MI MN MS MO MT NE NV NH NJ

1071 2314 1347 685 1456 266 513 419 331 1670

NM NY NC ND OH OK OR PA RI SC

377 2962 1961 197 2860 960 867 2794 169 938

SD TN TX UT VT VA WA WV WI WY

233 1384 4527 664 121 1717 1305 424 1478 168

❍

1.52 Election Results, continued Refer to Exercise 1.51. Listed here is the percentage of the popular vote received by President Bush in each of the 50 states:8

EX0152

AL AK AZ AR CA CO CT DE FL GA

62 61 55 54 44 52 44 46 52 58

HI ID IL IN IA KS KY LA ME MD

45 68 44 60 50 62 60 57 45 43

MA MI MN MS MO MT NE NV NH NJ

37 48 48 59 53 59 66 51 49 46

NM NY NC ND OH OK OR PA RI SC

50 40 56 63 51 66 47 48 39 58

SD TN TX UT VT VA WA WV WI WY

60 57 61 73 39 54 46 56 49 69

a. By just looking at the table, what shape do you think the data distribution for the percentage of the popular vote by state will have? b. Draw a relative frequency histogram to describe the distribution. Describe the shape of the distribution and look for outliers. Did the graph conﬁrm your answer to part a? 1.53 Election Results, continued Refer to Exercises 1.51 and 1.52. The accompanying stem and leaf plots were generated using MINITAB for the variables named “Popular Vote” and “Percent Vote.” Stem-and-Leaf Display: Popular Vote, Percent Vote Stem-and-leaf of Popular Vote N = 50 Leaf Unit = 100

Stem-and-leaf of Percent Vote N = 50 Leaf Unit = 1.0

7 12 18 22 25 25 18 15 12 10 8 8 6 6 5

3 8 19 (9) 22 13 5 1

0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 HI

1111111 22333 444555 6667 899 0001111 333 444 67 99

3 4 4 5 5 6 6 7

799 03444 55666788899 001122344 566778899 00011223 6689 3

33 7 89 39, 45, 55

46 ❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

percentage of popular votes by state is moundshaped. 1.54 Student Heights The self-reported heights of 105 students in a biostatistics class are described in the relative frequency histogram below.

EX0154

Relative frequency

10/105

5/105

b. Superimpose another line chart on the one drawn in part a to describe the percentage that do not approve. c. The following line chart was created using MINITAB. Does it differ from the graph that you drew? Use the line chart to summarize changes in the polls just after the terrorist attacks in Spain on March 11, 2004 and in England in July of 2005. d. A plot to bring down domestic ﬂights from England to the United States was foiled by British undercover agents, and the arrest of 12 suspects followed on August 9, 2006. Summarize any changes in approval rating that may have been brought about following the August 9th arrests.

0 60

66 Heights

Approve or Disapprove of the President’s Handling of Terrorism and Homeland Security? 70

a. Describe the shape of the distribution. b. Do you see any unusual feature in this histogram? c. Can you think of an explanation for the two peaks in the histogram? Is there some other factor that is causing the heights to mound up in two separate peaks? What is it?

Response

30 ’ M ar 04 25 ,’ 04 A pr ’0 4 M ar ’0 5 A ug Se ’05 pt 8 Se , ’0 5 pt 29 ,’ 05 N ov ’0 5 M ar ’0 M 6 ay ’ A ug 06 10 ,’ 06

Fe b

EX0155

Date 8/10–11/06 5/11–12/06 3/16–17/06 11/10–11/05 9/29–30/05 9/8–9/05 8/2–4/05 3/17–18/05 4/8–9/04 3/25–26/04 2/19–20/04

Approve %

Disapprove %

Unsure %

55 44 44 45 51 46 51 57 59 57 65

40 50 50 49 44 48 41 35 35 38 28

5 6 6 6 5 6 8 8 6 5 7

a. Draw a line chart to describe the percentage that approve of Bush’s handling of terrorism and homeland security. Use time as the horizontal axis.

1.55 Fear of Terrorism Many opinion polls

have tracked opinions regarding the fear of terrorist attacks following the September 11, 2001, attacks on the World Trade Center. A Newsweek poll conducted by Princeton Survey Research Associates International presented the results of several polls over a two-year period that asked, “Do you approve or disapprove of the way Bush is handling terrorism and homeland security?” The data are shown in the table below:13

Opinion Approve Disapprove

Date

1.56 Pulse Rates A group of 50 biomedical students recorded their pulse rates by counting the number of beats for 30 seconds and multiplying by 2.

EX0156

80 52 60 84 84

70 72 82 84 72

88 90 88 60 62

70 70 54 84 90

84 96 66 88 72

66 84 66 58 84

84 96 80 72 72

82 86 88 84 110

66 62 56 68 100

42 78 104 74 58

a. Why are all of the measurements even numbers? b. Draw a stem and leaf plot to describe the data, splitting each stem into two lines. c. Construct a relative frequency histogram for the data. d. Write a short paragraph describing the distribution of the student pulse rates. 1.57 Internet On-the-Go The mobile Internet is growing, with users accessing sites such as Yahoo! Mail,

SUPPLEMENTARY EXERCISES

the Weather Channel, ESPN, Google, Hotmail, and Mapquest from their cell phones. The most popular web browsers are shown in the table below, along with the percentage market share for each.14 Market Share

Browser Openwave Motorola Nokia Access Net Front

Market Share

Browser

27% 24% 13% 9%

Teleca AU Sony Ericsson RIM Blazer

6% 5% 5% 4%

Source: www.clickz.com

a. Do the percentages add up to 100%? If not, create a category called “Other” to account for the missing percentages. b. Use a pie chart to describe the market shares for the various mobile web browsers. 1.58 How Much Can You Save? An advertisement in a recent Time magazine claimed that Geico Insurance will help you save an average of $200 per year on your automobile insurance.15

EX0158

WA $178 OR $180 ID $189

UT $191

IN $203

NE $189

IL $149

CA $144

NM $146

OK $189

11.6 11.1 13.4 12.4 13.1 12.7 12.5

Caldicot

Island Thorns

Ashley Rails

11.8 11.6

18.3 15.8 18.0 18.0 20.8

17.7 18.3 16.7 14.8 19.1

a. Construct a relative frequency histogram to describe the aluminum oxide content in the 26 pottery samples. b. What unusual feature do you see in this graph? Can you think of an explanation for this feature? c. Draw a dotplot for the data, using a letter (L, C, I, or A) to locate the data point on the horizontal scale. Does this help explain the unusual feature in part b? 1.60 The Great Calorie Debate Want to lose weight? You can do it by cutting calories, as long as you get enough nutritional value from the foods that you do eat! Below you will see a visual representation of the number of calories in some of America’s favorite foods adapted from an article in The Press-Enterprise.17

CT $268

PA $194

Number of calories

OH $208

MO $174

AZ $188

14.4 13.8 14.6 11.5 13.8 10.9 10.1

NY $237

WI $189

WY $189

NV $239

Llanederyn

❍

MD $240

VA $215 NC $127

TN $235 AL $189

GA $209

TX $183 A SAMPLING OF SAVINGS

FL $130

a. Construct a relative frequency histogram to describe the average savings for the 27 states shown on the United States map. Do you see any unusual features in the histogram? b. Construct a stem and leaf plot for the data provided by Geico Insurance. c. How do you think that Geico selected the 27 states for inclusion in this advertisement? 1.59 An Archeological Find An article in

Archaeometry involved an analysis of 26 samples of Romano-British pottery, found at four different kiln sites in the United Kingdom.16 The samples were analyzed to determine their chemical composition, and the percentage of aluminum oxide in each of the 26 samples is shown in the following table.

EX0159

140

145

330

800

Hershey's Kiss

Oreo cookie

12-ounce can of Coke

12-ounce bottle of Budweiser beer

Slice of a large Papa John's pepperoni pizza

Burger King Whopper with cheese

a. Comment on the accuracy of the graph shown above. Do the sizes, heights, and volumes of the six items accurately represent the number of calories in the item? b. Draw an actual bar chart to describe the number of calories in these six food favorites. 1.61 Laptops and Learning An informal

experiment was conducted at McNair Academic High School in Jersey City, New Jersey, to investigate the use of laptop computers as a learning tool in the study of algebra.18 A freshman class of 20 students was given laptops to use at school and at home, while another freshman class of 27 students was not given laptops; however, many of these students

EX0161

48 ❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

were able to use computers at home. The ﬁnal exam scores for the two classes are shown below. Laptops

No Laptops

98 97 88 100 100 78 68 47 90 94

63 93 83 86 99 80 78 74 67

84 93 57 84 81 83 84 93 57 83

83 52 63 81 91 81 29 72 89

97 74 88 84 49 89 64 89 70

18.0 8.0 18.0 20.0 18.0 22.0 25.0 23.0 20.0 14.9 10.0

HI ID IL IN IA KS KY LA ME MD

16.0 25.0 19.0 18.0 21.0 24.0 19.0 20.0 25.9 23.5

MA MI MN MS MO MT NE NV NH NJ

21.0 19.0 20.0 18.0 17.0 27.0 26.1 23.0 18.0 10.5

NM NY NC ND OH OK OR PA RI SC

17.0 23.9 29.9 23.0 28.0 20.0 24.0 32.0 30.0 16.0

SD TN TX UT VT VA WA WV WI WY

20.0 20.0 20.0 24.5 19.0 17.5 31.0 20.5 32.9 14.0

Source: The World Almanac and Book of Facts 2007

a. Construct a stem and leaf display for the data.

The histograms below show the distribution of ﬁnal exam scores for the two groups. 30 40 50 60 70 80 90 100 Laptops

.40

AL AK AZ AR CA CO CT DE DC FL GA

b. How would you describe the shape of this distribution? c. Are there states with unusually high or low gasoline taxes? If so, which states are they?

No laptops

1.64 Hydroelectric Plants The following data represent the planned rated capacities in megawatts (millions of watts) for the world’s 20 largest hydroelectric plants.5

Relative frequency

EX0164 .30

.20

18,200 14,000 10,000 8,370 6,400 6,300 6,000

.10

0 30 40 50 60 70 80 90 100

4,500 4,200 4,200 3,840 3,230 3,300 3,100

3,000 2,940 2,715 2,700 2,541 2,512

Source: The World Almanac and Book of Facts 2007

Write a summary paragraph describing and comparing the distribution of ﬁnal exam scores for the two groups of students. 1.62 Old Faithful The data below are the waiting times between eruptions of the Old EX0162 Faithful geyser in Yellowstone National Park.19 Use one of the graphical methods from this chapter to describe the distribution of waiting times. If there are any unusual features in your graph, see if you can think of any practical explanation for them. 56 69 55 59 76 79 75 65 68 93

89 75 87 86 94 72 78 75 87 50

51 77 53 78 75 78 64 77 61 87

79 53 85 71 50 77 80 69 81 77

58 80 61 77 83 79 49 92 55 74

82 54 93 89 82 72 49 91 93 89

52 79 54 45 72 82 88 53 53 87

88 74 76 93 77 74 51 86 84 76

52 65 80 72 75 80 78 49 70 59

78 78 81 71 65 49 85 79 73 80

1.63 Gasoline Tax The following are the

2006 state gasoline tax rates in cents per gallon for the 50 United States and the District of Columbia.5

EX0163

a. Construct a stem and leaf display for the data. b. How would you describe the shape of this distribution? 1.65 Car Colors The most popular colors for

compact and sports cars in a recent year are given in the table.5

EX0165

Color

Percentage

Silver Gray Blue Black White

20 17 16 14 10

Color Red Green Light Brown Yellow/Gold Other

Percentage 9 6 5 1 2

Source: The World Almanac and Book of Facts 2007

Use an appropriate graphical display to describe these data. 1.66 Starbucks The number of Starbucks coffee shops in cities within 20 miles of the University of California, Riverside is shown in the following table.20

EX0166

MYAPPLET EXERCISES

Starbucks

Riverside Grand Terrace Rialto Colton San Bernardino Redlands Corona Yucaipa Chino

16 1 3 2 5 7 7 2 1

City

Starbucks

Ontario Norco Fontana Mira Loma Perris Highland Rancho Cucamonga Lake Elsinore Moreno Valley

.25

11 4 6 1 1 1 12 1 4

.20 Relative frequency

City

❍

.15

.10

.05

Source: www.starbucks.com 0

a. Draw a dotplot to describe the data. b. Describe the shape of the distribution. c. Is there another variable that you could measure that might help to explain why some cities have more Starbucks than others? Explain. 1.67 What’s Normal? The 98.6 degree standard for human body temperature was derived by a German doctor in 1868. In an attempt to verify his claim, Mackowiak, Wasserman, and Levine21 took temperatures from 148 healthy people over a three-day period. A data set closely matching the one in Mackowiak’s article was derived by Allen Shoemaker, and appears in the Journal of Statistics Education.22 The body temperatures for these 130 individuals are shown in the relative frequency histogram that follows.

EX0167

96.8

97.6

99.2 98.4 Temperature

100.0

100.8

a. Describe the shape of the distribution of temperatures. b. Are there any unusual observations? Can you think of any explanation for these? c. Locate the 98.6-degree standard on the horizontal axis of the graph. Does it appear to be near the center of the distribution?

Exercises 1.68 If you have not yet done so, use the ﬁrst applet

in Building a Dotplot to create a dotplot for the following data set: 2, 3, 9, 6, 7, 6. 1.69 Cheeseburgers Use the second applet in Building a Dotplot to create a dotplot for the number of cheeseburgers consumed in a given week by 10 college students: 4 3

5 3

4 4

2 2

1 7

a. How would you describe the shape of the distribution? b. What proportion of the students ate more than 4 cheeseburgers that week? 1.70 Social Security Numbers A group of

70 students were asked to record the last digit of their Social Security number.

EX0170

1 0 3 0 6 5 3

6 7 2 0 6 1 4

9 3 0 9 9 7 1

1 4 0 9 0 7 9

5 2 2 5 2 7 3

9 3 1 3 6 8 8

0 5 2 8 2 7 6

2 8 7 4 9 5 6

8 4 7 7 5 1 6

4 2 4 4 8 8 6

a. Before graphing the data, use your common sense to guess the shape of the data distribution. Explain your reasoning. b. Use the second applet in Building a Dotplot to create a dotplot to describe the data. Was your intuition correct in part a? 1.71 If you have not yet done so, use the ﬁrst applet

in Building a Histogram to create a histogram for the data in Example 1.11, the number of visits to Starbucks during a typical week.

❍

CHAPTER 1 DESCRIBING DATA WITH GRAPHS

1.72 The United Fund The following data set records the yearly charitable contributions (in dollars) to the United Fund for a group of employees at a public university.

EX0172

41 28 77 42

81 51 75 78

80 112 59 81

65 71 63 90

47 83 63 103

56 84 80 125

80 82 101 92

69 103 115 79

79 80 99 24

63 70 67 93

Use the second applet in Building a Histogram to construct a relative frequency histogram for the data. What is the shape of the distribution? Can you see any obvious outliers? 1.73 Survival Times Altman and Bland report the survival times for patients with active hepatitis, half treated with prednisone and half receiving no treatment.23 The data that follow are adapted from their data for those treated with prednisone. The survival times are recorded to the nearest month:

EX0173

CASE STUDY Blood Pressure

8 11 52 57 65 87 93 97 109 120

127 133 139 142 144 147 148 157 162 165

a. Look at the data. Can you guess the approximate shape of the data distribution? b. Use the second applet in Building a Histogram to construct a relative frequency histogram for the data. What is the shape of the distribution? c. Are there any outliers in the set? If so, which survival times are unusually short?

How Is Your Blood Pressure? Blood pressure is the pressure that the blood exerts against the walls of the arteries. When physicians or nurses measure your blood pressure, they take two readings. The systolic blood pressure is the pressure when the heart is contracting and therefore pumping. The diastolic blood pressure is the pressure in the arteries when the heart is relaxing. The diastolic blood pressure is always the lower of the two readings. Blood pressure varies from one person to another. It will also vary for a single individual from day to day and even within a given day. If your blood pressure is too high, it can lead to a stroke or a heart attack. If it is too low, blood will not get to your extremities and you may feel dizzy. Low blood pressure is usually not serious. So, what should your blood pressure be? A systolic blood pressure of 120 would be considered normal. One of 150 would be high. But since blood pressure varies with gender and increases with age, a better gauge of the relative standing of your blood pressure would be obtained by comparing it with the population of blood pressures of all persons of your gender and age in the United States. Of course, we cannot supply you with that data set, but we can show you a very large sample selected from it. The blood pressure data on 1910 persons, 965 men and 945 women between the ages of 15 and 20, are found at the Student Companion Website. The data are part of a health survey conducted by the National Institutes of Health (NIH). Entries for each person include that person’s age and systolic and diastolic blood pressures at the time the blood pressure was recorded. 1. Describe the variables that have been measured in this survey. Are the variables quantitative or qualitative? Discrete or continuous? Are the data univariate, bivariate, or multivariate? 2. What types of graphical methods are available for describing this data set? What types of questions could be answered using various types of graphical techniques?

CASE STUDY

❍

3. Using the systolic blood pressure data set, construct a relative frequency histogram for the 965 men and another for the 945 women. Use a statistical software package if you have access to one. Compare the two histograms. 4. Consider the 965 men and 945 women as the entire population of interest. Choose a sample of n 50 men and n 50 women, recording their systolic blood pressures and their ages. Draw two relative frequency histograms to graphically display the systolic blood pressures for your two samples. Do the shapes of the histograms resemble the population histograms from part 3? 5. How does your blood pressure compare with that of others of your same gender? Check your systolic blood pressure against the appropriate histogram in part 3 or 4 to determine whether your blood pressure is “normal” or whether it is unusually high or low.

Describing Data with Numerical Measures © Joe Sohm-VisionsofAmerica/Photodisc/Getty

GENERAL OBJECTIVES Graphs are extremely useful for the visual description of a data set. However, they are not always the best tool when you want to make inferences about a population from the information contained in a sample. For this purpose, it is better to use numerical measures to construct a mental picture of the data.

CHAPTER INDEX ● Box plots (2.7) ● Measures of center: mean, median, and mode (2.2) ● Measures of relative standing: z-scores, percentiles, quartiles, and the interquartile range (2.6) ● Measures of variability: range, variance, and standard deviation (2.3)

The Boys of Summer Are the baseball champions of today better than those of “yesteryear”? Do players in the National League hit better than players in the American League? The case study at the end of this chapter involves the batting averages of major league batting champions. Numerical descriptive measures can be used to answer these and similar questions.

● Tchebysheff’s Theorem and the Empirical Rule (2.4)

How Do I Calculate Sample Quartiles?

2.2 MEASURES OF CENTER

❍

DESCRIBING A SET OF DATA WITH NUMERICAL MEASURES

2.1

Graphs can help you describe the basic shape of a data distribution; “a picture is worth a thousand words.” There are limitations, however, to the use of graphs. Suppose you need to display your data to a group of people and the bulb on the data projector blows out! Or you might need to describe your data over the telephone—no way to display the graphs! You need to ﬁnd another way to convey a mental picture of the data to your audience. A second limitation is that graphs are somewhat imprecise for use in statistical inference. For example, suppose you want to use a sample histogram to make inferences about a population histogram. How can you measure the similarities and differences between the two histograms in some concrete way? If they were identical, you could say “They are the same!” But, if they are different, it is difficult to describe the “degree of difference.” One way to overcome these problems is to use numerical measures, which can be calculated for either a sample or a population of measurements. You can use the data to calculate a set of numbers that will convey a good mental picture of the frequency distribution. These measures are called parameters when associated with the population, and they are called statistics when calculated from sample measurements. Definition Numerical descriptive measures associated with a population of measurements are called parameters; those computed from sample measurements are called statistics.

MEASURES OF CENTER

2.2

In Chapter 1, we introduced dotplots, stem and leaf plots, and histograms to describe the distribution of a set of measurements on a quantitative variable x. The horizontal axis displays the values of x, and the data are “distributed” along this horizontal line. One of the ﬁrst important numerical measures is a measure of center—a measure along the horizontal axis that locates the center of the distribution. The birth weight data presented in Table 1.9 ranged from a low of 5.6 to a high of 9.4, with the center of the histogram located in the vicinity of 7.5 (see Figure 2.1). Let’s consider some rules for locating the center of a distribution of measurements.

Center of the birth weight data

● 8/30 7/30 Relative Frequency

FIGU R E 2 .1

6/30 5/30 4/30 3/30 2/30 1/30 0 5.6

6.1

6.6

7.1

7.6 8.1 Center Birth Weights

8.6

9.1

9.6

54 ❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

The arithmetic average of a set of measurements is a very common and useful measure of center. This measure is often referred to as the arithmetic mean, or simply the mean, of a set of measurements. To distinguish between the mean for the sample and the mean for the population, we will use the symbol x苶 (x-bar) for a sample mean and the symbol m (Greek lowercase mu) for the mean of a population. The arithmetic mean or average of a set of n measurements is equal to the sum of the measurements divided by n. Definition

Since statistical formulas often involve adding or “summing” numbers, we use a shorthand symbol to indicate the process of summing. Suppose there are n measurements on the variable x—call them x1, x2, . . . , xn. To add the n measurements together, we use this shorthand notation: n

冱 xi i1

which means x1 x2 x3 xn

The Greek capital sigma (S) tells you to add the items that appear to its right, beginning with the number below the sigma (i 1) and ending with the number above (i n). However, since the typical sums in statistical calculations are almost always made on the total set of n measurements, you can use a simpler notation: Sxi

which means “the sum of all the x measurements”

Using this notation, we write the formula for the sample mean: NOTATION Sx Sample mean: 苶x i n Population mean: m EXAMPLE

Draw a dotplot for the n 5 measurements 2, 9, 11, 5, 6. Find the sample mean and compare its value with what you might consider the “center” of these observations on the dotplot.

2.1

The dotplot in Figure 2.2 seems to be centered between 6 and 8. To ﬁnd the sample mean, calculate

Solution

Sxi 2 9 11 5 6 6.6 苶x n 5 FI GU R E 2 .2

Dotplot for Example 2.1

●

6 Measurements

The statistic x苶 6.6 is the balancing point or fulcrum shown on the dotplot. It does seem to mark the center of the data.

2.2 MEASURES OF CENTER

mean balancing point or fulcrum

❍

Remember that samples are measurements drawn from a larger population that is usually unknown. An important use of the sample mean x苶 is as an estimator of the unknown population mean m. The birth weight data in Table 1.9 are a sample from a larger population of birth weights, and the distribution is shown in Figure 2.1. The mean of the 30 birth weights is Sxi 227.2 7.57 苶x 30 30 shown in Figure 2.1; it marks the balancing point of the distribution. The mean of the entire population of newborn birth weights is unknown, but if you had to guess its value, your best estimate would be 7.57. Although the sample mean 苶x changes from sample to sample, the population mean m stays the same. A second measure of central tendency is the median, which is the value in the middle position in the set of measurements ordered from smallest to largest. The median m of a set of n measurements is the value of x that falls in the middle position when the measurements are ordered from smallest to largest. Definition

EXAMPLE

2.2

Find the median for the set of measurements 2, 9, 11, 5, 6. Solution

Rank the n 5 measurements from smallest to largest: 2

6 9 앖

The middle observation, marked with an arrow, is in the center of the set, or m 6. EXAMPLE

2.3

Find the median for the set of measurements 2, 9, 11, 5, 6, 27. Solution

Rank the measurements from smallest to largest: 2

Roughly 50% of the measurements are smaller, 50% are larger than the median.

6 9 앖

Now there are two “middle” observations, shown in the box. To ﬁnd the median, choose a value halfway between the two middle observations: 69 m 7.5 2 The value .5(n 1) indicates the position of the median in the ordered data set. If the position of the median is a number that ends in the value .5, you need to average the two adjacent values.

EXAMPLE

2.4

For the n 5 ordered measurements from Example 2.2, the position of the median is .5(n 1) .5(6) 3, and the median is the 3rd ordered observation, or m 6. For the n 6 ordered measurements from Example 2.3, the position of the median is .5(n 1) .5(7) 3.5, and the median is the average of the 3rd and 4th ordered observations, or m (6 9)/2 7.5.

56 ❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

Although both the mean and the median are good measures of the center of a distribution, the median is less sensitive to extreme values or outliers. For example, the value x 27 in Example 2.3 is much larger than the other ﬁve measurements. The median, m 7.5, is not affected by the outlier, whereas the sample average,

symmetric: mean median skewed right: mean median skewed left: mean median

Sx 60 x苶 i 10 n 6 is affected; its value is not representative of the remaining ﬁve observations. When a data set has extremely small or extremely large observations, the sample mean is drawn toward the direction of the extreme measurements (see Figure 2.3).

●

(b)

(a) .25

.25 Relative Frequency

Relative frequency distributions showing the effect of extreme values on the mean and median

Relative Frequency

FI GU R E 2 .3

.19 .12 .06

.19 .12 .06 0

0 Mean Median

Mean ⬎ Median

If a distribution is skewed to the right, the mean shifts to the right; if a distribution is skewed to the left, the mean shifts to the left. The median is not affected by these extreme values because the numerical values of the measurements are not used in its calculation. When a distribution is symmetric, the mean and the median are equal. If a distribution is strongly skewed by one or more extreme values, you should use the median rather than the mean as a measure of center.

You can see the effect of extreme values on both the mean and the median using the How Extreme Values Affect the Mean and Median applet. The ﬁrst of three applets (Figure 2.4) shows a dotplot of the data in Example 2.2. Use your mouse to move the largest observation (x 11) even further to the right. How does this larger observation affect the mean? How does it affect the median? We will use this applet again for the MyApplet Exercises at the end of the chapter. FI GU R E 2 .4

How Extreme Values Affect the Mean and Median applet

●

2.2 MEASURES OF CENTER

❍

Another way to locate the center of a distribution is to look for the value of x that occurs with the highest frequency. This measure of the center is called the mode. Definition The mode is the category that occurs most frequently, or the most frequently occurring value of x. When measurements on a continuous variable have been grouped as a frequency or relative frequency histogram, the class with the highest peak or frequency is called the modal class, and the midpoint of that class is taken to be the mode.

The mode is generally used to describe large data sets, whereas the mean and median are used for both large and small data sets. From the data in Example 1.11, the mode of the distribution of the number of reported weekly visits to Starbucks for 30 Starbucks customers is 5. The modal class and the value of x occurring with the highest frequency are the same, as shown in Figure 2.5(a). For the data in Table 1.9, a birth weight of 7.7 occurs four times, and therefore the mode for the distribution of birth weights is 7.7. Using the histogram to find the modal class, you find that the class with the highest peak is the fifth class, from 7.6 to 8.1. Our choice for the mode would be the midpoint of this class, or 7.85. See Figure 2.5(b). It is possible for a distribution of measurements to have more than one mode. These modes would appear as “local peaks” in the relative frequency distribution. For example, if we were to tabulate the length of fish taken from a lake during one season, we might get a bimodal distribution, possibly reflecting a mixture of young and old fish in the population. Sometimes bimodal distributions of sizes or weights reflect a mixture of measurements taken on males and females. In any case, a set or distribution of measurements may have more than one mode.

Remember that there can be several modes or no mode (if each observation occurs only once).

●

(b)

(a) 8/25

Relative Frequency

Relative frequency histograms for the Starbucks and birth weight data

Relative Frequency

FIGU R E 2 .5

6/25 4/25 2/25 0 1

2.2

4 5 Visits

8/30 7/30 6/30 5/30 4/30 3/30 2/30 1/30 0 5.6 6.1 6.6 7.1 7.6 8.1 8.6 9.1 9.6 Birth Weights

EXERCISES

BASIC TECHNIQUES 2.1 You are given n 5 measurements: 0, 5, 1, 1, 3.

a. Draw a dotplot for the data. (HINT: If two measurements are the same, place one dot above the other.) Guess the approximate “center.”

b. Find the mean, median, and mode. c. Locate the three measures of center on the dotplot in part a. Based on the relative positions of the mean and median, are the measurements symmetric or skewed?

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

2.2 You are given n 8 measurements: 3, 2, 5, 6, 4,

4, 3, 5. a. Find 苶x. b. Find m. c. Based on the results of parts a and b, are the measurements symmetric or skewed? Draw a dotplot to conﬁrm your answer. 2.3 You are given n 10 measurements: 3, 5, 4, 6,

10, 5, 6, 9, 2, 8. a. Calculate 苶x. b. Find m. c. Find the mode. APPLICATIONS 2.4 Auto Insurance The cost of automobile insur-

ance has become a sore subject in California because insurance rates are dependent on so many different variables, such as the city in which you live, the number of cars you insure, and the company with which you are insured. The website www.insurance.ca.gov reports the annual 2006–2007 premium for a single male, licensed for 6–8 years, who drives a Honda Accord 12,600 to 15,000 miles per year and has no violations or accidents.1 City Long Beach Pomona San Bernardino Moreno Valley

Allstate

21st Century

$2617 2305 2286 2247

$2228 2098 2064 1890

Source: www.insurance.ca.gov

a. What is the average premium for Allstate Insurance? b. What is the average premium for 21st Century Insurance? c. If you were a consumer, would you be interested in the average premium cost? If not, what would you be interested in? 2.5 DVD Players The DVD player is a common ﬁxture in most American households. In fact, most American households have DVDs, and many have more than one. A sample of 25 households produced the following measurements on x, the number of DVDs in the household:

EX0205

1 1 0 1 3

0 0 1 1 1

2 2 2 1 0

1 1 3 0 1

1 0 2 1 1

a. Is the distribution of x, the number of DVDs in a household, symmetric or skewed? Explain. b. Guess the value of the mode, the value of x that occurs most frequently. c. Calculate the mean, median, and mode for these measurements. d. Draw a relative frequency histogram for the data set. Locate the mean, median, and mode along the horizontal axis. Are your answers to parts a and b correct? 2.6 Fortune 500 Revenues Ten of the 50 largest businesses in the United States, randomly selected from the Fortune 500, are listed below along with their revenues (in millions of dollars):2 Company General Motors IBM Bank of America Home Depot Boeing

Revenues $192,604 91,134 83,980 81,511 54,848

Company Target Morgan Stanley Johnson & Johnson Intel Safeway

Revenues $52,620 52,498 50,514 38,826 38,416

Source: Time Almanac 2007

a. Draw a stem and leaf plot for the data. Are the data skewed? b. Calculate the mean revenue for these 10 businesses. Calculate the median revenue. c. Which of the two measures in part b best describes the center of the data? Explain. 2.7 Birth Order and Personality Does birth order have any effect on a person’s personality? A report on a study by an MIT researcher indicates that later-born children are more likely to challenge the establishment, more open to new ideas, and more accepting of change.3 In fact, the number of later-born children is increasing. During the Depression years of the 1930s, families averaged 2.5 children (59% later born), whereas the parents of baby boomers averaged 3 to 4 children (68% later born). What does the author mean by an average of 2.5 children?

2.2 MEASURES OF CENTER

2.8 Tuna Fish An article in Consumer

Reports gives the price—an estimated average for a 6-ounce can or a 7.06-ounce pouch—for 14 different brands of water-packed light tuna, based on prices paid nationally in supermarkets:4

EX0208

.99 1.12

1.92 .63

1.23 .85 .65 .53 1.41 .67 .69 .60 .60 .66

a. Find the average price for the 14 different brands of tuna. b. Find the median price for the 14 different brands of tuna. c. Based on your ﬁndings in parts a and b, do you think that the distribution of prices is skewed? Explain. 2.9 Sports Salaries As professional sports teams become a more and more lucrative business for their owners, the salaries paid to the players have also increased. In fact, sports superstars are paid astronomical salaries for their talents. If you were asked by a sports management ﬁrm to describe the distribution of players’ salaries in several different categories of professional sports, what measure of center would you choose? Why? 2.10 Time on Task In a psychological experiment,

the time on task was recorded for 10 subjects under a 5-minute time constraint. These measurements are in seconds: 175 200

190 185

250 190

230 225

240 265

a. Find the average time on task. b. Find the median time on task. c. If you were writing a report to describe these data, which measure of central tendency would you use? Explain.

❍

2.11 Starbucks The number of Starbucks coffee shops in 18 cities within 20 miles of the University of California, Riverside is shown in the following table (www.starbucks.com).5

EX0211

16 1 3 5

7 7 2 1

2 1 11 4

6 1 1 12

4 1

a. Find the mean, the median, and the mode. b. Compare the median and the mean. What can you say about the shape of this distribution? c. Draw a dotplot for the data. Does this conﬁrm your conclusion about the shape of the distribution from part b? 2.12 HDTVs The cost of televisions exhibits

huge variation—from $100–200 for a standard TV to $8,000–10,000 for a large plasma screen TV. Consumer Reports gives the prices for the top 10 LCD high deﬁnition TVs (HDTVs) in the 30- to 40-inch category:6

EX0212

Brand

Price

JVC LT-40FH96 Sony Bravia KDL-V32XBR1 Sony Bravia KDL-V40XBR1 Toshiba 37HLX95 Sharp Aquos LC-32DA5U Sony Bravia KLV-S32A10 Panasonic Viera TC-32LX50 JVC LT-37X776 LG 37LP1D Samsung LN-R328W

$2900 1800 2600 3000 1300 1500 1350 2000 2200 1200

a. What is the average price of these 10 HDTVs? b. What is the median price of these 10 HDTVs? c. As a consumer, would you be interested in the average cost of an HDTV? What other variables would be important to you?

60 ❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

MEASURES OF VARIABILITY

2.3

Data sets may have the same center but look different because of the way the numbers spread out from the center. Consider the two distributions shown in Figure 2.6. Both distributions are centered at x 4, but there is a big difference in the way the measurements spread out, or vary. The measurements in Figure 2.6(a) vary from 3 to 5; in Figure 2.6(b) the measurements vary from 0 to 8. ●

(b)

(a)

Relative Frequency

Variability or dispersion of data

Relative Frequency

FI GU R E 2 .6

Variability or dispersion is a very important characteristic of data. For example, if you were manufacturing bolts, extreme variation in the bolt diameters would cause a high percentage of defective products. On the other hand, if you were trying to discriminate between good and poor accountants, you would have trouble if the examination always produced test grades with little variation, making discrimination very difficult. Measures of variability can help you create a mental picture of the spread of the data. We will present some of the more important ones. The simplest measure of variation is the range. The range, R, of a set of n measurements is deﬁned as the difference between the largest and smallest measurements. Definition

For the birth weight data in Table 1.9, the measurements vary from 5.6 to 9.4. Hence, the range is 9.4 5.6 3.8. The range is easy to calculate, easy to interpret, and is an adequate measure of variation for small sets of data. But, for large data sets, the range is not an adequate measure of variability. For example, the two relative frequency distributions in Figure 2.7 have the same range but very different shapes and variability. ●

(b)

(a) Relative Frequency

Distributions with equal range and unequal variability

Relative Frequency

FI GU R E 2 .7

2.3 MEASURES OF VARIABILITY

❍

Is there a measure of variability that is more sensitive than the range? Consider, as an example, the sample measurements 5, 7, 1, 2, 4, displayed as a dotplot in Figure 2.8. The mean of these ﬁve measurements is Sx 19 x苶 i 3.8 n 5 FIGU R E 2 .8

Dotplot showing the deviations of points from the mean

● x = 3.8

(xi – x)

as indicated on the dotplot. The horizontal distances between each dot (measurement) and the mean x苶 will help you to measure the variability. If the distances are large, the data are more spread out or variable than if the distances are small. If xi is a particular dot (measurement), then the deviation of that measurement from the mean is (xi x苶). Measurements to the right of the mean produce positive deviations, and those to the left produce negative deviations. The values of x and the deviations for our example are listed in the ﬁrst and second columns of Table 2.1. TABLE 2.1

●

2 Computation of S(xi x 苶) x (xi x苶 ) (xi x苶 )2

5 7 1 2 4

1.2 3.2 2.8 1.8 .2

1.44 10.24 7.84 3.24 .04

0.0

22.80

Because the deviations in the second column of the table contain information on variability, one way to combine the ﬁve deviations into one numerical measure is to average them. Unfortunately, the average will not work because some of the deviations are positive, some are negative, and the sum is always zero (unless round-off errors have been introduced into the calculations). Note that the deviations in the second column of Table 2.1 sum to zero. Another possibility might be to disregard the signs of the deviations and calculate the average of their absolute values.† This method has been used as a measure of variability in exploratory data analysis and in the analysis of time series data. We prefer, however, to overcome the difficulty caused by the signs of the deviations by working The absolute value of a number is its magnitude, ignoring its sign. For example, the absolute value of 2, represented by the symbol 兩2兩, is 2. The absolute value of 2—that is, 兩2兩—is 2.

†

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

with their sum of squares. From the sum of squared deviations, a single measure called the variance is calculated. To distinguish between the variance of a sample and the variance of a population, we use the symbol s2 for a sample variance and s 2 (Greek lowercase sigma) for a population variance. The variance will be relatively large for highly variable data and relatively small for less variable data. The variance of a population of N measurements is the average of the squares of the deviations of the measurements about their mean m. The population variance is denoted by s 2 and is given by the formula

Definition

S(xi m)2 s2 N Most often, you will not have all the population measurements available but will need to calculate the variance of a sample of n measurements. Definition The variance of a sample of n measurements is the sum of the squared deviations of the measurements about their mean x苶 divided by (n 1). The sample variance is denoted by s 2 and is given by the formula

S(xi x苶 )2 s2 n1 For the set of n 5 sample measurements presented in Table 2.1, the square of the deviation of each measurement is recorded in the third column. Adding, we obtain The variance and the standard deviation cannot be negative numbers.

S(xi 苶x )2 22.80 and the sample variance is S(xi x苶 )2 22.80 5.70 s2 n1 4 The variance is measured in terms of the square of the original units of measurement. If the original measurements are in inches, the variance is expressed in square inches. Taking the square root of the variance, we obtain the standard deviation, which returns the measure of variability to the original units of measurement. The standard deviation of a set of measurements is equal to the positive square root of the variance. Definition

NOTATION n: number of measurements in the sample 2 s : sample variance 2 sample standard s 兹s苶: deviation

N: number of measurements in the population 2 s : population variance 2 苶: s 兹s population standard deviation

2.3 MEASURES OF VARIABILITY

❍

For the set of n 5 sample measurements in Table 2.1, the sample variance is 苶 2.39. The more s 2 5.70, so the sample standard deviation is s 兹s苶2 兹5.70 variable the data set is, the larger the value of s. For the small set of measurements we used, the calculation of the variance is not too difficult. However, for a larger set, the calculations can become very tedious. Most scientiﬁc calculators have built-in programs that will calculate 苶x and s or m and s, so that your computational work will be minimized. The sample or population mean key is usually marked with 苶x. The sample standard deviation key is usually marked with s, sx, or sxn1, and the population standard deviation key with s, sx, or sxn. In using any calculator with these built-in function keys, be sure you know which calculation is being carried out by each key! If you need to calculate s 2 and s by hand, it is much easier to use the alternative computing formula given next. This computational form is sometimes called the shortcut method for calculating s 2.

If you are using your calculator, make sure to choose the correct key for the sample standard deviation.

THE COMPUTING FORMULA FOR CALCULATING s 2 (Sxi)2 Sx2i n s 2 n1 The symbols (Sxi)2 and Sx 2i in the computing formula are shortcut ways to indicate the arithmetic operation you need to perform. You know from the formula for the sample mean that Sxi is the sum of all the measurements. To ﬁnd Sx 2i , you square each individual measurement and then add them together. Sx 2i Sum of the squares of the individual measurements (Sxi)2 Square of the sum of the individual measurements The sample standard deviation, s, is the positive square root of s 2. EXAMPLE

TABLE 2.2

2.5

Calculate the variance and standard deviation for the ﬁve measurements in Table 2.2, which are 5, 7, 1, 2, 4. Use the computing formula for s 2 and compare your results with those obtained using the original deﬁnition of s 2.

●

Table for Simplified Calculation of s2 and s xi

x i2

5 7 1 2 4

25 49 1 4 16

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

The entries in Table 2.2 are the individual measurements, xi, and their squares, x 2i , together with their sums. Using the computing formula for s 2, you have

Solution Don’t round off partial results as you go along!

(Sxi)2 Sx 2i n s 2 n1 (19)2 95 5 22.80 5.70 4 4 苶 2.39, as before. and s 兹s苶2 兹5.70 You may wonder why you need to divide by (n 1) rather than n when computing the sample variance. Just as we used the sample mean 苶x to estimate the population mean m, you may want to use the sample variance s 2 to estimate the population variance s 2. It turns out that the sample variance s 2 with (n 1) in the denominator provides better estimates of s 2 than would an estimator calculated with n in the denominator. For this reason, we always divide by (n 1) when computing the sample variance s 2 and the sample standard deviation s.

Why Divide by n 1? applet

●

2.3 MEASURES OF VARIABILITY

❍

At this point, you have learned how to compute the variance and standard deviation of a set of measurements. Remember these points: • • • •

The value of s is always greater than or equal to zero. The larger the value of s 2 or s, the greater the variability of the data set. If s 2 or s is equal to zero, all the measurements must have the same value. In order to measure the variability in the same units as the original 2 observations, we compute the standard deviation s 兹s苶.

This information allows you to compare several sets of data with respect to their locations and their variability. How can you use these measures to say something more speciﬁc about a single set of data? The theorem and rule presented in the next section will help answer this question.

2.3

EXERCISES

BASIC TECHNIQUES

2.16 You are given n 8 measurements: 3, 1, 5, 6,

2.13 You are given n 5 measurements: 2, 1, 1,

3, 5. a. Calculate the sample mean, 苶x. b. Calculate the sample variance, s 2, using the formula given by the deﬁnition. c. Find the sample standard deviation, s. d. Find s 2 and s using the computing formula. Compare the results with those found in parts b and c.

4, 4, 3, 5. a. Calculate the range. b. Calculate the sample mean. c. Calculate the sample variance and standard deviation. d. Compare the range and the standard deviation. The range is approximately how many standard deviations?

2.14 Refer to Exercise 2.13.

APPLICATIONS

a. Use the data entry method in your scientific calculator to enter the five measurements. Recall the proper memories to find the sample mean and standard deviation. b. Verify that the calculator provides the same values for x苶 and s as in Exercise 2.13, parts a and c.

2.17 An Archeological Find, again An article in

2.15 You are given n 8 measurements: 4, 1, 3, 1, 3,

1, 2, 2. a. Find the range. b. Calculate 苶x. c. Calculate s 2 and s using the computing formula. d. Use the data entry method in your calculator to ﬁnd x苶, s, and s 2. Verify that your answers are the same as those in parts b and c.

Archaeometry involved an analysis of 26 samples of Romano-British pottery found at four different kiln sites in the United Kingdom.7 The samples were analyzed to determine their chemical composition. The percentage of iron oxide in each of ﬁve samples collected at the Island Thorns site was: 1.28,

2.39,

1.50,

1.88,

1.51

a. Calculate the range. b. Calculate the sample variance and the standard deviation using the computing formula. c. Compare the range and the standard deviation. The range is approximately how many standard deviations?

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

2.18 Utility Bills in Southern California

The monthly utility bills for a household in Riverside, California, were recorded for 12 consecutive months starting in January 2006:

EX0218

Month

Amount ($)

January February March April May June

$266.63 163.41 219.41 162.64 187.16 289.17

Month July August September October November December

Amount ($) $306.55 335.48 343.50 226.80 208.99 230.46

a. Calculate the range of the utility bills for the year 2006. b. Calculate the average monthly utility bill for the year 2006. c. Calculate the standard deviation for the 2006 utility bills.

ON THE PRACTICAL SIGNIFICANCE OF THE STANDARD DEVIATION

2.4

We now introduce a useful theorem developed by the Russian mathematician Tchebysheff. Proof of the theorem is not difficult, but we are more interested in its application than its proof. Given a number k greater than or equal to 1 and a set of n measurements, at least [1 (1/k 2)] of the measurements will lie within k standard deviations of their mean.

Tchebysheff’s Theorem

Tchebysheff’s Theorem applies to any set of measurements and can be used to describe either a sample or a population. We will use the notation appropriate for populations, but you should realize that we could just as easily use the mean and the standard deviation for the sample. The idea involved in Tchebysheff’s Theorem is illustrated in Figure 2.10. An interval is constructed by measuring a distance ks on either side of the mean m. The number k can be any number as long as it is greater than or equal to 1. Then Tchebysheff’s Theorem states that at least 1 (1/k 2) of the total number n measurements lies in the constructed interval.

Illustrating Tchebysheff’s Theorem

●

Relative Frequency

FI GU R E 2 .1 0

At least 1 – (1/k2)

µ kσ

x kσ

2.4 ON THE PRACTICAL SIGNIFICANCE OF THE STANDARD DEVIATION

❍

In Table 2.3, we choose a few numerical values for k and compute [1 (1/k2)]. TABLE 2.3

●

Illustrative Values of [1 (1/k2)] k

1 (1/k 2)

1 2 3

110 1 1/4 3/4 1 1/9 8/9

From the calculations in Table 2.3, the theorem states: • • •

At least none of the measurements lie in the interval m s to m s. At least 3/4 of the measurements lie in the interval m 2s to m 2s. At least 8/9 of the measurements lie in the interval m 3s to m 3s.

Although the ﬁrst statement is not at all helpful, the other two values of k provide valuable information about the proportion of measurements that fall in certain intervals. The values k 2 and k 3 are not the only values of k you can use; for example, the proportion of measurements that fall within k 2.5 standard deviations of the mean is at least 1 [1/(2.5)2] .84. EXAMPLE

The mean and variance of a sample of n 25 measurements are 75 and 100, respectively. Use Tchebysheff’s Theorem to describe the distribution of measurements.

2.6

You are given x苶 75 and s 2 100. The standard deviation is 苶 10. The distribution of measurements is centered about x苶 75, and s 兹100 Tchebysheff’s Theorem states: Solution

• •

At least 3/4 of the 25 measurements lie in the interval 苶x 2s 75 2(10) —that is, 55 to 95. At least 8/9 of the measurements lie in the interval 苶x 3s 75 3(10)—that is, 45 to 105.

Since Tchebysheff’s Theorem applies to any distribution, it is very conservative. This is why we emphasize “at least 1 (1/k 2 )” in this theorem. Another rule for describing the variability of a data set does not work for all data sets, but it does work very well for data that “pile up” in the familiar mound shape shown in Figure 2.11. The closer your data distribution is to the mound-shaped curve in Figure 2.11, the more accurate the rule will be. Since mound-shaped data distributions occur quite frequently in nature, the rule can often be used in practical applications. For this reason, we call it the Empirical Rule. Mound-shaped distribution

● Relative Frequency

FIGU R E 2 .1 1

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

Empirical Rule

Given a distribution of measurements that is approximately

mound-shaped: The interval (m s) contains approximately 68% of the measurements. The interval (m 2s) contains approximately 95% of the measurements. The interval (m 3s) contains approximately 99.7% of the measurements. Remember these three numbers:

The mound-shaped distribution shown in Figure 2.11 is commonly known as the normal distribution and will be discussed in detail in Chapter 6.

68—95—99.7 EXAMPLE

2.7

In a time study conducted at a manufacturing plant, the length of time to complete a speciﬁed operation is measured for each of n 40 workers. The mean and standard deviation are found to be 12.8 and 1.7, respectively. Describe the sample data using the Empirical Rule. Solution

To describe the data, calculate these intervals:

(x苶 s) 12.8 1.7 ( 苶x 2s) 12.8 2(1.7)

11.1 to 14.5

9.4 to 16.2

7.7 to 17.9 (x苶 3s) 12.8 3(1.7) or According to the Empirical Rule, you expect approximately 68% of the measurements to fall into the interval from 11.1 to 14.5, approximately 95% to fall into the interval from 9.4 to 16.2, and approximately 99.7% to fall into the interval from 7.7 to 17.9. If you doubt that the distribution of measurements is mound-shaped, or if you wish for some other reason to be conservative, you can apply Tchebysheff’s Theorem and be absolutely certain of your statements. Tchebysheff’s Theorem tells you that at least 3/4 of the measurements fall into the interval from 9.4 to 16.2 and at least 8/9 into the interval from 7.7 to 17.9.

EXAMPLE

TABLE 2.4

2.8

Student teachers are trained to develop lesson plans, on the assumption that the written plan will help them to perform successfully in the classroom. In a study to assess the relationship between written lesson plans and their implementation in the classroom, 25 lesson plans were scored on a scale of 0 to 34 according to a Lesson Plan Assessment Checklist. The 25 scores are shown in Table 2.4. Use Tchebysheff’s Theorem and the Empirical Rule (if applicable) to describe the distribution of these assessment scores. ●

Lesson Plan Assessment Scores 26.1 22.1 15.9 25.6 29.0

26.0 21.2 20.8 26.5 21.3

14.5 26.6 20.2 15.7 23.5

29.3 31.9 17.8 22.1 22.1

19.7 25.0 13.3 13.8 10.2

Use your calculator or the computing formulas to verify that 苶x 21.6 and s 5.5. The appropriate intervals are calculated and listed in Table 2.5. We have also referred back to the original 25 measurements and counted the actual number of measurements that fall into each of these intervals. These frequencies and relative frequencies are shown in Table 2.5.

Solution

2.4 ON THE PRACTICAL SIGNIFICANCE OF THE STANDARD DEVIATION

TABLE 2.5

●

❍

Intervals x ៮ ks for the Data of Table 2.4 Interval x苵 ks

k 1 2 3

16.1–27.1 10.6–32.6 5.1–38.1

Frequency in Interval

Relative Frequency

16 24 25

.64 .96 1.00

Is Tchebysheff’s Theorem applicable? Yes, because it can be used for any set of data. According to Tchebysheff’s Theorem,

Empirical Rule ⇔ mound-shaped data

• •

Tchebysheff ⇔ any shaped data

at least 3/4 of the measurements will fall between 10.6 and 32.6. at least 8/9 of the measurements will fall between 5.1 and 38.1.

You can see in Table 2.5 that Tchebysheff’s Theorem is true for these data. In fact, the proportions of measurements that fall into the speciﬁed intervals exceed the lower bound given by this theorem. Is the Empirical Rule applicable? You can check for yourself by drawing a graph— either a stem and leaf plot or a histogram. The MINITAB histogram in Figure 2.12 shows that the distribution is relatively mound-shaped, so the Empirical Rule should work relatively well. That is, • • •

approximately 68% of the measurements will fall between 16.1 and 27.1. approximately 95% of the measurements will fall between 10.6 and 32.6. approximately 99.7% of the measurements will fall between 5.1 and 38.1.

The relative frequencies in Table 2.5 closely approximate those speciﬁed by the Empirical Rule. MINITAB histogram for Example 2.8

● 6/25

Relative Frequency

F I GU R E 2 .1 2

4/25

2/25

0 8.5

14.5

20.5 Scores

26.5

32.5

USING TCHEBYSHEFF’S THEOREM AND THE EMPIRICAL RULE Tchebysheff’s Theorem can be proven mathematically. It applies to any set of measurements—sample or population, large or small, mound-shaped or skewed. Tchebysheff’s Theorem gives a lower bound to the fraction of measurements to be found in an interval constructed as 苶x ks. At least 1 (1/k 2) of the measurements will fall into this interval, and probably more!

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

The Empirical Rule is a “rule of thumb” that can be used as a descriptive tool only when the data tend to be roughly mound-shaped (the data tend to pile up near the center of the distribution). When you use these two tools for describing a set of measurements, Tchebysheff’s Theorem will always be satisﬁed, but it is a very conservative estimate of the fraction of measurements that fall into a particular interval. If it is appropriate to use the Empirical Rule (mound-shaped data), this rule will give you a more accurate estimate of the fraction of measurements that fall into the interval.

A CHECK ON THE CALCULATION OF s

2.5

Tchebysheff’s Theorem and the Empirical Rule can be used to detect gross errors in the calculation of s. Roughly speaking, these two tools tell you that most of the time, measurements lie within two standard deviations of their mean. This interval is marked off in Figure 2.13, and it implies that the total range of the measurements, from smallest to largest, should be somewhere around four standard deviations. This is, of course, a very rough approximation, but it can be very useful in checking for large errors in your calculation of s. If the range, R, is about four standard deviations, or 4s, you can write R ⬇ 4s

R s ⬇ 4

The computed value of s using the shortcut formula should be of roughly the same order as the approximation.

FI GU R E 2 .1 3

Range approximation to s

●

x – 2s

EXAMPLE

2.9

2s x + 2s

Use the range approximation to check the calculation of s for Table 2.2. Solution

The range of the ﬁve measurements—5, 7, 1, 2, 4—is

R716 Then R 6 s ⬇ 1.5 4 4 This is the same order as the calculated value s 2.4.

s ⬇ R/4 gives only an approximate value for s.

The range approximation is not intended to provide an accurate value for s. Rather, its purpose is to detect gross errors in calculating, such as the failure to divide the sum of squares of deviations by (n 1) or the failure to take the square root of s 2. If you make one of these mistakes, your answer will be many times larger than the range approximation of s.

2.5 A CHECK ON THE CALCULATION OF s

EXAMPLE

2.10

❍

Use the range approximation to determine an approximate value for the standard deviation for the data in Table 2.4. Solution

The range R 31.9 10.2 21.7. Then

R 21.7 s ⬇ 5.4 4 4 Since the exact value of s is 5.5 for the data in Table 2.4, the approximation is very close. The range for a sample of n measurements will depend on the sample size, n. For larger values of n, a larger range of the x values is expected. The range for large samples (say, n 50 or more observations) may be as large as 6s, whereas the range for small samples (say, n 5 or less) may be as small as or smaller than 2.5s. The range approximation for s can be improved if it is known that the sample is drawn from a mound-shaped distribution of data. Thus, the calculated s should not differ substantially from the range divided by the appropriate ratio given in Table 2.6. TABLE 2.6

●

Divisor for the Range Approximation of s Number of Measurements 5 10 25

2.5

Expected Ratio of Range to s 2.5 3 4

EXERCISES

BASIC TECHNIQUES 2.19 A set of n 10 measurements consists of the

values 5, 2, 3, 6, 1, 2, 4, 5, 1, 3. a. Use the range approximation to estimate the value of s for this set. (HINT: Use the table at the end of Section 2.5.) b. Use your calculator to ﬁnd the actual value of s. Is the actual value close to your estimate in part a? c. Draw a dotplot of this data set. Are the data moundshaped? d. Can you use Tchebysheff’s Theorem to describe this data set? Why or why not? e. Can you use the Empirical Rule to describe this data set? Why or why not? 2.20 Suppose you want to create a mental picture

of the relative frequency histogram for a large data set consisting of 1000 observations, and you know

that the mean and standard deviation of the data set are 36 and 3, respectively. a. If you are fairly certain that the relative frequency distribution of the data is mound-shaped, how might you picture the relative frequency distribution? (HINT: Use the Empirical Rule.) b. If you have no prior information concerning the shape of the relative frequency distribution, what can you say about the relative frequency histogram? (HINT: Construct intervals 苶x ks for several choices of k.) 2.21 A distribution of measurements is relatively

mound-shaped with mean 50 and standard deviation 10. a. What proportion of the measurements will fall between 40 and 60? b. What proportion of the measurements will fall between 30 and 70?

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

c. What proportion of the measurements will fall between 30 and 60? d. If a measurement is chosen at random from this distribution, what is the probability that it will be greater than 60? 2.22 A set of data has a mean of 75 and a standard

deviation of 5. You know nothing else about the size of the data set or the shape of the data distribution. a. What can you say about the proportion of measurements that fall between 60 and 90? b. What can you say about the proportion of measurements that fall between 65 and 85? c. What can you say about the proportion of measurements that are less than 65?

2.25 Breathing Rates Is your breathing rate nor-

mal? Actually, there is no standard breathing rate for humans. It can vary from as low as 4 breaths per minute to as high as 70 or 75 for a person engaged in strenuous exercise. Suppose that the resting breathing rates for college-age students have a relative frequency distribution that is mound-shaped, with a mean equal to 12 and a standard deviation of 2.3 breaths per minute. What fraction of all students would have breathing rates in the following intervals? a. 9.7 to 14.3 breaths per minute b. 7.4 to 16.6 breaths per minute c. More than 18.9 or less than 5.1 breaths per minute 2.26 Ore Samples A geologist collected 20 different ore samples, all the same weight, and randomly divided them into two groups. She measured the titanium (Ti) content of the samples using two different methods.

EX0226

APPLICATIONS 2.23 Driving Emergencies The length of time re-

quired for an automobile driver to respond to a particular emergency situation was recorded for n 10 drivers. The times (in seconds) were .5, .8, 1.1, .7, .6, .9, .7, .8, .7, .8. a. Scan the data and use the procedure in Section 2.5 to ﬁnd an approximate value for s. Use this value to check your calculations in part b. b. Calculate the sample mean x苶 and the standard deviation s. Compare with part a.

Method 1

Method 2

.011 .013 .013 .015 .014 .013 .010 .013 .011 .012

.011 .016 .013 .012 .015 .012 .017 .013 .014 .015

a. Construct stem and leaf plots for the two data sets. Visually compare their centers and their ranges. b. Calculate the sample means and standard deviations for the two sets. Do the calculated values conﬁrm your visual conclusions from part a?

2.24 Packaging Hamburger Meat The data listed here are the weights (in pounds) of 27 packages of ground beef in a supermarket meat display:

2.27 Social Security Numbers The data from Exercise 1.70 (see data set EX0170), reproduced below, show the last digit of the Social Security number for a group of 70 students.

1.08 1.06 .89 .89

1 0 3 0 6 5 3

EX0224

.99 1.14 .89 .98

.97 1.38 .96 1.14

1.18 .75 1.12 .92

1.41 .96 1.12 1.18

1.28 1.08 .93 1.17

.83 .87 1.24

a. Construct a stem and leaf plot or a relative frequency histogram to display the distribution of weights. Is the distribution relatively moundshaped? b. Find the mean and standard deviation of the data set. c. Find the percentage of measurements in the intervals 苶x s, 苶x 2s, and 苶x 3s. d. How do the percentages obtained in part c compare with those given by the Empirical Rule? Explain. e. How many of the packages weigh exactly 1 pound? Can you think of any explanation for this?

6 7 2 0 6 1 4

9 3 0 9 9 7 1

1 4 0 9 0 7 9

5 2 2 5 2 7 3

9 3 1 3 6 8 8

0 5 2 8 2 7 6

2 8 7 4 9 5 6

8 4 7 7 5 1 6

4 2 4 4 8 8 6

a. You found in Exercise 1.70 that the distribution of this data was relatively “ﬂat,” with each different value from 0 to 9 occurring with nearly equal frequency. Using this fact, what would be your best estimate for the mean of the data set? b. Use the range approximation to guess the value of s for this set. c. Use your calculator to ﬁnd the actual values of 苶x and s. Compare with your estimates in parts a and b.

2.5 A CHECK ON THE CALCULATION OF s

2.28 Social Security Numbers, continued Refer

to the data set in Exercise 2.27. a. Find the percentage of measurements in the intervals x苶 s, 苶x 2s, and 苶x 3s. b. How do the percentages obtained in part a compare with those given by the Empirical Rule? Should they be approximately the same? Explain. 2.29 Survival Times A group of experimental animals is infected with a particular form of bacteria, and their survival time is found to average 32 days, with a standard deviation of 36 days. a. Visualize the distribution of survival times. Do you think that the distribution is relatively moundshaped, skewed right, or skewed left? Explain. b. Within what limits would you expect at least 3/4 of the measurements to lie? 2.30 Survival Times, continued Refer to Exercise 2.29. You can use the Empirical Rule to see why the distribution of survival times could not be moundshaped. a. Find the value of x that is exactly one standard deviation below the mean. b. If the distribution is in fact mound-shaped, approximately what percentage of the measurements should be less than the value of x found in part a? c. Since the variable being measured is time, is it possible to ﬁnd any measurements that are more than one standard deviation below the mean? d. Use your answers to parts b and c to explain why the data distribution cannot be mound-shaped. 2.31 Timber Tracts To estimate the amount of lumber in a tract of timber, an owner EX0231 decided to count the number of trees with diameters exceeding 12 inches in randomly selected 50-by-50foot squares. Seventy 50-by-50-foot squares were chosen, and the selected trees were counted in each tract. The data are listed here: 7 9 3 10 9 6 10

8 6 9 2 6 11 8

7 4 5 7 8 9 8

10 9 9 4 8 11 5

4 10 9 8 8 7 9

8 9 8 5 7 7 9

6 8 7 10 8 11 8

8 8 5 7 9 7 5

9 7 8 7 6 9 9

10 9 8 7 8 13 8

a. Construct a relative frequency histogram to describe the data. b. Calculate the sample mean x苶 as an estimate of m, the mean number of timber trees for all 50-by-50-foot squares in the tract.

❍

c. Calculate s for the data. Construct the intervals 苶x s, x苶 2s, and 苶x 3s. Calculate the percentage of squares falling into each of the three intervals, and compare with the corresponding percentages given by the Empirical Rule and Tchebysheff’s Theorem. 2.32 Tuna Fish, again Refer to Exercise 2.8 and

data set EX0208. The prices of a 6-ounce can or a 7.06 pouch for 14 different brands of water-packed light tuna, based on prices paid nationally in supermarkets are reproduced here.4 .99 1.12

1.92 .63

1.23 .85 .65 .53 1.41 .67 .69 .60 .60 .66

a. Use the range approximation to ﬁnd an estimate of s. b. How does it compare to the computed value of s? 2.33 Old Faithful The data below are 30 waiting times between eruptions of the Old Faithful geyser in Yellowstone National Park.8

EX0233

56 89 51 79 58 82 52 88 52 78 69 75 77 72 71 55 87 53 85 61 93 54 76 80 81 59 86 78 71 77

a. Calculate the range. b. Use the range approximation to approximate the standard deviation of these 30 measurements. c. Calculate the sample standard deviation s. d. What proportion of the measurements lie within two standard deviations of the mean? Within three standard deviations of the mean? Do these proportions agree with the proportions given in Tchebysheff’s Theorem? 2.34 The President’s Kids The table below shows the names of the 42 presidents of the United States along with the number of their children.2

EX0234

Washington Adams Jefferson Madison Monroe J.Q. Adams Jackson

0 5 6 0 2 4 0

Van Buren 4 W.H. Harrison 10 Tyler* 15 Polk 0 Taylor 6 Fillmore* 2 Pierce 3

Buchanan Lincoln A. Johnson Grant Hayes Garﬁeld Arthur

0 4 5 4 8 7 3

Cleveland B. Harrison* McKinley T. Roosevelt* Taft Wilson* Harding

5 3 2 6 3 3 0

Coolidge Hoover F.D. Roosevelt Truman Eisenhower Kennedy L.B. Johnson

Nixon Ford Carter Reagan* G.H.W. Bush Clinton G.W. Bush

2 4 4 4 6 1 2

*Married twice

2 2 6 1 2 3 2

Source: Time Almanac 2007

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

a. Construct a relative frequency histogram to describe the data. How would you describe the shape of this distribution? b. Calculate the mean and the standard deviation for the data set. c. Construct the intervals 苶x s, 苶x 2s, and 苶x 3s. Find the percentage of measurements falling into these three intervals and compare with the corresponding percentages given by Tchebysheff’s Theorem and the Empirical Rule. 2.35 An Archeological Find, again Refer to Exer-

cise 2.17. The percentage of iron oxide in each of ﬁve pottery samples collected at the Island Thorns site was: 1.28

2.39

1.50

1.88

1.51

a. Use the range approximation to ﬁnd an estimate of s, using an appropriate divisor from Table 2.6. b. Calculate the standard deviation s. How close did your estimate come to the actual value of s? 2.36 Brett Favre The number of passes

completed by Brett Favre, quarterback for the Green Bay Packers, was recorded for each of the 16 regular season games in the fall of 2006 (www.espn.com).9

EX0236

15 17 22

31 28 20

25 24 26

22 5 21

22 22

19 24

a. Draw a stem and leaf plot to describe the data. b. Calculate the mean and standard deviation for Brett Favre’s per game pass completions. c. What proportion of the measurements lie within two standard deviations of the mean? CALCULATING THE MEAN AND STANDARD DEVIATION FOR GROUPED DATA (OPTIONAL) 2.37 Suppose that some measurements occur more

than once and that the data x1, x2, . . . , xk are arranged in a frequency table as shown here: Observations

Frequency fi

x1 x2 . . . xk

f1 f2 . . . fk

The formulas for the mean and variance for grouped data are

Sxi fi x苶 , n

where n Sfi

and

(Sx fi)2 Sx2i fi i n s 2 n1 Notice that if each value occurs once, these formulas reduce to those given in the text. Although these formulas for grouped data are primarily of value when you have a large number of measurements, demonstrate their use for the sample 1, 0, 0, 1, 3, 1, 3, 2, 3, 0, 0, 1, 1, 3, 2. a. Calculate 苶x and s 2 directly, using the formulas for ungrouped data. b. The frequency table for the n 15 measurements is as follows: x

0 1 2 3

4 5 2 4

Calculate 苶x and s 2 using the formulas for grouped data. Compare with your answers to part a. 2.38 International Baccalaureate The International Baccalaureate (IB) program is an accelerated academic program offered at a growing number of high schools throughout the country. Students enrolled in this program are placed in accelerated or advanced courses and must take IB examinations in each of six subject areas at the end of their junior or senior year. Students are scored on a scale of 1–7, with 1–2 being poor, 3 mediocre, 4 average, and 5–7 excellent. During its ﬁrst year of operation at John W. North High School in Riverside, California, 17 juniors attempted the IB economics exam, with these results: Exam Grade

Number of Students

7 6 5 4 3

1 4 4 4 4

2.6 MEASURES OF RELATIVE STANDING

2.39 A Skewed Distribution To illustrate the util-

ity of the Empirical Rule, consider a distribution that is heavily skewed to the right, as shown in the accompanying ﬁgure. a. Calculate 苶x and s for the data shown. (NOTE: There are 10 zeros, 5 ones, and so on.) b. Construct the intervals 苶x s, 苶x 2s, and 苶x 3s and locate them on the frequency distribution. c. Calculate the proportion of the n 25 measurements that fall into each of the three intervals. Compare with Tchebysheff’s Theorem and the Empirical Rule. Note that, although the proportion that falls into the interval 苶x s does not agree closely with the Empirical Rule, the

2.6

proportions that fall into the intervals 苶x 2s and 苶x

3s agree very well. Many times this is true, even for non-mound-shaped distributions of data. Distribution for Exercise 2.39 10

9 8

7 Frequency

Calculate the mean and standard deviation for these scores.

❍

5 4

3 2

1 0 0

n 25

MEASURES OF RELATIVE STANDING Sometimes you need to know the position of one observation relative to others in a set of data. For example, if you took an examination with a total of 35 points, you might want to know how your score of 30 compared to the scores of the other students in the class. The mean and standard deviation of the scores can be used to calculate a z-score, which measures the relative standing of a measurement in a data set. Definition

Positive z-score ⇔ x is above the mean. Negative z-score ⇔ x is below the mean.

The sample z-score is a measure of relative standing deﬁned by x 苶x z-score s

A z-score measures the distance between an observation and the mean, measured in units of standard deviation. For example, suppose that the mean and standard deviation of the test scores (based on a total of 35 points) are 25 and 4, respectively. The z-score for your score of 30 is calculated as follows: x 苶x 30 25 1.25 z-score s 4 Your score of 30 lies 1.25 standard deviations above the mean (30 苶x 1.25s).

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

The z-score is a valuable tool for determining whether a particular observation is likely to occur quite frequently or whether it is unlikely and might be considered an outlier. According to Tchebysheff’s Theorem and the Empirical Rule, •

•

z-scores above 3 in absolute value are very unusual. EXAMPLE

2.11

at least 75% and more likely 95% of the observations lie within two standard deviations of their mean: their z-scores are between 2 and 2. Observations with z-scores exceeding 2 in absolute value happen less than 5% of the time and are considered somewhat unlikely. at least 89% and more likely 99.7% of the observations lie within three standard deviations of their mean: their z-scores are between 3 and 3. Observations with z-scores exceeding 3 in absolute value happen less than 1% of the time and are considered very unlikely.

You should look carefully at any observation that has a z-score exceeding 3 in absolute value. Perhaps the measurement was recorded incorrectly or does not belong to the population being sampled. Perhaps it is just a highly unlikely observation, but a valid one nonetheless! Consider this sample of n 10 measurements: 1, 1, 0, 15, 2, 3, 4, 0, 1, 3 The measurement x 15 appears to be unusually large. Calculate the z-score for this observation and state your conclusions. Solution Calculate x苶 3.0 and s 4.42 for the n 10 measurements. Then the z-score for the suspected outlier, x 15, is calculated as

x 苶x 15 3 2.71 z-score s 4.42 Hence, the measurement x 15 lies 2.71 standard deviations above the sample mean, 苶x 3.0. Although the z-score does not exceed 3, it is close enough so that you might suspect that x 15 is an outlier. You should examine the sampling procedure to see whether x 15 is a faulty observation. A percentile is another measure of relative standing and is most often used for large data sets. (Percentiles are not very useful for small data sets.) Definition A set of n measurements on the variable x has been arranged in order of magnitude. The pth percentile is the value of x that is greater than p% of the measurements and is less than the remaining (100 p)%. EXAMPLE

2.12

Suppose you have been notiﬁed that your score of 610 on the Verbal Graduate Record Examination placed you at the 60th percentile in the distribution of scores. Where does your score of 610 stand in relation to the scores of others who took the examination? Scoring at the 60th percentile means that 60% of all the examination scores were lower than your score and 40% were higher. Solution

2.6 MEASURES OF RELATIVE STANDING

❍

In general, the 60th percentile for the variable x is a point on the horizontal axis of the data distribution that is greater than 60% of the measurements and less than the others. That is, 60% of the measurements are less than the 60th percentile and 40% are greater (see Figure 2.14). Since the total area under the distribution is 100%, 60% of the area is to the left and 40% of the area is to the right of the 60th percentile. Remember that the median, m, of a set of data is the middle measurement; that is, 50% of the measurements are smaller and 50% are larger than the median. Thus, the median is the same as the 50th percentile!

The 60th percentile shown on the relative frequency histogram for a data set

●

Relative Frequency

FIGU R E 2 .1 4

60%

40%

x 60th percentile

The 25th and 75th percentiles, called the lower and upper quartiles, along with the median (the 50th percentile), locate points that divide the data into four sets, each containing an equal number of measurements. Twenty-ﬁve percent of the measurements will be less than the lower (ﬁrst) quartile, 50% will be less than the median (the second quartile), and 75% will be less than the upper (third) quartile. Thus, the median and the lower and upper quartiles are located at points on the x-axis so that the area under the relative frequency histogram for the data is partitioned into four equal areas, as shown in Figure 2.15.

Location of quartiles

●

Relative Frequency

FIGU R E 2 .1 5

25%

25% x

Median, m Lower quartile, Q1

Upper quartile, Q3

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

Definition A set of n measurements on the variable x has been arranged in order of magnitude. The lower quartile (ﬁrst quartile), Q1, is the value of x that is greater than one-fourth of the measurements and is less than the remaining three-fourths. The second quartile is the median. The upper quartile (third quartile), Q3, is the value of x that is greater than three-fourths of the measurements and is less than the remaining one-fourth.

For small data sets, it is often impossible to divide the set into four groups, each of which contains exactly 25% of the measurements. For example, when n 10, you would need to have 212 measurements in each group! Even when you can perform this task (for example, if n 12), there are many numbers that would satisfy the preceding deﬁnition, and could therefore be considered “quartiles.” To avoid this ambiguity, we use the following rule to locate sample quartiles. CALCULATING SAMPLE QUARTILES •

•

EXAMPLE

2.13

When the measurements are arranged in order of magnitude, the lower quartile, Q1, is the value of x in position .25(n 1), and the upper quartile, Q3, is the value of x in position .75(n 1). When .25(n 1) and .75(n 1) are not integers, the quartiles are found by interpolation, using the values in the two adjacent positions.†

Find the lower and upper quartiles for this set of measurements: 16, 25, 4, 18, 11, 13, 20, 8, 11, 9 Solution

Rank the n 10 measurements from smallest to largest:

4, 8, 9, 11, 11, 13, 16, 18, 20, 25 Calculate Position of Q1 .25(n 1) .25(10 1) 2.75 Position of Q3 .75(n 1) .75(10 1) 8.25 Since these positions are not integers, the lower quartile is taken to be the value 3/4 of the distance between the second and third ordered measurements, and the upper quartile is taken to be the value 1/4 of the distance between the eighth and ninth ordered measurements. Therefore, Q1 8 .75(9 8) 8 .75 8.75 and Q3 18 .25(20 18) 18 .5 18.5 Because the median and the quartiles divide the data distribution into four parts, each containing approximately 25% of the measurements, Q1 and Q3 are the upper and lower boundaries for the middle 50% of the distribution. We can measure the range of this “middle 50%” of the distribution using a numerical measure called the interquartile range. †

This deﬁnition of quartiles is consistent with the one used in the MINITAB package. Some textbooks use ordinary rounding when ﬁnding quartile positions, whereas others compute sample quartiles as the medians of the upper and lower halves of the data set.

2.6 MEASURES OF RELATIVE STANDING

❍

The interquartile range (IQR) for a set of measurements is the difference between the upper and lower quartiles; that is, IQR Q3 Q1. Definition

For the data in Example 2.13, IQR Q3 Q1 18.50 8.75 9.75. We will use the IQR along with the quartiles and the median in the next section to construct another graph for describing data sets.

How Do I Calculate Sample Quartiles? 1. Arrange the data set in order of magnitude from smallest to largest. 2. Calculate the quartile positions: •

Position of Q1: .25(n 1)

•

Position of Q3: .75(n 1)

Sorted

Position of Q1

Position of Q3

Lower Quartile, Q1

Upper Quartile, Q3

2, 5, 7, 1, 1, 2, 8

1, 1, 2, 2, 5, 7, 8

2nd

6th

5, 0, 1, 3, 1, 5, 5, 2, 4, 4, 1

Position of Q1

Measurements Above and Below

0, 1, 4, 4, 5, 9

1.75

0 and 1

0, 1, 3, 3, 4, 7, 7, 8

2.25

and

6.75

and

1, 1, 2, 5, 6, 6, 7, 9, 9

2.5

and

7.5

and

Q1 0 .75(1) .75

Position of Q3

Measurements Above and Below

5.25

5 and 9

Q3 5 .25(4) 6

Progress Report •

Still having trouble? Try again using the Exercise Reps at the end of this section.

•

Mastered sample quartiles? You can skip the Exercise Reps at the end of this section!

Answers are located on the perforated card at the back of this book.

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

Many of the numerical measures that you have learned are easily found using computer programs or even graphics calculators. The MINITAB command Stat 씮 Basic Statistics 씮 Display Descriptive Statistics (see the section “My MINITAB ” at the end of this chapter) produces output containing the mean, the standard deviation, the median, and the lower and upper quartiles, as well as the values of some other statistics that we have not discussed yet. The data from Example 2.13 produced the MINITAB output shown in Figure 2.16. Notice that the quartiles are identical to the handcalculated values in that example.

FI GU R E 2 .1 6

MINITAB output for the data in Example 2.13

2.7

● Descriptive Statistics: x Variable X

N N* Mean SE Mean 10 0 13.50 1.98

StDev Minimum 6.28 4.00

Q1 Median Q3 Maximum 8.75 12.00 18.50 25.00

THE FIVE-NUMBER SUMMARY AND THE BOX PLOT The median and the upper and lower quartiles shown in Figure 2.15 divide the data into four sets, each containing an equal number of measurements. If we add the largest number (Max) and the smallest number (Min) in the data set to this group, we will have a set of numbers that provide a quick and rough summary of the data distribution. The ﬁve-number summary consists of the smallest number, the lower quartile, the median, the upper quartile, and the largest number, presented in order from smallest to largest: Min

Median Q3

Max

By deﬁnition, one-fourth of the measurements in the data set lie between each of the four adjacent pairs of numbers. The ﬁve-number summary can be used to create a simple graph called a box plot to visually describe the data distribution. From the box plot, you can quickly detect any skewness in the shape of the distribution and see whether there are any outliers in the data set. An outlier may result from transposing digits when recording a measurement, from incorrectly reading an instrument dial, from a malfunctioning piece of equipment, or from other problems. Even when there are no recording or observational errors, a data set may contain one or more valid measurements that, for one reason or another, differ markedly from the others in the set. These outliers can cause a marked distortion in commonly used numerical measures such as 苶x and s. In fact, outliers may themselves contain important information not shared with the other measurements in the set. Therefore, isolating outliers, if they are present, is an important step in any preliminary analysis of a data set. The box plot is designed expressly for this purpose.

2.7 THE FIVE-NUMBER SUMMARY AND THE BOX PLOT

❍

TO CONSTRUCT A BOX PLOT • •

Calculate the median, the upper and lower quartiles, and the IQR for the data set. Draw a horizontal line representing the scale of measurement. Form a box just above the horizontal line with the right and left ends at Q1 and Q3. Draw a vertical line through the box at the location of the median.

A box plot is shown in Figure 2.17. FIGU R E 2 .1 7

●

Box plot

Lower fence

Upper fence

In Section 2.6, the z-score provided boundaries for ﬁnding unusually large or small measurements. You looked for z-scores greater than 2 or 3 in absolute value. The box plot uses the IQR to create imaginary “fences” to separate outliers from the rest of the data set: DETECTING OUTLIERS—OBSERVATIONS THAT ARE BEYOND: • •

Lower fence: Q1 1.5(IQR) Upper fence: Q3 1.5(IQR)

The upper and lower fences are shown with broken lines in Figure 2.17, but they are not usually drawn on the box plot. Any measurement beyond the upper or lower fence is an outlier; the rest of the measurements, inside the fences, are not unusual. Finally, the box plot marks the range of the data set using “whiskers” to connect the smallest and largest measurements (excluding outliers) to the box. TO FINISH THE BOX PLOT • •

EXAMPLE

2.14

Mark any outliers with an asterisk (*) on the graph. Extend horizontal lines called “whiskers” from the ends of the box to the smallest and largest observations that are not outliers.

As American consumers become more careful about the foods they eat, food processors try to stay competitive by avoiding excessive amounts of fat, cholesterol, and sodium in the foods they sell. The following data are the amounts of sodium per slice (in milligrams) for each of eight brands of regular American cheese. Construct a box plot for the data and look for outliers. 340, 300, 520,

340,

320,

290,

260,

330

❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

Solution

The n 8 measurements are ﬁrst ranked from smallest to largest:

260, 290,

300,

320,

330,

340, 340,

520

The positions of the median, Q1, and Q3 are .5(n 1) .5(9) 4.5 .25(n 1) .25(9) 2.25 .75(n 1) .75(9) 6.75 so that m (320 330)/2 325, Q1 290 .25(10) 292.5, and Q3 340. The interquartile range is calculated as IQR Q3 Q1 340 292.5 47.5 Calculate the upper and lower fences: Lower fence: 292.5 1.5(47.5) 221.25 Upper fence: 340 1.5(47.5) 411.25 The value x 520, a brand of cheese containing 520 milligrams of sodium, is the only outlier, lying beyond the upper fence. The box plot for the data is shown in Figure 2.18. The outlier is marked with an asterisk (*). Once the outlier is excluded, we ﬁnd (from the ranked data set) that the smallest and largest measurements are x 260 and x 340. These are the two values that form the whiskers. Since the value x 340 is the same as Q3, there is no whisker on the right side of the box.

FI GU R E 2 .1 8

Box plot for Example 2.14

●

250

300

350

400 Sodium

450

500

550

Now would be a good time to try the Building a Box Plot applet. The applet in Figure 2.19 shows a dotplot of the data in Example 2.14. Using the button, you will see a step-by-step description explaining how the box plot is constructed. We will use this applet again for the MyApplet Exercises at the end of the chapter.

2.7 THE FIVE-NUMBER SUMMARY AND THE BOX PLOT

FIGU R E 2 .1 9

Building a Box Plot applet

❍

●

You can use the box plot to describe the shape of a data distribution by looking at the position of the median line compared to Q1 and Q3, the left and right ends of the box. If the median is close to the middle of the box, the distribution is fairly symmetric, providing equal-sized intervals to contain the two middle quarters of the data. If the median line is to the left of center, the distribution is skewed to the right; if the median is to the right of center, the distribution is skewed to the left. Also, for most skewed distributions, the whisker on the skewed side of the box tends to be longer than the whisker on the other side. We used the MINITAB command Graph 씮 Boxplot to draw two box plots, one for the sodium contents of the eight brands of cheese in Example 2.14, and another for ﬁve brands of fat-free cheese with these sodium contents: 300,

300,

320,

290,

180

The two box plots are shown together in Figure 2.20. Look at the long whisker on the left side of both box plots and the position of the median lines. Both distributions are skewed to the left; that is, there are a few unusually small measurements. The regular cheese data, however, also show one brand (x 520) with an unusually large amount of sodium. In general, it appears that the sodium content of the fat-free brands is lower than that of the regular brands, but the variability of the sodium content for regular cheese (excluding the outlier) is less than that of the fat-free brands.

FIGU R E 2 .2 0

MINITAB output for regular and fat-free cheese

●

Fat-Free

Type

Regular

200

250

300

350 Sodium

400

450

500

550

84 ❍

2.7

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

EXERCISES EXERCISE REPS These exercises refer back to the MyPersonal Trainer section on page 79. 2.40 Below you will ﬁnd two practice data sets. Fill in the blanks to ﬁnd the necessary

quartiles. Data Set

Sorted

Position of Q1

Position of Q3

Lower Quartile, Q1

Upper Quartile, Q3

.13, .76, .34, .88, .21, .16, .28 2.3, 1.0, 2.1, 6.5, 2.8, 8.8, 1.7, 2.9, 4.4, 5.1, 2.0

2.41 Below you will ﬁnd three data sets that have already been sorted. Fill in the blanks

to ﬁnd the upper and lower quartiles. Sorted Data Set

Position of Q1

Measurements Above and Below

Position of Q3

Measurements Above and Below

1, 1.5, 2, 2, 2.2

and

0, 1.7, 1.8, 3.1, 3.2,

and

7, 8, 8.8, 8.9, 9, 10 .23, .30, .35, .41, .56, .58, .76, .80

BASIC TECHNIQUES

APPLICATIONS

2.42 Given the following data set: 8, 7, 1, 4, 6, 6, 4,

2.46 If you scored at the 69th percentile on a place-

5, 7, 6, 3, 0 a. Find the ﬁve-number summary and the IQR. b. Calculate 苶x and s. c. Calculate the z-score for the smallest and largest observations. Is either of these observations unusually large or unusually small?

ment test, how does your score compare with others?

2.43 Find the ﬁve-number summary and the IQR for

these data: 19, 12, 16, 0, 14, 9, 6, 1, 12, 13, 10, 19, 7, 5, 8 2.44 Construct a box plot for these data and identify

any outliers: 25, 22, 26, 23, 27, 26, 28, 18, 25, 24, 12 2.45 Construct a box plot for these data and identify

any outliers: 3, 9, 10, 2, 6, 7, 5, 8, 6, 6, 4, 9, 22

2.47 Mercury Concentration in Dolphins

Environmental scientists are increasingly concerned with the accumulation of toxic elements in marine mammals and the transfer of such elements to the animals’ offspring. The striped dolphin (Stenella coeruleoalba), considered to be the top predator in the marine food chain, was the subject of one such study. The mercury concentrations (micrograms/gram) in the livers of 28 male striped dolphins were as follows:

EX0247

1.70 1.72 8.80 5.90 101.00 85.40 118.00

183.00 168.00 218.00 180.00 264.00 481.00 485.00

221.00 406.00 252.00 329.00 316.00 445.00 278.00

286.00 315.00 241.00 397.00 209.00 314.00 318.00

❍

2.7 THE FIVE-NUMBER SUMMARY AND THE BOX PLOT

a. b. c. d.

Calculate the ﬁve-number summary for the data. Construct a box plot for the data. Are there any outliers? If you knew that the ﬁrst four dolphins were all less than 3 years old, while all the others were more than 8 years old, would this information help explain the difference in the magnitude of those four observations? Explain.

2.48 Hamburger Meat The weights (in pounds) of

the 27 packages of ground beef from Exercise 2.24 (see data set EX0224) are listed here in order from smallest to largest: .75 .93 1.08 1.18

.83 .96 1.08 1.18

.87 .96 1.12 1.24

.89 .97 1.12 1.28

.89 .98 1.14 1.38

.89 .99 1.14 1.41

.92 1.06 1.17

a. Conﬁrm the values of the mean and standard deviation, calculated in Exercise 2.24 as 苶x 1.05 and s .17. b. The two largest packages of meat weigh 1.38 and 1.41 pounds. Are these two packages unusually heavy? Explain. c. Construct a box plot for the package weights. What does the position of the median line and the length of the whiskers tell you about the shape of the distribution? 2.49 Comparing NFL Quarterbacks How

2.50 Presidential Vetoes The set of presidential

vetoes in Exercise 1.47 and data set EX0147 is listed here, along with a box plot generated by MINITAB. Use the box plot to describe the shape of the distribution and identify any outliers. Washington 2 J. Adams 0 Jefferson 0 Madison 5 Monroe 1 J. Q. Adams 0 Jackson 5 Van Buren 0 W. H. Harrison 0 Tyler 6 Polk 2 Taylor 0 Fillmore 0 Pierce 9 Buchanan 4 Lincoln 2 A. Johnson 21 Grant 45 Hayes 12 Garﬁeld 0 Arthur 4 Cleveland 304

B. Harrison Cleveland McKinley T. Roosevelt Taft Wilson Harding Coolidge Hoover F. D. Roosevelt Truman Eisenhower Kennedy L. Johnson Nixon Ford Carter Reagan G. H. W. Bush Clinton G. W. Bush

19 42 6 42 30 33 5 20 21 372 180 73 12 16 26 48 13 39 29 36 1

Source: The World Almanac and Book of Facts 2007

Box plot for Exercise 2.50

does Brett Favre, quarterback for the Green Bay Packers, compare to Peyton Manning, quarterback for the Indianapolis Colts? The table below shows the number of completed passes for each athlete during the 2006 NFL football season:9

EX0249

Brett Favre

Peyton Manning

15 31 25 22 22 19

25 26 14 21 20 25

17 28 24 5 22 24

22 20 26 21

32 30 27 20 14 21

25 29 21 22

a. Calculate ﬁve-number summaries for the number of passes completed by both Brett Favre and Peyton Manning. b. Construct box plots for the two sets of data. Are there any outliers? What do the box plots tell you about the shapes of the two distributions? c. Write a short paragraph comparing the number of pass completions for the two quarterbacks.

100

200 Vetoes

300

400

2.51 Survival Times Altman and Bland report the survival times for patients with active hepatitis, half treated with prednisone and half receiving no treatment.10 The survival times (in months) (Exercise 1.73 and EX0173) are adapted from their data for those treated with prednisone.

8 11 52 57 65 87 93 97 109 120

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

127 133 139 142 144 147 148 157 162 165

a. Construct a box plot for the monthly utility costs. b. What does the box plot tell you about the distribution of utility costs for this household? 2.53 What’s Normal? again Refer to Exercise

a. Can you tell by looking at the data whether it is roughly symmetric? Or is it skewed? b. Calculate the mean and the median. Use these measures to decide whether or not the data are symmetric or skewed. c. Draw a box plot to describe the data. Explain why the box plot conﬁrms your conclusions in part b. 2.52 Utility Bills in Southern California,

again The monthly utility bills for a household in Riverside, California, were recorded for 12 consecutive months starting in January 2006:

EX0252

Month

Amount ($)

January February March April May June

$266.63 163.41 219.41 162.64 187.16 289.17

Month July August September October November December

1.67 and data set EX0167. In addition to the normal body temperature in degrees Fahrenheit for the 130 individuals, the data record the gender of the individuals. Box plots for the two groups, male and female, are shown below:11 Box plots for Exercise 2.53

Male

Gender

86 ❍

Female

Amount ($) $306.55 335.48 343.50 226.80 208.99 230.46

100

101

Temperature

How would you describe the similarities and differences between male and female temperatures in this data set?

CHAPTER REVIEW

❍

CHAPTER REVIEW Key Concepts and Formulas I.

Measures of the Center of a Data Distribution

1. Arithmetic mean (mean) or average a. Population: m

Sx b. Sample of n measurements: x苶 i n

2. Median; position of the median .5(n 1) 3. Mode 4. The median may be preferred to the mean if the data are highly skewed. II. Measures of Variability

1. Range: R largest smallest 2. Variance a. Population of N measurements:

S(xi m)2 s2 N b. Sample of n measurements: (Sxi)2 2 Sx i n S(xi x苶 )2 s2 n1 n1

3. Standard deviation 苶2 a. Population: s 兹s b. Sample: s 兹苶 s2 4. A rough approximation for s can be calculated as s ⬇ R/4. The divisor can be adjusted depending on the sample size. III. Tchebysheff’s Theorem and the Empirical Rule

1. Use Tchebysheff’s Theorem for any data set, regardless of its shape or size. a. At least 1 (1/k2) of the measurements lie within k standard deviations of the mean. b. This is only a lower bound; there may be more measurements in the interval. 2. The Empirical Rule can be used only for relatively mound-shaped data sets. Approximately

68%, 95%, and 99.7% of the measurements are within one, two, and three standard deviations of the mean, respectively. IV. Measures of Relative Standing

x 苶x 1. Sample z-score: z s 2. pth percentile; p% of the measurements are smaller, and (100 p)% are larger. 3. Lower quartile, Q1; position of Q1 .25 (n 1)

4. Upper quartile, Q3; position of Q3 .75 (n 1) 5. Interquartile range: IQR Q3 Q1 V. The Five-Number Summary and Box Plots

1. The ﬁve-number summary: Min

Median Q3

Max

One-fourth of the measurements in the data set lie between each of the four adjacent pairs of numbers. 2. Box plots are used for detecting outliers and shapes of distributions. 3. Q1 and Q3 form the ends of the box. The median line is in the interior of the box. 4. Upper and lower fences are used to ﬁnd outliers, observations that lie outside these fences. a. Lower fence: Q1 1.5(IQR)

b. Upper fence: Q3 1.5(IQR) 5. Outliers are marked on the box plot with an asterisk (*). 6. Whiskers are connected to the box from the smallest and largest observations that are not outliers. 7. Skewed distributions usually have a long whisker in the direction of the skewness, and the median line is drawn away from the direction of the skewness.

88 ❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

62.0, 62.0, 64.5, 48.5, 57.5, 61.0, 45.5, 47.0, 33.0 27.0, 27.0, 24.0, 16.5, 25.0, 27.5, 14.0, 18.5, 17.0

●

Now click on the Variables box and select both columns from the list on the left. (You can click on the Graphs option and choose one of several graphs if you like. You may also click on the Statistics option to select the statistics you would like to see displayed.) Click OK. A display of descriptive statistics for both columns will appear in the Session window (see Figure 2.22). You may print this output using File 씮 Print Session Window if you choose. To examine the distribution of the two variables and look for outliers, you can create box plots using the command Graph 씮 Boxplot 씮 One Y 씮 Simple. Click OK. Select the appropriate column of measurements in the Dialog box (see Figure 2.23). You can change the appearance of the box plot in several ways. Scale 씮 Axes and Ticks will allow you to transpose the axes and orient the box plot horizontally, when you check the box marked “Transpose value and category scales.” Multiple Graphs

MY MINITAB

❍

●

FIGU R E 2 .2 3

●

FIGU R E 2 .2 4

●

90 ❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

Supplementary Exercises 2.54 Raisins The number of raisins in each

of 14 miniboxes (1/2-ounce size) was counted for a generic brand and for Sunmaid brand raisins. The two data sets are shown here:

EX0254

Generic Brand

Sunmaid

25 26 26 26

25 28 25 28

26 25 28 28 28 27 27 24 25 26

29 24 24 24 28 22 28 30 27 24

a. What are the mean and standard deviation for the generic brand? b. What are the mean and standard deviation for the Sunmaid brand? c. Compare the centers and variabilities of the two brands using the results of parts a and b. 2.55 Raisins, continued Refer to Exercise 2.54.

a. Find the median, the upper and lower quartiles, and the IQR for each of the two data sets. b. Construct two box plots on the same horizontal scale to compare the two sets of data. c. Draw two stem and leaf plots to depict the shapes of the two data sets. Do the box plots in part b verify these results? d. If we can assume that none of the boxes of raisins are being underﬁlled (that is, they all weigh approximately 1/2 ounce), what do your results say about the average number of raisins for the two brands? 2.56 TV Viewers The number of television

viewing hours per household and the prime viewing times are two factors that affect television advertising income. A random sample of 25 households in a particular viewing area produced the following estimates of viewing hours per household:

EX0256

3.0 6.5 5.0 7.5 9.0

6.0 8.0 12.0 5.0 2.0

7.5 4.0 1.0 10.0 6.5

15.0 5.5 3.5 8.0 1.0

12.0 6.0 3.0 3.5 5.0

a. Scan the data and use the range to ﬁnd an approximate value for s. Use this value to check your calculations in part b. b. Calculate the sample mean x苶 and the sample standard deviation s. Compare s with the approximate value obtained in part a.

c. Find the percentage of the viewing hours per household that falls into the interval x苶 2s. Compare with the corresponding percentage given by the Empirical Rule. 2.57 A Recurring Illness Refer to Exercise 1.26 and data set EX0126. The lengths of time (in months) between the onset of a particular illness and its recurrence were recorded: 2.1 9.0 14.7 19.2 4.1 7.4 14.1 8.7 1.6 3.7

4.4 2.0 9.6 6.9 18.4 .2 1.0 24.0 3.5 12.6

2.7 6.6 16.7 4.3 .2 8.3 2.4 1.4 11.4 23.1

32.3 3.9 7.4 3.3 6.1 .3 2.4 8.2 18.0 5.6

9.9 1.6 8.2 1.2 13.5 1.3 18.0 5.8 26.7 .4

a. Find the range. b. Use the range approximation to ﬁnd an approximate value for s. c. Compute s for the data and compare it with your approximation from part b. 2.58 A Recurring Illness, continued Refer to Exercise 2.57. a. Examine the data and count the number of observations that fall into the intervals x苶 s, 苶x 2s, and 苶x 3s. b. Do the percentages that fall into these intervals agree with Tchebysheff’s Theorem? With the Empirical Rule? c. Why might the Empirical Rule be unsuitable for describing these data? 2.59 A Recurring Illness, again Find the median and the lower and upper quartiles for the data on times until recurrence of an illness in Exercise 2.57. Use these descriptive measures to construct a box plot for the data. Use the box plot to describe the data distribution. 2.60 Tuna Fish, again Refer to Exercise 2.8. The

prices of a 6-ounce can or a 7.06-ounce pouch for 14 different brands of water-packed light tuna, based on prices paid nationally in supermarkets, are reproduced here.4 .99 1.12

1.92 .63

1.23 .67

.85 .69

.65 .60

.53 .60

1.41 .66

❍

SUPPLEMENTARY EXERCISES

a. Calculate the ﬁve-number summary. b. Construct a box plot for the data. Are there any outliers? c. The value x 1.92 looks large in comparison to the other prices. Use a z-score to decide whether this is an unusually expensive brand of tuna. 2.61 Electrolysis An analytical chemist wanted to use electrolysis to determine the number of moles of cupric ions in a given volume of solution. The solution was partitioned into n 30 portions of .2 milliliter each, and each of the portions was tested. The average number of moles of cupric ions for the n 30 portions was found to be .17 mole; the standard deviation was .01 mole. a. Describe the distribution of the measurements for the n 30 portions of the solution using Tchebysheff’s Theorem. b. Describe the distribution of the measurements for the n 30 portions of the solution using the Empirical Rule. (Do you expect the Empirical Rule to be suitable for describing these data?) c. Suppose the chemist had used only n 4 portions of the solution for the experiment and obtained the readings .15, .19, .17, and .15. Would the Empirical Rule be suitable for describing the n 4 measurements? Why? 2.62 Chloroform According to the EPA, chloroform, which in its gaseous form is suspected of being a cancer-causing agent, is present in small quantities in all of the country’s 240,000 public water sources. If the mean and standard deviation of the amounts of chloroform present in the water sources are 34 and 53 micrograms per liter, respectively, describe the distribution for the population of all public water sources. 2.63 Aptitude Tests In contrast to aptitude tests, which are predictive measures of what one can accomplish with training, achievement tests tell what an individual can do at the time of the test. Mathematics achievement test scores for 400 students were found to have a mean and a variance equal to 600 and 4900, respectively. If the distribution of test scores was mound-shaped, approximately how many of the scores would fall into the interval 530 to 670? Approximately how many scores would be expected to fall into the interval 460 to 740? 2.64 Sleep and the College Student How much sleep do you get on a typical school night? A group of 10 college students were asked to report the number of

hours that they slept on the previous night with the following results: 7,

7.25,

8.5,

6.75,

a. Find the mean and the standard deviation of the number of hours of sleep for these 10 students. b. Calculate the z-score for the largest value (x 8.5). Is this an unusually sleepy college student? c. What is the most frequently reported measurement? What is the name for this measure of center? d. Construct a box plot for the data. Does the box plot conﬁrm your results in part b? [HINT: Since the z-score and the box plot are two unrelated methods for detecting outliers, and use different types of statistics, they do not necessarily have to (but usually do) produce the same results.] 2.65 Gas Mileage The miles per gallon (mpg)

for each of 20 medium-sized cars selected from a production line during the month of March follow.

EX0265

23.1 20.2 24.7 25.9 24.9

21.3 24.4 22.7 24.7 22.2

23.6 25.3 26.2 24.4 22.9

23.7 27.0 23.2 24.2 24.6

a. What are the maximum and minimum miles per gallon? What is the range? b. Construct a relative frequency histogram for these data. How would you describe the shape of the distribution? c. Find the mean and the standard deviation. d. Arrange the data from smallest to largest. Find the z-scores for the largest and smallest observations. Would you consider them to be outliers? Why or why not? e. What is the median? f. Find the lower and upper quartiles. 2.66 Gas Mileage, continued Refer to Exercise

2.65. Construct a box plot for the data. Are there any outliers? Does this conclusion agree with your results in Exercise 2.65? 2.67 Polluted Seawater Petroleum pollution in seas and oceans stimulates the growth of some types of bacteria. A count of petroleumlytic micro-organisms (bacteria per 100 milliliters) in ten portions of seawater gave these readings: 49,

70,

54,

67,

59,

40,

61,

69,

71,

92 ❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

a. Guess the value for s using the range approximation. b. Calculate 苶x and s and compare with the range approximation of part a. c. Construct a box plot for the data and use it to describe the data distribution. 2.68 Basketball Attendances at a high school’s

basketball games were recorded and found to have a sample mean and variance of 420 and 25, respectively. Calculate 苶x s, 苶x 2s, and 苶x 3s and then state the approximate fractions of measurements you would expect to fall into these intervals according to the Empirical Rule. 2.69 SAT Tests The College Board’s verbal and mathematics scholastic aptitude tests are scored on a scale of 200 to 800. Although the tests were originally designed to produce mean scores of approximately 500, the mean verbal and math scores in recent years have been as low as 463 and 493, respectively, and have been trending downward. It seems reasonable to assume that a distribution of all test scores, either verbal or math, is mound-shaped. If s is the standard deviation of one of these distributions, what is the largest value (approximately) that s might assume? Explain. 2.70 Summer Camping A favorite summer pastime

for many Americans is camping. In fact, camping has become so popular at the California beaches that reservations must sometimes be made months in advance! Data from a USA Today Snapshot is shown below.13 Favorite Camping Activity 50% 40% 30%

2.71 Long-Stemmed Roses A strain of long-

stemmed roses has an approximate normal distribution with a mean stem length of 15 inches and standard deviation of 2.5 inches. a. If one accepts as “long-stemmed roses” only those roses with a stem length greater than 12.5 inches, what percentage of such roses would be unacceptable? b. What percentage of these roses would have a stem length between 12.5 and 20 inches? 2.72 Drugs for Hypertension A pharmaceutical company wishes to know whether an experimental drug being tested in its laboratories has any effect on systolic blood pressure. Fifteen randomly selected subjects were given the drug, and their systolic blood pressures (in millimeters) are recorded.

EX0272

172 140 123 130 115

148 108 129 137 161

123 152 133 128 142

a. Guess the value of s using the range approximation. b. Calculate 苶x and s for the 15 blood pressures. c. Find two values, a and b, such that at least 75% of the measurements fall between a and b. 2.73 Lumber Rights A company interested in lumbering rights for a certain tract of slash pine trees is told that the mean diameter of these trees is 14 inches with a standard deviation of 2.8 inches. Assume the distribution of diameters is roughly mound-shaped. a. What fraction of the trees will have diameters between 8.4 and 22.4 inches? b. What fraction of the trees will have diameters greater than 16.8 inches?

20% 2.74 Social Ambivalence The following data represent the social ambivalence scores for 15 people as measured by a psychological test. (The higher the score, the stronger the ambivalence.)

10%

EX0274

0% Gathering Enjoying at campfire scenery

Being outside

The Snapshot also reports that men go camping 2.9 times a year, women go 1.7 times a year; and men are more likely than women to want to camp more often. What does the magazine mean when they talk about 2.9 or 1.7 times a year?

9 14 10 8 11

13 15 4 19 17

12 11 10 13 9

a. Guess the value of s using the range approximation. b. Calculate 苶x and s for the 15 social ambivalence scores.

❍

SUPPLEMENTARY EXERCISES

c. What fraction of the scores actually lie in the interval 苶x 2s? 2.75 TV Commercials The mean duration of televi-

sion commercials on a given network is 75 seconds, with a standard deviation of 20 seconds. Assume that durations are approximately normally distributed. a. What is the approximate probability that a commercial will last less than 35 seconds? b. What is the approximate probability that a commercial will last longer than 55 seconds?

Number of Foxes, f

Mark McGwire’s record of 70 home runs hit in a single season. At the end of the 2003 major league baseball season, the number of home runs hit per season by each of four major league superstars over each player’s career were recorded, and are shown in the box plots below:14

Ruth

69 17

a. Construct a relative frequency histogram for x, the number of parasites per fox. b. Calculate 苶x and s for the sample. c. What fraction of the parasite counts fall within two standard deviations of the mean? Within three standard deviations? Do these results agree with Tchebysheff’s Theorem? With the Empirical Rule?

Player

McGwire

2.76 Parasites in Foxes A random sample of 100 foxes was examined by a team of veterinarians to determine the prevalence of a particular type of parasite. Counting the number of parasites per fox, the veterinarians found that 69 foxes had no parasites, 17 had one parasite, and so on. A frequency tabulation of the data is given here: Number of Parasites, x

Sosa

Bonds

40 Homers

Write a short paragraph comparing the home run hitting patterns of these four players. 2.80 Barry Bonds In the seasons that followed his 2001 record-breaking season, Barry Bonds hit 46, 45, 45, 5, and 26 homers, respectively (www.espn.com).14 Two boxplots, one of Bond’s homers through 2001, and a second including the years 2002–2006, follow.

EX0280

2.77 College Teachers Consider a population con-

sisting of the number of teachers per college at small 2-year colleges. Suppose that the number of teachers per college has an average m 175 and a standard deviation s 15. a. Use Tchebysheff’s Theorem to make a statement about the percentage of colleges that have between 145 and 205 teachers. b. Assume that the population is normally distributed. What fraction of colleges have more than 190 teachers? 2.78 Is It Accurate? From the following data, a student calculated s to be .263. On what grounds might we doubt his accuracy? What is the correct value (to the nearest hundredth)?

EX0278

17.2 17.1

17.1 17.0

17.0 17.1

17.1 16.9

16.9 17.0

17.0 17.1

17.1 17.3

17.0 17.2

17.3 17.4

17.2 17.1

2.79 Homerun Kings In the summer of EX0279

2001, Barry Bonds began his quest to break

2001

2006

30 40 50 Homers by Barry Bonds

The statistics used to construct these boxplots are given in the table. Years

Min

Median

IQR

Max

2001 2006

16 5

25.00 25.00

34.00 34.00

41.50 45.00

16.5 20.0

73 73

16 21

a. Calculate the upper fences for both of these boxplots. b. Can you explain why the record number of homers is an outlier in the 2001 boxplot, but not in the 2006 boxplot?

94 ❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

2.81 Ages of Pennies Here are the ages of 50 pen-

Stem-and-Leaf Display: Liters

nies from Exercise 1.45 and data set EX0145. The data have been sorted from smallest to largest.

Stem-and-leaf of Liters Leaf Unit 0.10

0 0 2 6 19

1 4 2 4 5 4 8 4 12 5 (4) 5 14 5 11 5 7 5 4 6 2 6 1 6

0 0 3 8 20

0 1 3 9 20

0 0 0 0 1 1 1 1 3 4 4 5 9 10 16 17 21 22 23 25

0 0 0 1 2 2 5 5 5 17 19 19 25 28 36

a. What is the average age of the pennies? b. What is the median age of the pennies? c. Based on the results of parts a and b, how would you describe the age distribution of these 50 pennies? d. Construct a box plot for the data set. Are there any outliers? Does the box plot conﬁrm your description of the distribution’s shape? 2.82 Snapshots Here are a few facts reported as Snapshots in USA Today. • The median hourly pay for salespeople in the building supply industry is $10.41.15

• Sixty-nine percent of U.S. workers ages 16 and older work at least 40 hours per week.16 • Seventy-ﬁve percent of all Associate Professors of Mathematics in the U.S. earn $91,823 or less.17

2.83 Breathing Patterns Research

psychologists are interested in ﬁnding out whether a person’s breathing patterns are affected by a particular experimental treatment. To determine the general respiratory patterns of the n 30 people in the study, the researchers collected some baseline measurements—the total ventilation in liters of air per minute adjusted for body size—for each person before the treatment. The data are shown here, along with some descriptive tools generated by MINITAB.

EX0283

5.23 5.92 4.67

4.79 5.38 5.77

5.83 6.34 5.84

5.37 5.12 6.19

4.35 5.14 5.58

5.54 4.72 5.72

6.04 5.17 5.16

5.48 4.99 5.32

6.58 4.82 4.51 5.70 4.96 5.63

Descriptive Statistics: Liters Variable Liters

N 30

N* 0

Mean 5.3953

SE Mean 0.0997

StDev 0.5462

Minimum Q1 Median Q3 Variable Maximum 4.3500 4.9825 5.3750 5.7850 Liters 6.5800

3 5 677 899 1111 2333 455 6777 889 01 3 5

a. Summarize the characteristics of the data distribution using the MINITAB output. b. Does the Empirical Rule provide a good description of the proportion of measurements that fall within two or three standard deviations of the mean? Explain. c. How large or small does a ventilation measurement have to be before it is considered unusual? 2.84 Arranging Objects The following data

are the response times in seconds for n 25 ﬁrst graders to arrange three objects by size.

EX0284

5.2 4.2 3.1 3.6 4.7

Identify the variable x being measured, and any percentiles you can determine from this information.

N 30

3.8 4.1 2.5 3.9 3.3

5.7 4.3 3.0 4.8 4.2

3.9 4.7 4.4 5.3 3.8

3.7 4.3 4.8 4.2 5.4

a. Find the mean and the standard deviation for these 25 response times. b. Order the data from smallest to largest. c. Find the z-scores for the smallest and largest response times. Is there any reason to believe that these times are unusually large or small? Explain. 2.85 Arranging Objects, continued Refer to

Exercise 2.84. a. Find the five-number summary for this data set. b. Construct a box plot for the data. c. Are there any unusually large or small response times identified by the box plot? d. Construct a stem and leaf display for the response times. How would you describe the shape of the distribution? Does the shape of the box plot confirm this result?

MYAPPLET EXERCISES

❍

Exercises 2.86 Refer to Data Set #1 in the How Extreme Val-

ues Affect the Mean and Median applet. This applet loads with a dotplot for the following n 5 observations: 2, 5, 10, 11, 12. a. Use your mouse to move the value x 12 to the left until it is smaller than the value x 11. b. As the value of x gets smaller, what happens to the sample mean? c. As the value of x gets smaller, at what point does the value of the median ﬁnally change? d. As you move the green dot, what are the largest and smallest possible values for the median? 2.88 Refer to Data Set #3 in the How Extreme

Values Affect the Mean and Median applet. This applet loads with a dotplot for the following n 5 observations: 27, 28, 32, 34, 37. a. What are the mean and median for this data set? b. Use your mouse to change the value x 27 (the moveable green dot) to x 25. What are the mean and median for the new data set? c. Use your mouse to move the green dot to x 5. When the smallest value is extremely small compared to the other observations, which is larger, the mean or the median? d. At what value of x does the mean equal the median? e. What are the smallest and largest possible values for the median? f. What effect does an extremely small value have on the mean? What effect does it have on the median? 2.89 Refer to the Why Divide by n 1 applet. The

ﬁrst applet on the page randomly selects sample of

second applet on the page randomly selects sample of n 10 from the same population in which the standard deviation is s 29.2. a. Repeat the instructions in part c and d of Exercise 2.89. b. Based on your simulation, when the sample size is larger, does it make as much difference whether you divide by n or n 1 when computing the sample standard deviation? 2.91 If you have not yet done so, use the ﬁrst Building a Box Plot applet to construct a box plot for the data in Example 2.14. a. Compare the ﬁnished box plot to the plot shown in Figure 2.18. b. How would you describe the shape of the data distribution? c. Are there any outliers? If so, what is the value of the unusual observation? 2.92 Use the second Building a Box Plot applet to

construct a box plot for the data in Example 2.13. a. How would you describe the shape of the data distribution? b. Use the box plot to approximate the values of the median, the lower quartile, and the upper quartile. Compare your results to the actual values calculated in Example 2.13.

96 ❍

CHAPTER 2 DESCRIBING DATA WITH NUMERICAL MEASURES

CASE STUDY Batting

The Boys of Summer Which baseball league has had the best hitters? Many of us have heard of baseball greats like Stan Musial, Hank Aaron, Roberto Clemente, and Pete Rose of the National League and Ty Cobb, Babe Ruth, Ted Williams, Rod Carew, and Wade Boggs of the American League. But have you ever heard of Willie Keeler, who batted .432 for the Baltimore Orioles, or Nap Lajoie, who batted .422 for the Philadelphia A’s? The batting averages for the batting champions of the National and American Leagues are given on the Student Companion Website. The batting averages for the National League begin in 1876 with Roscoe Barnes, whose batting average was .403 when he played with the Chicago Cubs. The last entry for the National League is for the year 2006, when Freddy Sanchez of the Pittsburgh Pirates averaged .344. The American League records begin in 1901 with Nap Lojoie of the Philadelphia A’s, who batted .422, and end in 2006 with Joe Mauer of the Minnesota Twins, who batted .347.18 How can we summarize the information in this data set? 1. Use MINITAB or another statistical software package to describe the batting averages for the American and National League batting champions. Generate any graphics that may help you in interpreting these data sets. 2. Does one league appear to have a higher percentage of hits than the other? Do the batting averages of one league appear to be more variable than the other? 3. Are there any outliers in either league? 4. Summarize your comparison of the two baseball leagues.

Describing Bivariate Data

GENERAL OBJECTIVES Sometimes the data that are collected consist of observations for two variables on the same experimental unit. Special techniques that can be used in describing these variables will help you identify possible relationships between them.

CHAPTER INDEX ● The best-ﬁtting line (3.4) ● Bivariate data (3.1) ● Covariance and the correlation coefficient (3.4) ● Scatterplots for two quantitative variables (3.3) ● Side-by-side pie charts, comparative line charts (3.2) ● Side-by-side bar charts, stacked bar charts (3.2)

How Do I Calculate the Correlation Coefficient? How Do I Calculate the Regression Line?

Do You Think Your Dishes Are Really Clean? Does the price of an appliance, such as a dishwasher, convey something about its quality? In the case study at the end of this chapter, we rank 20 different brands of dishwashers according to their prices, and then we rate them on various characteristics, such as how the dishwasher performs, how much noise it makes, its cost for either gas or electricity, its cycle time, and its water use. The techniques presented in this chapter will help to answer our question.

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

BIVARIATE DATA

3.1

Very often researchers are interested in more than just one variable that can be measured during their investigation. For example, an auto insurance company might be interested in the number of vehicles owned by a policyholder as well as the number of drivers in the household. An economist might need to measure the amount spent per week on groceries in a household and also the number of people in that household. A real estate agent might measure the selling price of a residential property and the square footage of the living area. When two variables are measured on a single experimental unit, the resulting data are called bivariate data. How should you display these data? Not only are both variables important when studied separately, but you also may want to explore the relationship between the two variables. Methods for graphing bivariate data, whether the variables are qualitative or quantitative, allow you to study the two variables together. As with univariate data, you use different graphs depending on the type of variables you are measuring.

“Bi” means “two.” Bivariate data generate pairs of measurements.

GRAPHS FOR QUALITATIVE VARIABLES

3.2

When at least one of the two variables is qualitative, you can use either simple or more intricate pie charts, line charts, and bar charts to display and describe the data. Sometimes you will have one qualitative and one quantitative variable that have been measured in two different populations or groups. In this case, you can use two sideby-side pie charts or a bar chart in which the bars for the two populations are placed side by side. Another option is to use a stacked bar chart, in which the bars for each category are stacked on top of each other. EXAMPLE

TABLE 3.1

3.1

Are professors in private colleges paid more than professors at public colleges? The data in Table 3.1 were collected from a sample of 400 college professors whose rank, type of college, and salary were recorded.1 The number in each cell is the average salary (in thousands of dollars) for all professors who fell into that category. Use a graph to answer the question posed for this sample.

●

Salaries of Professors by Rank and Type of College Full Professor

Associate Professor

Assistant Professor

94.8 118.1

65.9 76.0

56.4 65.1

Public Private

Source: Digest of Educational Statistics

To display the average salaries of these 400 professors, you can use a side-by-side bar chart, as shown in Figure 3.1. The height of the bars is the average salary, with each pair of bars along the horizontal axis representing a different professorial rank. Salaries are substantially higher for full professors in private colleges, however, there are less striking differences at the lower two ranks.

Solution

3.2 GRAPHS FOR QUALITATIVE VARIABLES

FIGU R E 3 .1

● School Public Private

120

Average Salary ($ Thousands)

Comparative bar charts for Example 3.1

❍

100 80 60 40 20

0 School Rank

EXAMPLE

3.2

TABLE 3.2

Public Private Full

Public Private Associate

Public Private Assistant

Along with the salaries for the 400 college professors in Example 3.1, the researcher recorded two qualitative variables for each professor: rank and type of college. Table 3.2 shows the number of professors in each of the 2 3 6 categories. Use comparative charts to describe the data. Do the private colleges employ as many highranking professors as the public colleges do?

●

Number of Professors by Rank and Type of College Full Professor

Associate Professor

Assistant Professor

Total

24 60

57 78

69 112

150 250

Public Private

The numbers in the table are not quantitative measurements on a single experimental unit (the professor). They are frequencies, or counts, of the number of professors who fall into each category. To compare the numbers of professors at public and private colleges, you might draw two pie charts and display them side by side, as in Figure 3.2.

Solution

FIGU R E 3 .2

Comparative pie charts for Example 3.2

● Private

Public

16.0%

24.0%

44.8%

Category Full Professor Associate Professor Assistant Professor

46.0%

31.2%

38.0%

100

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

Alternatively, you could draw either a stacked or a side-by-side bar chart. The stacked bar chart is shown in Figure 3.3.

FI GU R E 3 .3

Stacked bar chart for Example 3.2

● 200

School Public Private

Frequency

150

100

0 Rank

Full

Associate

Assistant

Although the graphs are not strikingly different, you can see that public colleges have fewer full professors and more associate professors than private colleges. The reason for these differences is not clear, but you might speculate that private colleges, with their higher salaries, are able to attract more full professors. Or perhaps public colleges are not as willing to promote professors to the higher-paying ranks. In any case, the graphs provide a means for comparing the two sets of data. You can also compare the distributions for public versus private colleges by creating conditional data distributions. These conditional distributions are shown in Table 3.3. One distribution shows the proportion of professors in each of the three ranks under the condition that the college is public, and the other shows the proportions under the condition that the college is private. These relative frequencies are easier to compare than the actual frequencies and lead to the same conclusions: • •

TABLE 3.3

●

The proportion of assistant professors is roughly the same for both public and private colleges. Public colleges have a smaller proportion of full professors and a larger proportion of associate professors.

Proportions of Professors by Rank for Public and Private Colleges

Public Private

Full Professor

Associate Professor

Assistant Professor

24 .16 150 60 .24 250

57 .38 150 78 .31 250

69 .46 150 112 .45 250

Total 1.00 1.00

❍

3.2 GRAPHS FOR QUALITATIVE VARIABLES

3.2

101

EXERCISES

BASIC TECHNIQUES

APPLICATIONS

3.1 Gender Differences Male and female respondents to a questionnaire about gender differences are categorized into three groups according to their answers to the ﬁrst question:

3.4 M&M’S The color distributions for two snacksize bags of M&M’S® candies, one plain and one peanut, are displayed in the table. Choose an appropriate graphical method and compare the distributions.

Men Women

Group 1

Group 2

Group 3

37 7

49 50

72 31

a. Create side-by-side pie charts to describe these data. b. Create a side-by-side bar chart to describe these data. c. Draw a stacked bar chart to describe these data. d. Which of the three charts best depicts the difference or similarity of the responses of men and women? 3.2 State-by-State A group of items are categorized according to a certain attribute—X, Y, Z—and according to the state in which they are produced: New York California

20 10

5 10

5 5

Yellow

Red

Orange

Green

Blue

15 6

14 2

12 2

4 3

5 3

6 5

3.5 How Much Free Time? When you were growing up, did you feel that you did not have enough free time? Parents and children have differing opinions on this subject. A research group surveyed 198 parents and 200 children and recorded their responses to the question, “How much free time does your child have?” or “How much free time do you have?” The responses are shown in the table below:2 Just the Right Amount

Not Enough

Too Much

Don’t Know

138 130

14 48

40 16

6 6

Parents Children

a. Create a comparative (side-by-side) bar chart to compare the numbers of items of each type made in California and New York. b. Create a stacked bar chart to compare the numbers of items of each type made in the two states. c. Which of the two types of presentation in parts a and b is more easily understood? Explain. d. What other graphical methods could you use to describe the data? 3.3 Consumer Spending The table below shows the average amounts spent per week by men and women in each of four spending categories: Men Women

Plain Peanut

Brown

$54 21

$27 85

$105 100

$22 75

a. What possible graphical methods could you use to compare the spending patterns of women and men? b. Choose two different methods of graphing and display the data in graphical form. c. What can you say about the similarities or differences in the spending patterns for men and women? d. Which of the two methods used in part b provides a better descriptive graph?

a. Deﬁne the sample and the population of interest to the researchers. b. Describe the variables that have been measured in this survey. Are the variables qualitative or quantitative? Are the data univariate or bivariate? c. What do the entries in the cells represent? d. Use comparative pie charts to compare the responses for parents and children. e. What other graphical techniques could be used to describe the data? Would any of these techniques be more informative than the pie charts constructed in part d? 3.6 Consumer Price Index The price of

living in the United States has increased dramatically in the past decade, as demonstrated by the consumer price indexes (CPIs) for housing and transportation. These CPIs are listed in the table for the years 1996 through the ﬁrst ﬁve months of 2007.3

EX0306

Year

1996

1997

1998

1999

2000

2001

Housing Transportation

152.8 143.0

156.8 144.3

160.4 141.6

163.9 144.4

169.6 153.3

176.4 154.3

Year

2002

2003

2004

2005

2006

2007

Housing Transportation

180.3 152.9

184.8 157.6

189.5 163.1

195.7 173.9

203.2 180.9

207.8 181.0

Source: www.bls.gov

102

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

a. Create side-by-side comparative bar charts to describe the CPIs over time. b. Draw two line charts on the same set of axes to describe the CPIs over time. c. What conclusions can you draw using the two graphs in parts a and b? Which is the most effective?

c. What conclusions can you draw using the graphs in parts a and b? 3.8 Charitable Contributions Charitable organizations count on support from both private donations and other sources. Here are the sources of income in a recent year for several well-known charitable organizations in the United States.4

EX0308

3.7 How Big Is the Household? A local

chamber of commerce surveyed 126 households within its city and recorded the type of residence and the number of family members in each of the households. The data are shown in the table.

EX0307

Type of Residence Family Members

Apartment

Duplex

Single Residence

8 15 9 6

10 4 5 1

2 14 24 28

1 2 3 4 or more

a. Use a side-by-side bar chart to compare the number of family members living in each of the three types of residences. b. Use a stacked bar chart to compare the number of family members living in each of the three types of residences.

3.3

Amounts ($ millions) Organization Salvation Army YMCA American Red Cross American Cancer Society American Heart Association Total

Private

Other

Total

$1545 773 557 868 436 $4179

$1559 4059 2509 58 157 $8342

$3104 4832 3066 926 593 $12,521

Source: The World Almanac and Book of Facts 2007

a. Construct a stacked bar chart to display the sources of income given in the table. b. Construct two comparative pie charts to display the sources of income given in the table. c. Write a short paragraph summarizing the information that can be gained by looking at these graphs. Which of the two types of comparative graphs is more effective?

SCATTERPLOTS FOR TWO QUANTITATIVE VARIABLES When both variables to be displayed on a graph are quantitative, one variable is plotted along the horizontal axis and the second along the vertical axis. The ﬁrst variable is often called x and the second is called y, so that the graph takes the form of a plot on the (x, y) axes, which is familiar to most of you. Each pair of data values is plotted as a point on this two-dimensional graph, called a scatterplot. It is the twodimensional extension of the dotplot we used to graph one quantitative variable in Section 1.4. You can describe the relationship between two variables, x and y, using the patterns shown in the scatterplot. •

• •

What type of pattern do you see? Is there a constant upward or downward trend that follows a straight-line pattern? Is there a curved pattern? Is there no pattern at all, but just a random scattering of points? How strong is the pattern? Do all of the points follow the pattern exactly, or is the relationship only weakly visible? Are there any unusual observations? An outlier is a point that is far from the cluster of the remaining points. Do the points cluster into groups? If so, is there an explanation for the observed groupings?

3.3 SCATTERPLOTS FOR TWO QUANTITATIVE VARIABLES

EXAMPLE

3.3

❍

103

The number of household members, x, and the amount spent on groceries per week, y, are measured for six households in a local area. Draw a scatterplot of these six data points. x

$95.75

$110.19

$118.33

$150.92

$85.86

$180.62

Solution Label the horizontal axis x and the vertical axis y. Plot the points using the coordinates (x, y) for each of the six pairs. The scatterplot in Figure 3.4 shows the six pairs marked as dots. You can see a pattern even with only six data pairs. The cost of weekly groceries increases with the number of household members in an apparent straight-line relationship. Suppose you found that a seventh household with two members spent $165 on groceries. This observation is shown as an X in Figure 3.4. It does not ﬁt the linear pattern of the other six observations and is classiﬁed as an outlier. Possibly these two people were having a party the week of the survey! FIGU R E 3 .4

●

Scatterplot for Example 3.3

180

160

140

120

100

80 1

EXAMPLE

TABLE 3.4

3.4

3 x

A distributor of table wines conducted a study of the relationship between price and demand using a type of wine that ordinarily sells for $10.00 per bottle. He sold this wine in 10 different marketing areas over a 12-month period, using ﬁve different price levels—from $10 to $14. The data are given in Table 3.4. Construct a scatterplot for the data, and use the graph to describe the relationship between price and demand. ●

Cases of Wine Sold at Five Price Levels Cases Sold per 10,000 Population

Price per Bottle

23, 21 19, 18 15, 17 19, 20 25, 24

$10 11 12 13 14

104

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

The 10 data points are plotted in Figure 3.5. As the price increases from $10 to $12 the demand decreases. However, as the price continues to increase, from $12 to $14, the demand begins to increase. The data show a curved pattern, with the relationship changing as the price changes. How do you explain this relationship? Possibly, the increased price is a signal of increased quality for the consumer, which causes the increase in demand once the cost exceeds $12. You might be able to think of other reasons, or perhaps some other variable, such as the income of people in the marketing areas, that may be causing the change.

Solution

FI GU R E 3 .5

Scatterplot for Example 3.4

● 25.0

Cases

22.5

20.0

17.5

15.0 10

12 Price

Now would be a good time for you to try creating a scatterplot on your own. Use the applets in Building a Scatterplot to create the scatterplots that you see in Figures 3.5 and 3.7. You will ﬁnd step-by-step instructions on the left-hand side of the applet (Figure 3.6), and you will be corrected if you make a mistake! FI GU R E 3 .6

Building a Scatterplot applet

●

3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA

❍

105

NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA

3.4

A constant rate of increase or decrease is perhaps the most common pattern found in bivariate scatterplots. The scatterplot in Figure 3.4 exhibits this linear pattern—that is, a straight line with the data points lying both above and below the line and within a ﬁxed distance from the line. When this is the case, we say that the two variables exhibit a linear relationship. 3.5

TABLE 3.5

FIGU R E 3 .7

Scatterplot of x versus y for Example 3.5

The data in Table 3.5 are the size of the living area (in square feet), x, and the selling price, y, of 12 residential properties. The MINITAB scatterplot in Figure 3.7 shows a linear pattern in the data. ●

Living Area and Selling Price of 12 Properties Residence

x (sq. ft.)

y (in thousands)

1 2 3 4 5 6 7 8 9 10 11 12

1360 1940 1750 1550 1790 1750 2230 1600 1450 1870 2210 1480

$278.5 375.7 339.5 329.8 295.6 310.3 460.5 305.2 288.6 365.7 425.3 268.8

● 450

400

EXAMPLE

350

300

250 1400

1600

1800 x

2000

2200

For the data in Example 3.5, you could describe each variable, x and y, individually using descriptive measures such as the means (x苶 and y苶) or the standard deviations (sx and sy). However, these measures do not describe the relationship between x and y for a particular residence—that is, how the size of the living space affects the selling price of the home. A simple measure that serves this purpose is called the correlation coefficient, denoted by r, and is deﬁned as sxy r sxsy

106

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

The quantities sx and sy are the standard deviations for the variables x and y, respectively, which can be found by using the statistics function on your calculator or the computing formula in Section 2.3. The new quantity sxy is called the covariance between x and y and is deﬁned as S(xi x苶 )( yi 苶 y) sxy n1 There is also a computing formula for the covariance: (Sxi)(Syi) Sxiyi n sxy n1 where Sxiyi is the sum of the products xiyi for each of the n pairs of measurements. How does this quantity detect and measure a linear pattern in the data? Look at the signs of the cross-products (xi x苶 )(yi y苶 ) in the numerator of r, or sxy. When a data point (x, y) is in either area I or III in the scatterplot shown in Figure 3.8, the cross-product will be positive; when a data point is in area II or IV, the cross-product will be negative. We can draw these conclusions: • • •

FI GU R E 3 .8

The signs of the crossproducts (xi x苶 )(yi y苶 ) in the covariance formula

●

If most of the points are in areas I and III (forming a positive pattern), sxy and r will be positive. If most of the points are in areas II and IV (forming a negative pattern), sxy and r will be negative. If the points are scattered across all four areas (forming no pattern), sxy and r will be close to 0.

y II : –

I:+

y II : –

I:+

y III : +

IV : –

x (a) Positive pattern

y III : +

I:+

IV : –

x (b) Negative pattern

III : +

IV : –

The applet called Exploring Correlation will help you to visualize how the pattern of points affects the correlation coefficient. Use your mouse to move the slider at the bottom of the scatterplot (Figure 3.9). You will see the value of r change as the pattern of the points changes. Notice that a positive pattern (a) results in a positive value of r; no pattern (c) gives a value of r close to zero; and a negative pattern (b) results in a negative value of r. What pattern do you see when r 1? When r 1? You will use this applet again for the MyApplet Exercises section at the end of the chapter.

3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA

FIGU R E 3 .9

Exploring Correlation applet

❍

107

●

r ⇔ positive linear relationship r 0 ⇔ negative linear relationship r ⬇ 0 ⇔ no relationship

Most scientiﬁc and graphics calculators can compute the correlation coefficient, r, when the data are entered in the proper way. Check your calculator manual for the proper sequence of entry commands. Computer programs such as MINITAB are also programmed to perform these calculations. The MINITAB output in Figure 3.10 shows the covariance and correlation coefficient for x and y in Example 3.5. In the covariance table, you will ﬁnd these values: sxy 15,545.20

s2x 79,233.33

s2y 3571.16

and in the correlation output, you ﬁnd r .924. However you decide to calculate the correlation coefficient, it can be shown that the value of r always lies between 1 and 1. When r is positive, x increases when y increases, and vice versa. When r is negative, x decreases when y increases, or x increases when y decreases. When r takes the value 1 or 1, all the points lie exactly on a straight line. If r 0, then there is no apparent linear relationship between the two variables. The closer the value of r is to 1 or 1, the stronger the linear relationship between the two variables.

FIGU R E 3 .1 0

●

MINITAB output of covariance and correlation for Example 3.5

EXAMPLE

Covariances: x, y x y

3.6

x 79233.33 15545.20

Correlations: x, y y

3571.16

Pearson correlation of x and y = 0.924 P-Value = 0.000

Find the correlation coefficient for the number of square feet of living area and the selling price of a home for the data in Example 3.5. Three quantities are needed to calculate the correlation coefficient. The standard deviations of the x and y variables are found using a calculator with a statistical function. You can verify that sx 281.4842 and sy 59.7592. Finally,

Solution

108

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

(Sxi)(Syi) Sxiyi n sxy n1 (20,980)(4043.5) 7,240,383 12 15,545.19697 11 This agrees with the value given in the MINITAB printout in Figure 3.10. Then sxy 15,545.19697 r .9241 sxsy (281.4842)(59.7592) which also agrees with the value of the correlation coefficient given in Figure 3.10. (You may wish to verify the value of r using your calculator.) This value of r is fairly close to 1, which indicates that the linear relationship between these two variables is very strong. Additional information about the correlation coefficient and its role in analyzing linear relationships, along with alternative calculation formulas, can be found in Chapter 12. Sometimes the two variables, x and y, are related in a particular way. It may be that the value of y depends on the value of x; that is, the value of x in some way explains the value of y. For example, the cost of a home (y) may depend on its amount of ﬂoor space (x); a student’s grade point average (x) may explain her score on an achievement test (y). In these situations, we call y the dependent variable, while x is called the independent variable. If one of the two variables can be classiﬁed as the dependent variable y and the other as x, and if the data exhibit a straight-line pattern, it is possible to describe the relationship relating y to x using a straight line given by the equation

x “explains” y or y “depends on” x. x is the explanatory or independent variable. y is the response or dependent variable.

y a bx as shown in Figure 3.11.

FI GU R E 3 .1 1

The graph of a straight line

●

y y = a + bx

b b a 0

As you can see, a is where the line crosses or intersects the y-axis: a is called the y-intercept. You can also see that for every one-unit increase in x, y increases by an amount b. The quantity b determines whether the line is increasing (b 0), decreasing (b 0), or horizontal (b 0) and is appropriately called the slope of the line.

3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA

❍

109

You can see the effect of changing the slope and the y-intercept of a line using the applet called How a Line Works. Use your mouse to move the slider on the right side of the scatterplot. As you move the slider, the slope of the line, shown as the vertical side of the green triangle (light gray in Figure 3.12), will change. Moving the slider on the left side of the applet causes the y-intercept, shown in red (blue in Figure 3.12), to change. What is the slope and y-intercept for the line shown in the applet in Figure 3.12? You will use this applet again for the MyApplet Exercises section at the end of the chapter. FIGU R E 3 .1 2

How a Line Works applet

●

Our points (x, y) do not all fall on a straight line, but they do show a trend that could be described as a linear pattern. We can describe this trend by ﬁtting a line as best we can through the points. This best-ﬁtting line relating y to x, often called the regression or least-squares line, is found by minimizing the sum of the squared differences between the data points and the line itself, as shown in Figure 3.13. The formulas for computing b and a, which are derived mathematically, are shown below. COMPUTING FORMULAS FOR THE LEAST-SQUARES REGRESSION LINE sy br sx

冢冣

and

a 苶y bx苶

and the least-squares regression line is: y a bx FIGU R E 3 .1 3

The best-ﬁtting line

● y y = a + bx 3 2 1

110

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

Since sx and sy are both positive, b and r have the same sign, so that: • • •

Remember that r and b have the same sign!

EXAMPLE

When r is positive, so is b, and the line is increasing with x. When r is negative, so is b, and the line is decreasing with x. When r is close to 0, then b is close to 0.

Find the best-ﬁtting line relating y starting hourly wage to x number of years of work experience for the following data. Plot the line and the data points on the same graph.

3.7

$6.00

7.50

8.00

12.00

13.00

15.50

Use the data entry method for your calculator to ﬁnd these descriptive statistics for the bivariate data set:

Solution

苶x 4.5

苶y 10.333

sx 1.871

sy 3.710

r .980

Then

Use the regression line to predict y for a given value of x.

sy 3.710 b r .980 1.9432389 ⬇ 1.943 sx 1.871

冢冣

冢

冣

and a 苶y bx苶 10.333 1.943(4.5) 1.590 Therefore, the best-fitting line is y 1.590 1.943x. The plot of the regression line and the actual data points are shown in Figure 3.14. The best-fitting line can be used to estimate or predict the value of the variable y when the value of x is known. For example, if a person applying for a job has 3 years of work experience (x), what would you predict his starting hourly wage (y) to be? From the best-fitting line in Figure 3.14, the best estimate would be y a bx 1.590 1.943(3) 7.419

Fitted line and data points for Example 3.7

● 15.0

12.5

FI GU R E 3 .1 4

10.0 y 1.590 1.943x 7.5

5.0 2

5 x

3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA

❍

111

How Do I Calculate the Correlation Coefficient? 1. First, create a table or use your calculator to ﬁnd Sx, Sy, and Sxy. 2. Calculate the covariance, sxy. 3. Use your calculator or the computing formula from Chapter 2 to calculate sx and sy. sxy 4. Calculate r . sxsy

How Do I Calculate the Regression Line? sxy 1. First, calculate 苶y and x苶. Then, calculate r . sxsy sy 2. Find the slope, b r and the y-intercept, a y苶 bx苶. sx

冢冣

3. Write the regression line by substituting the values for a and b into the equation: y a bx. Exercise Reps A. Below you will ﬁnd a simple set of bivariate data. Fill in the blanks to ﬁnd the correlation coefficient. x

Calculate:

Covariance

(Sx)(Sy) Sxy n sxy n1

Correlation Coefficient

sxy r sxsy

Sxy

B. Use the information from part A and ﬁnd the regression line. From Part A

From Part A

Calculate:

Slope

x苶 y苶

sy br sx

冢冣

y-intercept a y苶 bx苶

Regression Line: y

Answers are located on the perforated card at the back of this book.

When should you describe the linear relationship between x and y using the correlation coefficient r, and when should you use the regression line y a bx? The regression approach is used when the values of x are set in advance and then the corresponding value of y is measured. The correlation approach is used when an experimental unit is selected at random and then measurements are made on both

112

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

variables x and y. This technical point will be taken up in Chapter 12, which addresses regression analysis. Most data analysts begin any data-based investigation by examining plots of the variables involved. If the relationship between two variables is of interest, data analysts can also explore bivariate plots in conjunction with numerical measures of location, dispersion, and correlation. Graphs and numerical descriptive measures are only the ﬁrst of many statistical tools you will soon have at your disposal.

3.4

EXERCISES EXERCISE REPS These questions refer to the MyPersonal Trainer section on page 111. 3.9 Below you will ﬁnd a simple set of bivariate data. Fill in the blanks to ﬁnd the correlation coefficient. x

Calculate:

Covariance

(Sx)(Sy) Sxy n sxy n1

Correlation Coefficient

sxy r sxsy

Sxy

3.10 Use the information from Exercise 3.9 and ﬁnd the regression line. From Part A

From Part A

Calculate:

Slope

x苶 y苶

sy br sx

measurements on two variables, x and y: (5, 8)

(2, 6)

(1, 4)

a 苶y bx苶

3.12 Refer to Exercise 3.11.

3.11 A set of bivariate data consists of these

(3, 6)

冢冣

Regression Line: y

BASIC TECHNIQUES EX0311

y-intercept

(4, 7)

(4, 6)

a. Draw a scatterplot to describe the data. b. Does there appear to be a relationship between x and y? If so, how do you describe it? c. Calculate the correlation coefficient, r, using the computing formula given in this section. d. Find the best-ﬁtting line using the computing formulas. Graph the line on the scatterplot from part a. Does the line pass through the middle of the points?

a. Use the data entry method in your scientiﬁc calculator to enter the six pairs of measurements. Recall the proper memories to ﬁnd the correlation coefficient, r, the y-intercept, a, and the slope, b, of the line. b. Verify that the calculator provides the same values for r, a, and b as in Exercise 3.11. 3.13 Consider this set of bivariate data: EX0313

5.6

4.6

4.5

3.7

3.2

2.7

a. Draw a scatterplot to describe the data.

3.4 NUMERICAL MEASURES FOR QUANTITATIVE BIVARIATE DATA

b. Does there appear to be a relationship between x and y? If so, how do you describe it? c. Calculate the correlation coefficient, r. Does the value of r conﬁrm your conclusions in part b? Explain. 3.14 The value of a quantitative variable is

measured once a year for a 10-year period:

EX0314

Year

Measurement

Year

Measurement

1 2 3 4 5

61.5 62.3 60.7 59.8 58.0

6 7 8 9 10

58.2 57.5 57.5 56.1 56.0

MINITAB output for Exercise 3.14 x 9.16667 -6.42222

y 4.84933

d. Find the best-fitting line using the results of part c. Verify your answer using the data entry method in your calculator. e. Plot the best-fitting line on your scatterplot from part a. Describe the fit of the line.

APPLICATIONS 3.15 Grocery Costs These data relating the amount spent on groceries per week and the number of household members are from Example 3.3:

EX0315

$95.75

$110.19 $118.33 $150.92

$85.86

$180.62

a. Find the best-fitting line for these data. b. Plot the points and the best-fitting line on the same graph. Does the line summarize the information in the data points? c. What would you estimate a household of six to spend on groceries per week? Should you use the fitted line to estimate this amount? Why or why not? 3.16 Real Estate Prices The data relating EX0316

x (sq. ft.)

y (in thousands)

1360 1940 1750 1550 1790 1750 2230 1600 1450 1870 2210 1480

$278.5 375.7 339.5 329.8 295.6 310.3 460.5 305.2 288.6 365.7 425.3 268.8

1 2 3 4 5 6 7 8 9 10 11 12

3.17 Disabled Students A social skills training program, reported in Psychology in the Schools, was implemented for seven students with mild handicaps in a study to determine whether the program caused improvement in pre/post measures and behavior ratings.5 For one such test, these are the pretest and posttest scores for the seven students:

EX0317

C ovarianc es x y

113

price of 12 residential properties given in Example 3.5 are reproduced here. First, find the best-fitting line that describes these data, and then plot the line and the data points on the same graph. Comment on the goodness of the fitted line in describing the selling price of a residential property as a linear function of the square feet of living area. Residence

a. Draw a scatterplot to describe the variable as it changes over time. b. Describe the measurements using the graph constructed in part a. c. Use this MINITAB output to calculate the correlation coefficient, r:

❍

the square feet of living space and the selling

Student

Pretest

Posttest

Earl Ned Jasper Charlie Tom Susie Lori

101 89 112 105 90 91 89

113 89 121 99 104 94 99

a. Draw a scatterplot relating the posttest score to the pretest score. b. Describe the relationship between pretest and posttest scores using the graph in part a. Do you see any trend? c. Calculate the correlation coefficient and interpret its value. Does it reinforce any relationship that was apparent from the scatterplot? Explain. 3.18 Lexus, Inc. The makers of the Lexus

automobile have steadily increased their sales since their U.S. launch in 1989. However, the rate of increase changed in 1996 when Lexus introduced a line of trucks. The sales of Lexus from 1996 to 2005 are shown in the table.6

EX0318

114 Year

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Sales (thousands 80 of vehicles)

100

155

180

210

224

234

260

288

303

Source: Adapted from: Automotive News, January 26, 2004, and May 22, 2006.

a. Plot the data using a scatterplot. How would you describe the relationship between year and sales of Lexus? b. Find the least-squares regression line relating the sales of Lexus to the year being measured. c. If you were to predict the sales of Lexus in the year 2015, what problems might arise with your prediction? 3.19 HDTVs, again In Exercise 2.12, Con-

sumer Reports gave the prices for the top 10 LCD high deﬁnition TVs (HDTVs) in the 30- to 40-inch category. Does the price of an LCD TV depend on the size of the screen? The table below shows the 10 costs again, along with the screen size.6

EX0319

Brand

Price

Size

JVC LT-40FH96 Sony Bravia KDL-V32XBR1 Sony Bravia KDL-V40XBR1 Toshiba 37HLX95 Sharp Aquos LC-32DA5U Sony Bravia KLV-S32A10 Panasonic Viera TC-32LX50 JVC LT-37X776 LG 37LP1D Samsung LN-R328W

$2900 1800 2600 3000 1300 1500 1350 2000 2200 1200

40" 32" 40" 37" 32" 32" 32" 37" 37" 32"

a. Which of the two variables (price and size) is the independent variable, and which is the dependent variable? b. Construct a scatterplot for the data. Does the relationship appear to be linear? 3.20 HDTVs, continued Refer to Exercise 3.19.

Suppose we assume that the relationship between x and y is linear. a. Find the correlation coefficient, r. What does this value tell you about the strength and direction of the relationship between size and price? b. What is the equation of the regression line used to predict the price of the TV based on the size of the screen? c. The Sony Corporation is introducing a new 37" LCD TV. What would you predict its price to be? d. Would it be reasonable to try to predict the price of a 45" LCD TV? Explain.

CHAPTER REVIEW Key Concepts I.

Bivariate Data

1. Both qualitative and quantitative variables 2. Describing each variable separately 3. Describing the relationship between the two variables II. Describing Two Qualitative Variables

1. Side-by-side pie charts 2. Comparative line charts

3. Comparative bar charts a. Side-by-side b. Stacked 4. Relative frequencies to describe the relationship between the two variables III. Describing Two Quantitative Variables

1. Scatterplots a. Linear or nonlinear pattern

MY MINITAB

b. Strength of relationship

❍

115

3. The best-ﬁtting regression line

c. Unusual observations: clusters and outliers

a. Calculating the slope and y-intercept b. Graphing the line c. Using the line for prediction

2. Covariance and correlation coefficient

Describing Bivariate Data MINITAB provides different graphical techniques for qualitative and quantitative bivariate data, as well as commands for obtaining bivariate descriptive measures when the data are quantitative. To explore both types of bivariate procedures, you need to enter two different sets of bivariate data into a MINITAB worksheet. Once you are on the Windows desktop, double-click on the MINITAB icon or use the Start button to start MINITAB. Start a new project using File 씮 New 씮 Minitab Project. Then open the existing project called “Chapter 1.” We will use the college student data, which should be in Worksheet 1. Suppose that the 105 students already tabulated were from the University of California, Riverside, and that another 100 students from an introductory statistics class at UC Berkeley were also interviewed. Table 3.6 shows the status distribution for both sets of students. Create another variable in C3 of the worksheet called “College” and enter UCR for the ﬁrst ﬁve rows. Now enter the UCB data in columns C1–C3. You can use the familiar Windows cut-and-paste icons if you like.

TABLE 3.6

●

Frequency (UCR) Frequency (UCB)

Freshman

Sophomore

Junior

Senior

Grad Student

5 10

23 35

32 24

35 25

10 6

The other worksheet in “Chapter 1” is not needed and can be deleted by clicking on the X in the top right corner of the worksheet. We will use the worksheet called “Minivans” from Chapter 2, which you should open using File 씮 Open Worksheet and selecting “Minivans.mtw.” Now save this new project as “Chapter 3.” To graphically describe the UCR/UCB student data, you can use comparative pie charts—one for each school (see Chapter 1). Alternatively, you can use either stacked or side-by-side bar charts. Use Graph 씮 Bar Chart. In the “Bar Charts” Dialog box (Figure 3.15), select Values from a Table in the drop-down list and click either Stack or Cluster in the row marked “One Column of Values.” Click OK. In the next Dialog box (Figure 3.16), select “Frequency” for the Graph variables box and “Status” and “College” for the Categorical variable for grouping box. Click OK. Once the bar chart is displayed (Figure 3.17), you can right-click on various items in the bar chart to edit. If you right-click on the bars and select Update Graph Automatically, the bar chart will automatically update when you change the data in the Minitab worksheet.

116

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

FI GU R E 3 .1 5

●

FI GU R E 3 .1 6

●

MY MINITAB

FIGU R E 3 .1 7

❍

117

●

Turn to Worksheet 2, in which the bivariate minivan data from Chapter 2 are located. To examine the relationship between the second and third car seat lengths, you can plot the data and numerically describe the relationship with the correlation coefficient and the best-fitting line. Use Stat 씮 Regression 씮 Fitted Line Plot, and select “2nd Seat” and “3rd Seat” for Y and X, respectively (see Figure 3.18). Make sure that the dot next to Linear is selected, and click OK. The plot of the nine data points and the best-fitting line will be generated as in Figure 3.19.

FIGU R E 3 .1 8

●

118

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

FI GU R E 3 .1 9

●

To calculate the correlation coefficient, use Stat 씮 Basic Statistics 씮 Correlation, selecting “2nd Seat” and “3rd Seat” for the Variables box. To select both variables at once, hold the Shift key down as you highlight the variables and then click Select. Click OK, and the correlation coefficient will appear in the Session window (see Figure 3.20). Notice the relatively strong positive correlation and the positive slope of the regression line, indicating that a minivan with a long ﬂoor length behind the second seat will also tend to have a long ﬂoor length behind the third seat. Save “Chapter 3” before you exit MINITAB!

FI GU R E 3 .2 0

●

SUPPLEMENTARY EXERCISES

❍

119

Supplementary Exercises 3.21 Professor Asimov Professor Isaac Asimov was one of the most proliﬁc writers of all time. He wrote nearly 500 books during a 40-year career prior to his death in 1992. In fact, as his career progressed, he became even more productive in terms of the number of books written within a given period of time.8 These data are the times (in months) required to write his books, in increments of 100: Number of Books

100

200

300

400

490

Time (in months)

237

350

419

465

507

a. Plot the accumulated number of books as a function of time using a scatterplot. b. Describe the productivity of Professor Asimov in light of the data set graphed in part a. Does the relationship between the two variables seem to be linear? 3.22 Cheese, Please! Health-conscious Americans often consult the nutritional information on food packages in an attempt to avoid foods with large amounts of fat, sodium, or cholesterol. The following information was taken from eight different brands of American cheese slices:

f. Write a paragraph to summarize the relationships you can see in these data. Use the correlations and the patterns in the four scatterplots to verify your conclusions. 3.23 Army versus Marine Corps Who are the men and women who serve in our armed forces? Are they male or female, officers or enlisted? What is their ethnic origin and their average age? An article in Time magazine provided some insight into the demographics of the U.S. armed forces.9 Two of the bar charts are shown below. U.S. Army 50% Enlisted Officers

40% 30% 20%

EX0322

Sodium (mg)

4.5

340

49 50

44 45

39 40

34 35

29 30

24 25

Age

Calories U.S. Marine Corps

5 8 4

3.5 5.0 2.5

15 25 15

300 520 340

70 100 60

2.0

320

5 5

3.5 3.0

15 15

290 260

70 60

3.5

330

50% Enlisted Officers

40% 30% 20% 10%

49 ov er d an

to 40

to 35

to 30

0% to

Kraft Deluxe American Kraft Velveeta Slices Private Selection Ralphs Singles Kraft 2% Milk Singles Kraft Singles American Borden Singles Lake to Lake American

Fat (g)

Cholesterol (mg)

Brand

Saturated Fat (g)

10%

Age

a. Which pairs of variables do you expect to be strongly related? b. Draw a scatterplot for fat and saturated fat. Describe the relationship. c. Draw a scatterplot for fat and calories. Compare the pattern to that found in part b. d. Draw a scatterplot for fat versus sodium and another for cholesterol versus sodium. Compare the patterns. Are there any clusters or outliers? e. For the pairs of variables that appear to be linearly related, calculate the correlation coefficients.

a. What variables have been measured in this study? Are the variables qualitative or quantitative? b. Describe the population of interest. Do these data represent a population or a sample drawn from the population? c. What type of graphical presentation has been used? What other type could have been used? d. How would you describe the similarities and differences in the age distributions of enlisted persons and officers?

120

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

e. How would you describe the similarities and differences in the age distributions of personnel in the U.S. Army and the Marine Corps? 3.24 Cheese, again! The demand for healthy foods that are low in fats and calories has resulted in a large number of “low-fat” and “fat-free” products at the supermarket. The table shows the numbers of calories and the amounts of sodium (in milligrams) per slice for ﬁve different brands of fat-free American cheese. Brand

Sodium (mg)

Calories

300 300 320 290 180

30 30 30 30 25

Kraft Fat Free Singles Ralphs Fat Free Singles Borden Fat Free Healthy Choice Fat Free Smart Beat American

a. Draw a scatterplot to describe the relationship between the amount of sodium and the number of calories. b. Describe the plot in part a. Do you see any outliers? Do the rest of the points seem to form a pattern? c. Based only on the relationship between sodium and calories, can you make a clear decision about which of the ﬁve brands to buy? Is it reasonable to base your choice on only these two variables? What other variables should you consider? 3.25 Peak Current Using a chemical proce-

dure called differential pulse polarography, a chemist measured the peak current generated (in microamperes) when a solution containing a given amount of nickel (in parts per billion) is added to a buffer. The data are shown here:

EX0325

x Ni (ppb)

y Peak Current (mA)

19.1 38.2 57.3 76.2 95 114 131 150 170

.095 .174 .256 .348 .429 .500 .580 .651 .722

Use a graph to describe the relationship between x and y. Add any numerical descriptive measures that are appropriate. Write a paragraph summarizing your results. 3.26 Movie Money How much money do

movies make on a single weekend? Does this amount in any way predict the movie’s success or failure, or is the movie’s total monetary success more

EX0326

dependent on the number of weeks that the movie remains in the movie theaters? In a recent week, the following data was collected for the top 10 movies in theaters that weekend.10

Top Movies

Weekend Gross (in millions)

1 The Prestige $14.8 Disney 2 The Departed $13.7 Warner Bros. 3 Flags of Our Fathers $10.2 Paramount/DreamWorks 4 Open Season $8.0 Sony 5 Flicka $7.7 20th Century Fox 5 The Grudge 2 $7.7 Sony 7 Man of the Year $7.0 Universal 8 Marie Antoinette $5.3 Sony 9 The Texas Chainsaw Massacre: The Beginning $3.8 New Line 10 The Marine $3.7 20th Century Fox

Number of Per-Screen Weeks in Screens Average Release

Gross to Date (in millions)

2281

$6496

$14.8

3005

$4550

$77.1

1876

$5437

$10.2

3379

$2367

$69.6

2877

$2676

$7.7

3124

$2395

$31.4

2522

$2789

$22.5

859

$6169

$5.3

2569

$1496

$36.0

2545

$1463

$12.5

a. Which pairs of variables in the table do you think will have a positive correlation? Which pairs will have a negative correlation? Explain. b. Draw a scatterplot relating the gross to date to the number of weeks in release. How would you describe the relationship between these two variables? c. Draw a scatterplot relating the weekend gross to the number of screens on which the movie is being shown. How would you describe the relationship between these two variables? d. Draw a scatterplot relating the per-screen average to the number of screens on which the movie is being shown. How would you describe the relationship between these two variables? 3.27 Movie Money, continued The data from Exercise 3.26 were entered into a MINITAB worksheet, and the following output was obtained. Covariances:

Weekend gross Screens Avg/screen Weeks Gross to date

Weekend gross

Screens

Avg/ screen

Weeks

Gross to date

14.2 403.8 4635.5 -0.5 28.3

521835.8 -884350.0 493.2 11865.1

3637602.0 -1110.9 -10929.2

1.1 23.8

655.6

a. Use the MINITAB output or the original data to ﬁnd the correlation between the number of weeks in release and the gross to date.

SUPPLEMENTARY EXERCISES

b. For the pair of variables described in part a, which of the variables would you classify as the independent variable? The dependent variable? c. Use the MINITAB output or the original data to ﬁnd the correlation between the weekend gross and the number of screens on which the movie is being shown. Find the correlation between the number of screens on which the movie is being shown and the per-screen average. d. Do the correlations found in part c conﬁrm your answer to Exercise 3.26a? What might be the practical reasons for the direction and strength of the correlations in part c? 3.28 Heights and Gender Refer to Exercise 1.54

and data set EX0154. When the heights of these 105 students were recorded, their gender was also recorded. a. What variables have been measured in this experiment? Are they qualitative or quantitative? b. Look at the histogram from Exercise 1.54 along with the comparative box plots shown below. Do the box plots help to explain the two local peaks in the histogram? Explain. Histogram of Heights 10

Frequency

❍

121

3.29 Hazardous Waste The data in Exercise 1.37 gave the number of hazardous waste sites in each of the 50 states and the District of Columbia in 2005.4 Suspecting that there might be a relationship between the number of waste sites and the size of the state (in thousands of square miles), researchers recorded both variables and generated a scatterplot with MINITAB.

EX0329

State AL AK AZ AR CA CO CT DE DC FL GA HI ID IL IN IA KS KY LA ME MD MA MI MN MS MO

Sites

Area

State

Sites

Area

15 6 9 10 95 19 16 15 1 50 17 3 9 48 30 12 11 14 14 12 18 33 68 24 5 26

52 663 114 53 164 104 6 2 0 66 59 11 84 58 36 56 82 40 52 35 12 11 97 87 48 70

MT NE NV NH NJ NM NY NC ND OH OK OR PA RI SC SD TN TX UT VT VA WA WV WI WY

15 14 1 21 117 13 87 31 0 37 11 11 96 12 26 2 14 44 18 11 28 47 9 38 2

147 77 111 9 9 122 55 54 71 45 70 98 46 2 32 77 42 269 85 10 43 71 24 65 98

MINITAB printout for Exercise 3.29 2

Covariances: Sites, Area

0 60

66 Heights

Sites 682.641 -98.598

Sites Area

Area 9346.603

120 100 F

Sites

Gender

80 60 40 M 20 0 60

68 Height

100

200

300 400 Area

500

600

700

122

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

a. Is there any clear pattern in the scatterplot? Describe the relationship between number of waste sites and the size of the state. b. Use the MINITAB output to calculate the correlation coefficient. Does this conﬁrm your answer to part a? c. Are there any outliers or clusters in the data? If so, can you explain them?

3.31 Pottery, continued In Exercise 1.59, we analyzed the percentage of aluminum oxide in 26 samples of Romano-British pottery found at four different kiln sites in the United Kingdom.12 Since one of the sites only provided two measurements, that site is eliminated, and comparative box plots of aluminum oxide at the other three sites are shown.

d. What other variables could you consider in trying to understand the distribution of hazardous waste sites in the United States?

Week

Completions

Total Yards

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

15 31 25 22 22 19 17 28 24 5 22 24 22 20 26 21

170 340 340 205 220 206 180 287 347 73 266 214 293 174 285 285

Source: www.espn.com

a. Draw a scatterplot to describe the relationship between number of completions and total passing yards for Brett Favre. b. Describe the plot in part a. Do you see any outliers? Do the rest of the points seem to form a pattern? c. Calculate the correlation coefficient, r, between the number of completions and total passing yards. d. What is the regression line for predicting total number of passing yards y based on the total number of completions x? e. If Brett Favre had 20 pass completions in his next game, what would you predict his total number of passing yards to be?

Site

3.30 Brett Favre, again The number of passes completed and the total number of passing yards was recorded for Brett Favre for each of the 16 regular season games in the fall of 2006.11

EX0330

16 Aluminum Oxide

a. What two variables have been measured in this experiment? Are they qualitative or quantitative? b. How would you compare the amount of aluminum oxide in the samples at the three sites? 3.32 Pottery, continued Here is the percentage of aluminum oxide, the percentage of iron oxide, and the percentage of magnesium oxide in ﬁve samples collected at Ashley Rails in the United Kingdom.

EX0332

Sample 1 2 3 4 5

17.7 18.3 16.7 14.8 19.1

1.12 1.14 0.92 2.74 1.64

0.56 0.67 0.53 0.67 0.60

a. Find the correlation coefficients describing the relationships between aluminum and iron oxide content, between iron oxide and magnesium oxide, and between aluminum oxide and magnesium oxide. b. Write a sentence describing the relationships between these three chemicals in the pottery samples. 3.33 Computer Networks at Home The table below (Exercise 1.50) shows the predicted rise of home networking of PCs in the next few years.13

EX0333

SUPPLEMENTARY EXERCISES

Wired

Wireless

2002 2003 2004 2005 2006 2007 2008

6.1 6.5 6.2 5.7 4.9 4.1 3.4

1.7 4.5 8.7 13.7 19.1 24.0 28.2

123

the back with arms outstretched to make a “T”) is roughly equal to the person’s height. To test this claim, we measured eight people with the following results:

U.S Home Networks (in millions) Year

❍

Source: Jupiter Research

a. What variables have been measured in this experiment? Are they qualitative or quantitative?

Person

Armspan (inches) Height (inches)

68 69

62.25 62

65 65

69.5 70

Person

Armspan (inches) Height (inches)

68 67

69 67

62 63

60.25 62

b. Use one of the graphical methods given in this chapter to describe the data. c. Write a sentence describing the relationship between wired and wireless technology as it will be in the next few years. 3.34 Politics and Religion A survey was conducted prior to the 2004 presidential election to explore the relationship between a person’s religious fervor and their choice of a political candidate. Voters were asked how often they attended church and which of the two major presidential candidates (George W. Bush or his democratic opponent) they would favor in the 2004 election.14 The results are shown below. Church Attendance More than once a week Once a week Once or twice a month Once or twice a year Seldom/never

G. W. Bush

Democratic Candidate

63% 56% 52% 46% 38%

37% 44% 48% 54% 62%

Source: Press-Enterprise

a. What variables have been measured in this survey? Are they qualitative or quantitative? b. Draw side-by-side comparative bar charts to describe the percentages favoring the two candidates, categorized by church attendance. c. Draw two line charts on the same set of axes to describe the same percentages for the two candidates. d. What conclusions can you draw using the two graphs in parts b and c? Which is more effective? 3.35 Armspan and Height Leonardo

da Vinci (1452–1519) drew a sketch of a man, indicating that a person’s armspan (measuring across

EX0335

a. Draw a scatterplot for armspan and height. Use the same scale on both the horizontal and vertical axes. Describe the relationship between the two variables. b. Calculate the correlation coefficient relating armspan and height. c. If you were to calculate the regression line for predicting height based on a person’s armspan, how would you estimate the slope of this line? d. Find the regression line relating armspan to a person’s height. e. If a person has an armspan of 62 inches, what would you predict the person’s height to be? 3.36 Airline Revenues The number of passengers x (in millions) and the revenue y (in billions of dollars) for the top nine U.S. airlines in a recent year are given in the following table.4

EX0336

124

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

American

United

Delta

Northwest

Continental

98.0 20.7

66.7 17.4

86.0 16.2

56.5 12.3

42.8 11.2

U.S. Air

Southwest

x y

64.0 5.1

88.4 7.6

Alaska 16.7 3.0

SkyWest 16.6 2.0

Source: The World Almanac and Book of Facts 2007

a. Construct a scatterplot for the data. b. Describe the form, direction, and strength of the pattern in the scatterplot. 3.37 Test Interviews Of two personnel

evaluation techniques available, the ﬁrst requires a two-hour test-interview while the second can be completed in less than an hour. The scores for each of the eight individuals who took both tests are given in the next table.

EX0337

Applicant

Test 1 (x)

Test 2 (y)

1 2 3 4 5 6 7 8

75 89 60 71 92 105 55 87

38 56 35 45 59 70 31 52

a. Construct a scatterplot for the data. b. Describe the form, direction, and strength of the pattern in the scatterplot. 3.38 Test Interviews, continued Refer to Exercise 3.37. a. Find the correlation coefficient, r, to describe the relationship between the two tests.

b. Would you be willing to use the second and quicker test rather than the longer test-interview to evaluate personnel? Explain. 3.39 Rain and Snow Is there a correlation between the amount of rain and the amount of snow that falls in a particular location? The table below shows the average annual rainfall (inches) and the average annual snowfall (inches) for 10 cities in the United States.15

EX0339

City

Rainfall (inches)

Snowfall (inches)

14.77 13.03 37.60 21.19 37.98 58.33 54.65 49.69 37.07 35.56

56.9 77.8 64.5 40.8 19.9 97.0 5.1 28.6 6.5 23.2

Billings, MT Casper, WY Concord, NH Fargo, ND Kansas City, MO Juneau, AK Memphis, TN New York, NY Portland, OR Springﬁeld, IL Source: Time Almanac 2007

a. Construct a scatterplot for the data. b. Calculate the correlation coefficient r between rainfall and snowfall. Describe the form, direction, and strength of the relationship between rainfall and snowfall. c. Are there any outliers in the scatterplot? If so, which city does this outlier represent? d. Remove the outlier that you found in part c from the data set and recalculate the correlation coefficient r for the remaining nine cities. Does the correlation between rainfall and snowfall change, and, if so, in what way?

Exercises 3.40 If you have not yet done so, use the ﬁrst applet

in Building a Scatterplot to create a scatterplot for the data in Example 3.4. 3.41 If you have not yet done so, use the second

applet in Building a Scatterplot to create a scatterplot for the data in Example 3.5. 3.42 Cordless Phones The table below shows the prices of 8 single handset cordless phones along with their overall score (on a scale of

EX0342

0–100) in a consumer rating survey presented by Consumer Reports.16 Brand and Model Uniden EXI 4246 AT&T E2116 Panasonic KX-TG5621S GE 27831GE1 VTech V Mix VTech ia5829 Panasonic KX-TG2421W Clarity C410

Price

Overall Score

$25 30 50 20 30 30 40 70

77 74 74 73 72 71 70 65

SUPPLEMENTARY EXERCISES

a. Calculate the correlation coefficient r between price and overall score. How would you describe the relationship between price and overall score? b. Use the applet called Correlation and the Scatterplot to plot the eight data points. What is the correlation coefficient shown on the applet? Compare with the value you calculated in part a. c. Describe the pattern that you see in the scatterplot. What unexpected relationship do you see in the data? 3.43 Midterm Scores When a student per-

forms poorly on a midterm exam, the student sometimes is convinced that their score is an anomaly and that they will do much better on the second midterm. The data below show the midterm scores (out of 100 points) for eight students in an introductory statistics class.

EX0343

Student

Midterm 1

Midterm 2

1 2 3 4 5 6 7 8

70 58 85 82 70 40 85 85

88 52 84 74 80 36 48 96

a. Calculate the correlation coefficient r between the two midterm scores. How would you describe the relationship between scores on the ﬁrst and second midterm? b. Use the applet called Correlation and the Scatterplot to plot the eight data points. What is the correlation coefficient shown on the applet? Compare with the value you calculated in part a. c. Describe the pattern that you see in the scatterplot. Are there any clusters or outliers? If so, how would you explain them? 3.44 Acess the applet called Exploring Correlation.

a. Move the slider in the ﬁrst applet so that r ⬇ .75. Now switch the sign using the button at the bottom of the applet. Describe the change in the pattern of the points.

❍

125

b. Move the slider in the ﬁrst applet so that r ⬇ 0. Describe the pattern of points on the scatterplot. c. Refer to part b. In the second applet labeled Correlation and the Quadrants, with r ⬇ 0, count the number of points falling in each of the four quadrants of the scatterplot. Is the distribution of points in the quadrants relatively uniform, or do more points fall into certain quadrants than others? d. Use the second applet labeled Correlation and the Quadrants and change the correlation coefficient to r ⬇ 0.9. Is the distribution of points in the quadrants relatively uniform, or do more points fall into certain quadrants than others? What happens if r ⬇ 0.9? e. Use the third applet labeled Correlation and the Regression Line. Move the slider to see the relationship between the correlation coefficient r, the slope of the regression line and the direction of the relationship between x and y. Describe the relationship. 3.45 Suppose that the relationship between two vari-

ables x and y can be described by the regression line y 2.0 0.5x. Use the applet in How a Line Works to answer the following questions: a. What is the change in y for a one-unit change in x? b. Do the values of y increase or decrease as x increases? c. At what point does the line cross the y-axis? What is the name given to this value? d. If x 2.5, use the least squares equation to predict the value of y. What value would you predict for y if x 4.0? 3.46 Access the applet in How a Line Works. a. Use the slider to change the y-intercept of the line, but do not change the slope. Describe the changes that you see in the line. b. Use the slider to change the slope of the line, but do not change the y-intercept. Describe the changes that you see in the line.

126

❍

CHAPTER 3 DESCRIBING BIVARIATE DATA

CASE STUDY Dishwashers

Are Your Dishes Really Clean? Does the price of an appliance convey something about its quality? Thirty-six different dishwashers were ranked on characteristics ranging from an overall satisfaction score, washing (x1), energy use (x2), noise (x3), loading ﬂexibility (x4), ease of use (x5), and cycle time (in minutes).17 The Kenmore (1374[2]) had the highest performance score of 83 while the Whirlpool Gold GU3600XTS[Q] had the lowest at 76. Ratings pictograms were converted to numerical values for x1, . . . , x5 where 5 Excellent, 4 Very good, 3 Good, 2 Fair, and 1 Poor. Use a statistical computer package to explore the relationships between various pairs of variables in the table. Excellent

Very good

Brand & Model

Price

Score

Ariston LI670 Asko D3122XL Asko Encore D3531XLHD[SS] Bosch SHE45C0[2]UC Bosch SHE58C0[2]UC Frigidaire GLD2445RF[S] Frigidaire Gallery GLD4355RF[S] Frigidaire Professional PLD4555RF[C] GE GLD4600N[WW] GE GLD5900N[WW] GE GLD6700N[WW] GE Monogram ZBD0710N[SS] GE Proﬁle PDW8600N[WW] GE Proﬁle PDW9900N[WW] Haier ESD310 Kenmore (Sears) 1359[2] Kenmore (Sears) 1373[2] Kenmore (Sears) 1374[2] Kenmore (Sears) Elite 1376[2] Kenmore (Sears) Elite 1378[2] Kenmore (Sears) PRO 1387[3] KitchenAid Architect KUDD01DP[WH] KitchenAid KUDK03CT[WH] KitchenAid KUDS03CT[WH] KitchenAid KUDU03ST[SS] LG LDF7810[WW] LG LDS5811[W] Maytag MDB4651AW[W] Maytag MDB5601AW[W] Maytag MDB6601AW[W] Maytag MDB7601AW[W] Maytag MDB8751BW[W] Miele Advanta G2020SC Whirlpool DU1100XTP[Q] Whirlpool Gold GU2455XTS[Q] Whirlpool Gold GU3600XTS[Q]

$800 $850 $1600 $700 $900 $400 $500 $710 $460 $510 $550 $1500 $900 $1300 $600 $350 $580 $650 $800 $1000 $1400 $1400 $650 $850 $1400 $800 $650 $400 $450 $500 $560 $700 $1000 $500 $550 $750

48. 78. 81. 78. 78. 62. 71. 75. 75. 74. 68. 59. 68. 70. 56. 68. 79. 83. 79. 82. 78. 60. 76. 78. 79. 77. 74. 71. 68. 71. 71. 74. 74. 77. 77. 76.

Good

Fair

Poor

Cycle Time

190 115 145 125 135 105 110 120 115 115 115 115 115 120 125 110 125 125 130 125 130 115 130 115 125 110 140 110 120 120 130 120 125 125 125 130

4 5 5 5 5 3 4 4 4 4 4 3 4 4 4 5 5 5 5 5 5 4 5 5 5 4 4 4 5 5 5 5 5 5 5 5

2 4 5 4 4 4 4 4 4 4 4 4 4 4 3 3 4 4 4 4 4 4 4 4 4 5 4 3 3 3 2 3 4 4 4 4

3 3 5 4 4 3 3 3 3 4 4 3 4 4 3 2 3 4 4 4 4 2 4 4 4 4 5 3 4 3 4 3 3 3 3 3

3 3 4 3 4 3 3 4 3 3 4 4 3 4 4 3 4 4 5 5 5 4 3 5 5 5 5 3 4 4 5 5 4 4 3 5

3 3 4 3 4 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 5 5 4 4 4 5 5 3 5 5 4

Source: © 2007 by Consumers Union of U.S., Inc., Yonkers, NY 10703-1057, a nonproﬁt organization. Reprinted with permission from the September 2007 issue of Consumer Reports® for educational purposes only. No commercial use or reproduction permitted. www.ConsumerReports.org®.

1. Look at the variables Price, Score, and Cycle Time individually. What can you say about symmetry? About outliers? 2. Look at all the variables in pairs. Which pairs are positively correlated? Negatively correlated? Are there any pairs that exhibit little or no correlation? Are some of these results counterintuitive? 3. Does the price of an appliance, speciﬁcally a dishwasher, convey something about its quality? Which variables did you use in arriving at your answer?

Probability and Probability Distributions GENERAL OBJECTIVES Now that you have learned to describe a data set, how can you use sample data to draw conclusions about the sampled populations? The technique involves a statistical tool called probability. To use this tool correctly, you must first understand how it works. The first part of this chapter will teach you the new language of probability, presenting the basic concepts with simple examples. The variables that we measured in Chapters 1 and 2 can now be redefined as random variables, whose values depend on the chance selection of the elements in the sample. Using probability as a tool, you can create probability distributions that serve as models for discrete random variables, and you can describe these random variables using a mean and standard deviation similar to those in Chapter 2.

CHAPTER INDEX ● The Addition and Multiplication Rules (4.6) ● Bayes’ Rule and the Law of Total Probability (optional) (4.7) ● Conditional probability and independence (4.6)

Probability and Decision Making in the Congo In his exciting novel Congo, author Michael Crichton describes an expedition racing to ﬁnd boron-coated blue diamonds in the rain forests of eastern Zaire. Can probability help the heroine Karen Ross in her search for the Lost City of Zinj? The case study at the end of this chapter involves Ross’s use of probability in decision-making situations.

● Counting rules (optional) (4.4) ● Experiments and events (4.2) ● Intersections, unions, and complements (4.5) ● The mean and standard deviation for a discrete random variable (4.8) ● Probability distributions for discrete random variables (4.8) ● Random variables (4.8) ● Relative frequency deﬁnition of probability (4.3)

What’s the Difference between Mutually Exclusive and Independent Events?

127

128

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

4.1

THE ROLE OF PROBABILITY IN STATISTICS Probability and statistics are related in an important way. Probability is used as a tool; it allows you to evaluate the reliability of your conclusions about the population when you have only sample information. Consider these situations: •

•

When you toss a single coin, you will see either a head (H) or a tail (T). If you toss the coin repeatedly, you will generate an inﬁnitely large number of Hs and Ts—the entire population. What does this population look like? If the coin is fair, then the population should contain 50% Hs and 50% Ts. Now toss the coin one more time. What is the chance of getting a head? Most people would say that the “probability” or chance is 1/2. Now suppose you are not sure whether the coin is fair; that is, you are not sure whether the makeup of the population is 50–50. You decide to perform a simple experiment. You toss the coin n 10 times and observe 10 heads in a row. Can you conclude that the coin is fair? Probably not, because if the coin were fair, observing 10 heads in a row would be very unlikely; that is, the “probability” would be very small. It is more likely that the coin is biased.

As in the coin-tossing example, statisticians use probability in two ways. When the population is known, probability is used to describe the likelihood of observing a particular sample outcome. When the population is unknown and only a sample from that population is available, probability is used in making statements about the makeup of the population—that is, in making statistical inferences. In Chapters 4–7, you will learn many different ways to calculate probabilities. You will assume that the population is known and calculate the probability of observing various sample outcomes. Once you begin to use probability for statistical inference in Chapter 8, the population will be unknown and you will use your knowledge of probability to make reliable inferences from sample information. We begin with some simple examples to help you grasp the basic concepts of probability.

4.2

EVENTS AND THE SAMPLE SPACE Data are obtained by observing either uncontrolled events in nature or controlled situations in a laboratory. We use the term experiment to describe either method of data collection. An experiment is the process by which an observation (or measurement) is obtained. Definition

The observation or measurement generated by an experiment may or may not produce a numerical value. Here are some examples of experiments: • • •

Recording a test grade Measuring daily rainfall Interviewing a householder to obtain his or her opinion on a greenbelt zoning ordinance

4.2 EVENTS AND THE SAMPLE SPACE

• •

❍

129

Testing a printed circuit board to determine whether it is a defective product or an acceptable product Tossing a coin and observing the face that appears

When an experiment is performed, what we observe is an outcome called a simple event, often denoted by the capital E with a subscript. A simple event is the outcome that is observed on a single repetition of the experiment. Definition

EXAMPLE

4.1

Experiment: Toss a die and observe the number that appears on the upper face. List the simple events in the experiment. When the die is tossed once, there are six possible outcomes. There are the simple events, listed below.

Solution

Event E1: Observe a 1 Event E2: Observe a 2 Event E3: Observe a 3

Event E4: Observe a 4 Event E5: Observe a 5 Event E6: Observe a 6

We can now deﬁne an event as a collection of simple events, often denoted by a capital letter. Definition

An event is a collection of simple events.

EXAMPLE co n tin u ed

4.1

We can deﬁne the events A and B for the die tossing experiment: A: Observe an odd number B: Observe a number less than 4 Since event A occurs if the upper face is 1, 3, or 5, it is a collection of three simple events and we write A {E1, E3, E5}. Similarly, the event B occurs if the upper face is 1, 2, or 3 and is deﬁned as a collection or set of these three simple events: B {E1, E2, E3}. Sometimes when one event occurs, it means that another event cannot. Two events are mutually exclusive if, when one event occurs, the others cannot, and vice versa. Definition

In the die-tossing experiment, events A and B are not mutually exclusive, because they have two outcomes in common—if the number on the upper face of the die is a 1 or a 3. Both events A and B will occur if either E1 or E3 is observed when the experiment is performed. In contrast, the six simple events E1, E2, . . . , E6 form a set of all mutually exclusive outcomes of the experiment. When the experiment is performed once, one and only one of these simple events can occur. Definition

The set of all simple events is called the sample space, S.

130

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

Sometimes it helps to visualize an experiment using a picture called a Venn diagram, shown in Figure 4.1. The outer box represents the sample space, which contains all of the simple events, represented by labeled points. Since an event is a collection of one or more simple events, the appropriate points are circled and labeled with the event letter. For the die-tossing experiment, the sample space is S {E1, E2, E3, E4, E5, E6} or, more simply, S {1, 2, 3, 4, 5, 6}. The events A {1, 3, 5} and B {1, 2, 3} are circled in the Venn diagram.

FI GU R E 4 .1

Venn diagram for die tossing

● A

B E2

E1 E5

E3 E4

EXAMPLE

4.2

Experiment: Toss a single coin and observe the result. These are the simple events: E1: Observe a head (H) E2: Observe a tail (T) The sample space is S {E1, E2}, or, more simply, S {H, T}.

EXAMPLE

4.3

Experiment: Record a person’s blood type. The four mutually exclusive possible outcomes are these simple events: E1: Blood type A E2: Blood type B E3: Blood type AB E4: Blood type O The sample space is S {E1, E2, E3, E4}, or S {A, B, AB, O}. Some experiments can be generated in stages, and the sample space can be displayed in a tree diagram. Each successive level of branching on the tree corresponds to a step required to generate the ﬁnal outcome.

EXAMPLE

4.4

A medical technician records a person’s blood type and Rh factor. List the simple events in the experiment. For each person, a two-stage procedure is needed to record the two variables of interest. The tree diagram is shown in Figure 4.2. The eight simple events in the tree diagram form the sample space, S {A, A, B, B, AB, AB, O, O}.

Solution

4.3 CALCULATING PROBABILITIES USING SIMPLE EVENTS

❍

131

An alternative way to display the simple events is to use a probability table, as shown in Table 4.1. The rows and columns show the possible outcomes at the ﬁrst and second stages, respectively, and the simple events are shown in the cells of the table.

F IGU R E 4 .2

●

Tree diagram for Example 4.4

Blood type

Rh factor

Outcome

E1 : A+

A _

E2 : A–

E3 : B+

E4 : B– E5 : AB+

B + AB _ +

E6 : AB– E7 : O+

O _

TABLE 4.1

●

E8 : O–

Probability Table for Example 4.4 Blood Type

4.3

Rh Factor

Negative Positive

A A

B B

AB AB

O O

CALCULATING PROBABILITIES USING SIMPLE EVENTS The probability of an event A is a measure of our belief that the event A will occur. One practical way to interpret this measure is with the concept of relative frequency. Recall from Chapter 1 that if an experiment is performed n times, then the relative frequency of a particular occurrence—say, A—is Frequency Relative frequency n where the frequency is the number of times the event A occurred. If you let n, the number of repetitions of the experiment, become larger and larger (n 씮 ), you will eventually generate the entire population. In this population, the relative frequency of the event A is deﬁned as the probability of event A; that is, Frequency P(A) lim n씮 n Since P(A) behaves like a relative frequency, P(A) must be a proportion lying between 0 and 1; P(A) 0 if the event A never occurs, and P(A) 1 if the event A always occurs. The closer P(A) is to 1, the more likely it is that A will occur.

132

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

For example, if you tossed a balanced, six-sided die an infinite number of times, you would expect the relative frequency for any of the six values, x 1, 2, 3, 4, 5, 6, to be 1/6. Needless to say, it would be very time-consuming, if not impossible, to repeat an experiment an infinite number of times. For this reason, there are alternative methods for calculating probabilities that make use of the relative frequency concept. An important consequence of the relative frequency definition of probability involves the simple events. Since the simple events are mutually exclusive, their probabilities must satisfy two conditions. REQUIREMENTS FOR SIMPLE-EVENT PROBABILITIES • •

Each probability must lie between 0 and 1. The sum of the probabilities for all simple events in S equals 1.

When it is possible to write down the simple events associated with an experiment and to determine their respective probabilities, we can ﬁnd the probability of an event A by summing the probabilities for all the simple events contained in the event A. The probability of an event A is equal to the sum of the probabilities of the simple events contained in A. Definition

EXAMPLE

4.5

Toss two fair coins and record the outcome. Find the probability of observing exactly one head in the two tosses. To list the simple events in the sample space, you can use a tree diagram as shown in Figure 4.3. The letters H and T mean that you observed a head or a tail, respectively, on a particular toss. To assign probabilities to each of the four simple events, you need to remember that the coins are fair. Therefore, any of the four simple events is as likely as any other. Since the sum of the four simple events must be 1, each must have probability P(Ei) 1/4. The simple events in the sample space are shown in Table 4.2, along with their equally likely probabilities. To ﬁnd P(A) P(observe exactly one head), you need to ﬁnd all the simple events that result in event A—namely, E2 and E3: Solution

Probabilities must lie between 0 and 1.

P(A) P(E2) P(E3) 1 1 1 4 4 2

F IGU R E 4 .3

Tree diagram for Example 4.5

●

First coin

Second coin Head (H)

Outcome

E1 = (HH)

Head (H) Tail (T) The probabilities of all the simple events must add to 1.

Head (H)

E2 = (HT) E3 = (TH)

Tail (T) Tail (T)

E4 = (TT)

4.3 CALCULATING PROBABILITIES USING SIMPLE EVENTS

TABLE 4.2

EXAMPLE

●

4.6

❍

133

Simple Events and Their Probabilities Event

First Coin

Second Coin

P (Ei)

E1 E2 E3 E4

H H T T

H T H T

1/4 1/4 1/4 1/4

The proportions of blood phenotypes A, B, AB, and O in the population of all Caucasians in the United States are reported as .41, .10, .04, and .45, respectively.1 If a single Caucasian is chosen randomly from the population, what is the probability that he or she will have either type A or type AB blood? The four simple events, A, B, AB, and O, do not have equally likely probabilities. Their probabilities are found using the relative frequency concept as

Solution

P(A) .41

P(B) .10

P(AB) .04

P(O) .45

The event of interest consists of two simple events, so P(person is either type A or type AB) P(A) P(AB) .41 .04 .45 EXAMPLE

4.7

A candy dish contains one yellow and two red candies. You close your eyes, choose two candies one at a time from the dish, and record their colors. What is the probability that both candies are red? draw 2

Y R2

Since no probabilities are given, you must list the simple events in the sample space. The two-stage selection of the candies suggests a tree diagram, shown in Figure 4.4. There are two red candies in the dish, so you can use the letters R1, R2, and Y to indicate that you have selected the ﬁrst red, the second red, or the yellow candy, respectively. Since you closed your eyes when you chose the candies, all six choices should be equally likely and are assigned probability 1/6. If A is the event that both candies are red, then Solution

A {R1R2, R2R1} Thus, P(A) P(R1R2) P(R2R1) 1 1 1 6 6 3 FIGU R E 4 .4

Tree diagram for Example 4.7

A tree diagram helps to ﬁnd simple events. Branch step toward outcome Following branches ⇒ list of simple events

●

First choice R1

Second choice

Simple event

Probability

R1 R 2

1/6

R1 Y

1/6

R2 R 1

1/6

R2 Y

1/6

Y R1

1/6

Y R2

1/6

134

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

CALCULATING THE PROBABILITY OF AN EVENT 1. 2. 3. 4.

List all the simple events in the sample space. Assign an appropriate probability to each simple event. Determine which simple events result in the event of interest. Sum the probabilities of the simple events that result in the event of interest.

In your calculation, you must always be careful that you satisfy these two conditions: • •

Include all simple events in the sample space. Assign realistic probabilities to the simple events.

When the sample space is large, it is easy to unintentionally omit some of the simple events. If this happens, or if your assigned probabilities are wrong, your answers will not be useful in practice. One way to determine the required number of simple events is to use the counting rules presented in the next optional section. These rules can be used to solve more complex problems, which generally involve a large number of simple events. If you need to master only the basic concepts of probability, you may choose to skip the next section.

4.3

EXERCISES

BASIC TECHNIQUES

4.2 A sample space S consists of ﬁve simple events

4.1 Tossing a Die An experiment involves tossing a

with these probabilities:

single die. These are some events: A: Observe a 2 B: Observe an even number C: Observe a number greater than 2 D: Observe both A and B

P(E1) P(E2) .15 P(E3) .4 P(E4) 2P(E5) a. Find the probabilities for simple events E4 and E5. b. Find the probabilities for these two events:

E: Observe A or B or both

A {E1, E3, E4}

F: Observe both A and C

B {E2, E3}

a. List the simple events in the sample space. b. List the simple events in each of the events A through F. c. What probabilities should you assign to the simple events? d. Calculate the probabilities of the six events A through F by adding the appropriate simple-event probabilities.

c. List the simple events that are either in event A or event B or both. d. List the simple events that are in both event A and event B. 4.3 A sample space contains 10 simple events: E1,

E2, . . . , E10. If P(E1) 3P(E2) .45 and the remaining simple events are equiprobable, ﬁnd the probabilities of these remaining simple events.

4.3 CALCULATING PROBABILITIES USING SIMPLE EVENTS

4.4 Free Throws A particular basketball player hits

70% of her free throws. When she tosses a pair of free throws, the four possible simple events and three of their associated probabilities are as given in the table: Simple Event 1 2 3 4

Outcome of First Free Throw

Outcome of Second Free Throw

Hit Hit Miss Miss

Hit Miss Hit Miss

.49 ? .21 .09

4.5 Four Coins A jar contains four coins: a nickel, a dime, a quarter, and a half-dollar. Three coins are randomly selected from the jar. a. List the simple events in S. b. What is the probability that the selection will contain the half-dollar? c. What is the probability that the total amount drawn will equal 60¢ or more? 4.6 Preschool or Not? On the ﬁrst day of kindergarten, the teacher randomly selects 1 of his 25 students and records the student’s gender, as well as whether or not that student had gone to preschool. a. How would you describe the experiment? b. Construct a tree diagram for this experiment. How many simple events are there? c. The table below shows the distribution of the 25 students according to gender and preschool experience. Use the table to assign probabilities to the simple events in part b. Male

Female

8 6

9 2

135

containing three red and two yellow balls. Its color is noted, and the ball is returned to the bowl before a second ball is selected. List the additional ﬁve simple events that must be added to the sample space in Exercise 4.7.

Probability

a. Find the probability that the player will hit on the ﬁrst throw and miss on the second. b. Find the probability that the player will hit on at least one of the two free throws.

Preschool No Preschool

❍

d. What is the probability that the randomly selected student is male? What is the probability that the student is a female and did not go to preschool? 4.7 The Urn Problem A bowl contains three red

and two yellow balls. Two balls are randomly selected and their colors recorded. Use a tree diagram to list the 20 simple events in the experiment, keeping in mind the order in which the balls are drawn. 4.8 The Urn Problem, continued Refer to Exer-

cise 4.7. A ball is randomly selected from the bowl

APPLICATIONS 4.9 Need Eyeglasses? A survey classiﬁed a large number of adults according to whether they were judged to need eyeglasses to correct their reading vision and whether they used eyeglasses when reading. The proportions falling into the four categories are shown in the table. (Note that a small proportion, .02, of adults used eyeglasses when in fact they were judged not to need them.) Used Eyeglasses for Reading Judged to Need Eyeglasses

Yes

Yes No

.44 .02

.14 .40

If a single adult is selected from this large group, find the probability of each event: a. The adult is judged to need eyeglasses. b. The adult needs eyeglasses for reading but does not use them. c. The adult uses eyeglasses for reading whether he or she needs them or not. 4.10 Roulette The game of roulette uses a wheel containing 38 pockets. Thirty-six pockets are numbered 1, 2, . . . , 36, and the remaining two are marked 0 and 00. The wheel is spun, and a pocket is identified as the “winner.” Assume that the observance of any one pocket is just as likely as any other. a. Identify the simple events in a single spin of the roulette wheel. b. Assign probabilities to the simple events. c. Let A be the event that you observe either a 0 or a 00. List the simple events in the event A and find P(A). d. Suppose you placed bets on the numbers 1 through 18. What is the probability that one of your numbers is the winner? 4.11 Jury Duty Three people are randomly selected

from voter registration and driving records to report for jury duty. The gender of each person is noted by the county clerk.

136 ❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

a. Deﬁne the experiment. b. List the simple events in S. c. If each person is just as likely to be a man as a woman, what probability do you assign to each simple event? d. What is the probability that only one of the three is a man? e. What is the probability that all three are women? 4.12 Jury Duty II Refer to Exercise 4.11. Suppose

that there are six prospective jurors, four men and two women, who might be impaneled to sit on the jury in a criminal case. Two jurors are randomly selected from these six to ﬁll the two remaining jury seats. a. List the simple events in the experiment (HINT: There are 15 simple events if you ignore the order of selection of the two jurors.) b. What is the probability that both impaneled jurors are women? 4.13 Tea Tasters A food company plans to conduct an experiment to compare its brand of tea with that of two competitors. A single person is hired to taste and rank each of three brands of tea, which are unmarked except for identifying symbols A, B, and C. a. Deﬁne the experiment. b. List the simple events in S. c. If the taster has no ability to distinguish a difference in taste among teas, what is the probability that the taster will rank tea type A as the most desirable? As the least desirable?

4.15 Fruit Flies In a genetics experiment, the

researcher mated two Drosophila fruit ﬂies and observed the traits of 300 offspring. The results are shown in the table. Wing Size Eye Color

Normal

Miniature

Normal Vermillion

140 3

6 151

One of these offspring is randomly selected and observed for the two genetic traits. a. What is the probability that the fly has normal eye color and normal wing size? b. What is the probability that the fly has vermillion eyes? c. What is the probability that the fly has either vermillion eyes or miniature wings, or both? 4.16 Creation Which of the following comes closest to your views on the origin and development of human beings? Do you believe that human beings have developed over millions of years from less advanced forms of life, but that God has guided the process? Do you think that human beings have developed over millions of years, and that God had no part in the process? Or do you believe that God created humans in their present form within the last 10,000 years or so? When asked these questions, the proportions of Americans with varying opinions are approximately as shown in the table.2 Opinion Guided by God God had no part God created in present form No opinion

Proportion .36 .13 .46 .05

4.14 100-Meter Run Four equally qualiﬁed runners,

Source: Adapted from www.pollingreport.com

John, Bill, Ed, and Dave, run a 100-meter sprint, and the order of finish is recorded. a. How many simple events are in the sample space? b. If the runners are equally qualified, what probability should you assign to each simple event? c. What is the probability that Dave wins the race? d. What is the probability that Dave wins and John places second? e. What is the probability that Ed finishes last?

Suppose that one person is randomly selected and his or her opinion on this question is recorded. a. What are the simple events in the experiment? b. Are the simple events in part a equally likely? If not, what are the probabilities? c. What is the probability that the person feels that God had some part in the creation of humans? d. What is the probability that the person feels that God had no part in the process?

4.4 USEFUL COUNTING RULES (OPTIONAL)

4.4

❍

137

USEFUL COUNTING RULES (OPTIONAL) Suppose that an experiment involves a large number N of simple events and you know that all the simple events are equally likely. Then each simple event has probability 1/N, and the probability of an event A can be calculated as nA P(A) N where nA is the number of simple events that result in the event A. In this section, we present three simple rules that can be used to count either N, the number of simple events in the sample space, or nA, the number of simple events in event A. Once you have obtained these counts, you can find P(A) without actually listing all the simple events. THE mn RULE Consider an experiment that is performed in two stages. If the first stage can be accomplished in m ways and for each of these ways, the second stage can be accomplished in n ways, then there are mn ways to accomplish the experiment. For example, suppose that you can order a car in one of three styles and in one of four paint colors. To find out how many options are available, you can think of first picking one of the m 3 styles and then selecting one of the n 4 paint colors. Using the mn Rule, as shown in Figure 4.5, you have mn (3)(4) 12 possible options.

FIGU R E 4 .5

Style–color combinations

● Style

Color 1 2

3 4 1 2

3 4

EXAMPLE

4.8

Two dice are tossed. How many simple events are in the sample space S? Solution The ﬁrst die can fall in one of m 6 ways, and the second die can fall in one of n 6 ways. Since the experiment involves two stages, forming the pairs of numbers shown on the two faces, the total number of simple events in S is

mn (6)(6) 36

138

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

The Java applet called Tossing Dice gives a visual display of the 36 simple events described in Example 4.8. You can use this applet to ﬁnd probabilities for any event involving the tossing of two fair dice. By clicking on the appropriate dice combinations, we have found the probability of observing a sum of 3 on the upper faces to be 2/36 .056. What is the probability that the sum equals 4? You will use this applet for the MyApplet Exercises at the end of the chapter. FI GU R E 4 .6

Tossing Dice applet

EXAMPLE

●

4.9

A candy dish contains one yellow and two red candies. Two candies are selected one at a time from the dish, and their colors are recorded. How many simple events are in the sample space S? draw 2

Y R2

The ﬁrst candy can be chosen in m 3 ways. Since one candy is now gone, the second candy can be chosen in n 2 ways. The total number of simple events is

Solution

mn (3)(2) 6 These six simple events were listed in Example 4.7. We can extend the mn Rule for an experiment that is performed in more than two stages. THE EXTENDED mn RULE If an experiment is performed in k stages, with n1 ways to accomplish the ﬁrst stage, n2 ways to accomplish the second stage, . . . , and nk ways to accomplish the kth stage, then the number of ways to accomplish the experiment is n1n2n3 nk

4.4 USEFUL COUNTING RULES (OPTIONAL)

EXAMPLE

4.10

❍

139

How many simple events are in the sample space when three coins are tossed? Solution

Each coin can land in one of two ways. Hence, the number of simple

events is (2)(2)(2) 8

EXAMPLE

4.11

A truck driver can take three routes from city A to city B, four from city B to city C, and three from city C to city D. If, when traveling from A to D, the driver must drive from A to B to C to D, how many possible A-to-D routes are available? Solution

Let

n1 Number of routes from A to B 3 n2 Number of routes from B to C 4 n3 Number of routes from C to D 3 Then the total number of ways to construct a complete route, taking one subroute from each of the three groups, (A to B), (B to C), and (C to D), is n1n2n3 (3)(4)(3) 36 A second useful counting rule follows from the mn Rule and involves orderings or permutations. For example, suppose you have three books, A, B, and C, but you have room for only two on your bookshelf. In how many ways can you select and arrange the two books? There are three choices for the two books—A and B, A and C, or B and C—but each of the pairs can be arranged in two ways on the shelf. All the permutations of the two books, chosen from three, are listed in Table 4.3. The mn Rule implies that there are 6 ways, because the ﬁrst book can be chosen in m 3 ways and the second in n 2 ways, so the result is mn 6.

TABLE 4.3

●

Permutations of Two Books Chosen from Three Combinations of Two

Reordering of Combinations

AB AC BC

BA CA CB

In how many ways can you arrange all three books on your bookshelf? These are the six permutations: ABC BCA

ACB CAB

BAC CBA

Since the ﬁrst book can be chosen in n1 3 ways, the second in n2 2 ways, and the third in n3 1 way, the total number of orderings is n1n2n3 (3)(2)(1) 6. Rather than applying the mn Rule each time, you can ﬁnd the number of orderings using a general formula involving factorial notation.

140

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

A COUNTING RULE FOR PERMUTATIONS The number of ways we can arrange n distinct objects, taking them r at a time, is n! Prn (n r)! where n! n(n 1)(n 2) (3)(2)(1) and 0! 1. Since r objects are chosen, this is an r-stage experiment. The ﬁrst object can be chosen in n ways, the second in (n 1) ways, the third in (n 2) ways, and the rth in (n r 1) ways. We can simplify this awkward notation using the counting rule for permutations because n! n(n 1)(n 2) (n r 1)(n r) (2)(1) (n r)! (n r) (2)(1) n(n 1) (n r 1) A SPECIAL CASE: ARRANGING n ITEMS The number of ways to arrange an entire set of n distinct items is Pnn n!

EXAMPLE

4.12

Three lottery tickets are drawn from a total of 50. If the tickets will be distributed to each of three employees in the order in which they are drawn, the order will be important. How many simple events are associated with the experiment? Solution

The total number of simple events is

50! P 50 3 50(49)(48) 117,600 47! EXAMPLE

4.13

A piece of equipment is composed of ﬁve parts that can be assembled in any order. A test is to be conducted to determine the time necessary for each order of assembly. If each order is to be tested once, how many tests must be conducted? Solution

The total number of tests is

5! P 55 5(4)(3)(2)(1) 120 0! When we counted the number of permutations of the two books chosen for your bookshelf, we used a systematic approach: • •

First we counted the number of combinations or pairs of books to be chosen. Then we counted the number of ways to arrange the two chosen books on the shelf.

Sometimes the ordering or arrangement of the objects is not important, but only the objects that are chosen. In this case, you can use a counting rule for combinations. For example, you may not care in what order the books are placed on the shelf, but

4.4 USEFUL COUNTING RULES (OPTIONAL)

❍

141

only which books you are able to shelve. When a ﬁve-person committee is chosen from a group of 12 students, the order of choice is unimportant because all ﬁve students will be equal members of the committee. A COUNTING RULE FOR COMBINATIONS The number of distinct combinations of n distinct objects that can be formed, taking them r at a time, is n! C rn r!(n r)! The number of combinations and the number of permutations are related: Pn C rn r r! You can see that C rn results when you divide the number of permutations by r!, the number of ways of rearranging each distinct group of r objects chosen from the total n. EXAMPLE

4.14

A printed circuit board may be purchased from ﬁve suppliers. In how many ways can three suppliers be chosen from the ﬁve? Since it is important to know only which three have been chosen, not the order of selection, the number of ways is

Solution

(5)(4) 5! C 35 10 2 3!2! The next example illustrates the use of counting rules to solve a probability problem. EXAMPLE

4.15

Five manufacturers produce a certain electronic device, whose quality varies from manufacturer to manufacturer. If you were to select three manufacturers at random, what is the chance that the selection would contain exactly two of the best three? The simple events in this experiment consist of all possible combinations of three manufacturers, chosen from a group of five. Of these five, three have been designated as “best” and two as “not best.” You can think of a candy dish containing three red and two yellow candies, from which you will select three, as illustrated in Figure 4.7. The total number of simple events N can be counted as the number of ways to choose three of the five manufacturers, or

Solution

5! N C 35 10 3!2! FIGU R E 4 .7

Illustration for Example 4.15

● Choose 3

3 “best” 2 “not best”

142

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

Since the manufacturers are selected at random, any of these 10 simple events will be equally likely, with probability 1/10. But how many of these simple events result in the event A: Exactly two of the “best” three You can count nA, the number of events in A, in two steps because event A will occur when you select two of the “best” three and one of the two “not best.” There are 3! C 23 3 2!1! ways to accomplish the ﬁrst stage and 2! C 12 2 1!1! ways to accomplish the second stage. Applying the mn Rule, we ﬁnd there are nA (3)(2) 6 of the 10 simple events in event A and P(A) nA/N 6/10. Many other counting rules are available in addition to the three presented in this section. If you are interested in this topic, you should consult one of the many textbooks on combinatorial mathematics.

4.4

EXERCISES

BASIC TECHNIQUES 4.17 You have two groups of distinctly different

items, 10 in the ﬁrst group and 8 in the second. If you select one item from each group, how many different pairs can you form? 4.18 You have three groups of distinctly different

items, four in the ﬁrst group, seven in the second, and three in the third. If you select one item from each group, how many different triplets can you form? 4.19 Permutations Evaluate the following permutations. (HINT: Your scientiﬁc calculator may have a function that allows you to calculate permutations and combinations quite easily.)

a. P 35

b. P 10 9

c. P 66

d. P 20 1

4.22 Choosing People, again In how many ways can you select two people from a group of 20 if the order of selection is not important? 4.23 Dice Three dice are tossed. How many simple

events are in the sample space? 4.24 Coins Four coins are tossed. How many simple

events are in the sample space? 4.25 The Urn Problem, again Three balls are selected from a box containing 10 balls. The order of selection is not important. How many simple events are in the sample space?

APPLICATIONS 4.26 What to Wear? You own 4 pairs of jeans, 12

4.20 Combinations Evaluate these combinations:

C 35

C 10 9

C 66

C 20 1

4.21 Choosing People In how many ways can you select ﬁve people from a group of eight if the order of selection is important?

clean T-shirts, and 4 wearable pairs of sneakers. How many outﬁts (jeans, T-shirt, and sneakers) can you create? 4.27 Itineraries A businessman in New York is

preparing an itinerary for a visit to six major cities. The distance traveled, and hence the cost of the trip,

4.4 USEFUL COUNTING RULES (OPTIONAL)

will depend on the order in which he plans his route. How many different itineraries (and trip costs) are possible? 4.28 Vacation Plans Your family vacation involves

a cross-country air ﬂight, a rental car, and a hotel stay in Boston. If you can choose from four major air carriers, ﬁve car rental agencies, and three major hotel chains, how many options are available for your vacation accommodations? 4.29 A Card Game Three students are playing a

card game. They decide to choose the first person to play by each selecting a card from the 52-card deck and looking for the highest card in value and suit. They rank the suits from lowest to highest: clubs, diamonds, hearts, and spades. a. If the card is replaced in the deck after each student chooses, how many possible configurations of the three choices are possible? b. How many configurations are there in which each student picks a different card? c. What is the probability that all three students pick exactly the same card? d. What is the probability that all three students pick different cards? 4.30 Dinner at Gerard’s A French restaurant in Riverside, California, offers a special summer menu in which, for a fixed dinner cost, you can choose from one of two salads, one of two entrees, and one of two desserts. How many different dinners are available? 4.31 Playing Poker Five cards are selected from a

52-card deck for a poker hand. a. How many simple events are in the sample space? b. A royal flush is a hand that contains the A, K, Q, J, and 10, all in the same suit. How many ways are there to get a royal flush? c. What is the probability of being dealt a royal flush? 4.32 Poker II Refer to Exercise 4.31. You have a poker hand containing four of a kind.

a. How many possible poker hands can be dealt? b. In how many ways can you receive four cards of the same face value and one card from the other 48 available cards? c. What is the probability of being dealt four of a kind?

❍

143

4.33 A Hospital Survey A study is to be conducted in a hospital to determine the attitudes of nurses toward various administrative procedures. If a sample of 10 nurses is to be selected from a total of 90, how many different samples can be selected? (HINT: Is order important in determining the makeup of the sample to be selected for the survey?) 4.34 Traffic Problems Two city council members are to be selected from a total of ﬁve to form a subcommittee to study the city’s traffic problems. a. How many different subcommittees are possible? b. If all possible council members have an equal chance of being selected, what is the probability that members Smith and Jones are both selected? 4.35 The WNBA Professional basketball is now a reality for women basketball players in the United States. There are two conferences in the WNBA, each with seven teams, as shown in the table below. Western Conference

Eastern Conference

Houston Comets Minnesota Lynx Phoenix Mercury Sacramento Monarchs Los Angeles Sparks Seattle Storm San Antonio Silver Stars

Indiana Fever New York Liberty Washington Mystics Detroit Shock Charlotte Sting Connecticut Sun Chicago Sky

Two teams, one from each conference, are randomly selected to play an exhibition game. a. How many pairs of teams can be chosen? b. What is the probability that the two teams are Los Angeles and New York? c. What is the probability that the Western Conference team is from California? 4.36 100-Meter Run, again Refer to Exercise 4.14,

in which a 100-meter sprint is run by John, Bill, Ed, and Dave. Assume that all of the runners are equally qualified, so that any order of finish is equally likely. Use the mn Rule or permutations to answer these questions: a. How many orders of finish are possible? b. What is the probability that Dave wins the sprint? c. What is the probability that Dave wins and John places second? d. What is the probability that Ed finishes last?

144 ❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

4.37 Gender Bias? The following case occurred in Gainesville, Florida. The eight-member Human Relations Advisory Board considered the complaint of a woman who claimed discrimination, based on her gender, on the part of a local surveying company. The board, composed of five women and three men, voted 5–3 in favor of the plaintiff, the five women voting for the plaintiff and the three men against. The attorney representing the company appealed the board’s decision by claiming gender bias on the part of the board members. If the vote in favor of the plaintiff was 5–3 and the board members were not biased by gender, what is the probability that the vote would split along gender lines (five women for, three men against)?

4.5

4.38 Cramming A student prepares for an exam

by studying a list of 10 problems. She can solve 6 of them. For the exam, the instructor selects 5 questions at random from the list of 10. What is the probability that the student can solve all 5 problems on the exam? 4.39 Monkey Business A monkey is given 12 blocks: 3 shaped like squares, 3 like rectangles, 3 like triangles, and 3 like circles. If it draws three of each kind in order—say, 3 triangles, then 3 squares, and so on—would you suspect that the monkey associates identically shaped ﬁgures? Calculate the probability of this event.

EVENT RELATIONS AND PROBABILITY RULES Sometimes the event of interest can be formed as a combination of several other events. Let A and B be two events deﬁned on the sample space S. Here are three important relationships between events. The union of events A and B, denoted by A B, is the event that either A or B or both occur. Definition

The intersection of events A and B, denoted by A B, is the event that both A and B occur.† Definition

Definition

The complement of an event A, denoted by Ac, is the event that A does

not occur. Figures 4.8, 4.9, and 4.10 show Venn diagram representations of A B, A B, and Ac, respectively. Any simple event in the shaded area is a possible outcome resulting in the appropriate event. One way to ﬁnd the probabilities of the union, the intersection, or the complement is to sum the probabilities of all the associated simple events.

†

Some authors use the notation AB.

4.5 EVENT RELATIONS AND PROBABILITY RULES

FIGURE 4.8

FIGURE 4.9

●

Venn diagram of A B

The complement of an event

Union ⇔ “either . . . or . . . or both” or just “or”

FIGU R E 4 .1 0

145

●

Venn diagram A B S

Intersection ⇔ “both . . . and” or just “and”

❍

A∪B

A∩B

●

S Ac

EXAMPLE

4.16

Two fair coins are tossed, and the outcome is recorded. These are the events of interest: A: Observe at least one head B: Observe at least one tail Deﬁne the events A, B, A B, A B, and Ac as collections of simple events, and ﬁnd their probabilities. Solution

Recall from Example 4.5 that the simple events for this experiment are

E1: HH (head on ﬁrst coin, head on second) E2: HT E3: TH E4: TT and that each simple event has probability 1/4. Event A, at least one head, occurs if E1, E2, or E3 occurs, so that A {E1, E2, E3}

3 P(A) 4

and Ac {E4}

1 P(Ac) 4

146

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

Similarly, 3 P(B) 4

B {E2, E3, E4} A B {E2, E3}

1 P(A B) 2

A B {E1, E2, E3, E4}

4 P(A B) 1 4

Note that (A B) S, the sample space, and is thus certain to occur. The concept of unions and intersections can be extended to more than two events. For example, the union of three events A, B, and C, which is written as A B C, is the set of simple events that are in A or B or C or in any combination of those events. Similarly, the intersection of three events A, B, and C, which is written as A B C, is the collection of simple events that are common to the three events A, B, and C.

Calculating Probabilities for Unions and Complements When we can write the event of interest in the form of a union, a complement, or an intersection, there are special probability rules that can simplify our calculations. The ﬁrst rule deals with unions of events. THE ADDITION RULE Given two events, A and B, the probability of their union, A B, is equal to P(A B) P(A) P(B) P(A B) Notice in the Venn diagram in Figure 4.11 that the sum P(A) P(B) double counts the simple events that are common to both A and B. Subtracting P(A B) gives the correct result.

FIGU R E 4 .1 1

The Addition Rule

●

S A

A∩B

When two events A and B are mutually exclusive or disjoint, it means that when A occurs, B cannot, and vice versa. This means that the probability that they both

4.5 EVENT RELATIONS AND PROBABILITY RULES

❍

147

occur, P(A B), must be zero. Figure 4.12 is a Venn diagram representation of two such events with no simple events in common. F I GU R E 4 .1 2

●

Two disjoint events

When two events A and B are mutually exclusive, then P(A B) 0 and the Addition Rule simpliﬁes to

Remember, mutually exclusive ⇔ P (A B) 0.

P(A B) P(A) P(B) The second rule deals with complements of events. You can see from the Venn diagram in Figure 4.10 that A and Ac are mutually exclusive and that A Ac S, the entire sample space. It follows that P(A) P(Ac ) 1 and P(Ac ) 1 P(A) RULE FOR COMPLEMENTS P(Ac ) 1 P(A)

EXAMPLE

TABLE 4.4

An oil-prospecting ﬁrm plans to drill two exploratory wells. Past evidence is used to assess the possible outcomes listed in Table 4.4.

4.17

●

Outcomes for Oil-Drilling Experiment Event A B C

Description Neither well produces oil or gas Exactly one well produces oil or gas Both wells produce oil or gas

Probability .80 .18 .02

Find P(A B) and P(B C). Solution By their deﬁnition, events A, B, and C are jointly mutually exclusive because the occurrence of one event precludes the occurrence of either of the other two. Therefore,

P(A B) P(A) P(B) .80 .18 .98 and P(B C) P(B) P(C) .18 .02 .20

148

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

The event A B can be described as the event that at most one well produces oil or gas, and B C describes the event that at least one well produces gas or oil.

EXAMPLE

TABLE 4.5

4.18

In a telephone survey of 1000 adults, respondents were asked about the expense of a college education and the relative necessity of some form of ﬁnancial assistance. The respondents were classiﬁed according to whether they currently had a child in college and whether they thought the loan burden for most college students is too high, the right amount, or too little. The proportions responding in each category are shown in the probability table in Table 4.5. Suppose one respondent is chosen at random from this group. ●

Probability Table

Child in College (D) No Child in College (E)

Too High (A)

Right Amount (B)

Too Little (C )

.35 .25

.08 .20

.01 .11

1. What is the probability that the respondent has a child in college? 2. What is the probability that the respondent does not have a child in college? 3. What is the probability that the respondent has a child in college or thinks that the loan burden is too high? Table 4.5 gives the probabilities for the six simple events in the cells of the table. For example, the entry in the top left corner of the table is the probability that a respondent has a child in college and thinks the loan burden is too high (A D).

Solution

1. The event that a respondent has a child in college will occur regardless of his or her response to the question about loan burden. That is, event D consists of the simple events in the ﬁrst row: P(D) .35 .08 .01 .44 In general, the probabilities of marginal events such as D and A are found by summing the probabilities in the appropriate row or column. 2. The event that the respondent does not have a child in college is the complement of the event D denoted by D c. The probability of D c is found as P(D c ) 1 P(D) Using the result of part 1, we have P(D c ) 1 .44 .56 3. The event of interest is P(A D). Using the Addition Rule P(A D) P(A) P(D) P(A D) .60 .44 .35 .69

4.6 INDEPENDENCE, CONDITIONAL PROBABILITY, AND THE MULTIPLICATION RULE

4.6

❍

149

INDEPENDENCE, CONDITIONAL PROBABILITY, AND THE MULTIPLICATION RULE There is a probability rule that can be used to calculate the probability of the intersection of several events. However, this rule depends on the important statistical concept of independent or dependent events. Two events, A and B, are said to be independent if and only if the probability of event B is not inﬂuenced or changed by the occurrence of event A, or vice versa. Definition

Colorblindness Suppose a researcher notes a person’s gender and whether or not the person is colorblind to red and green. Does the probability that a person is colorblind change depending on whether the person is male or not? Deﬁne two events:

A: Person is a male B: Person is colorblind In this case, since colorblindness is a male sex-linked characteristic, the probability that a man is colorblind will be greater than the probability that a person chosen from the general population will be colorblind. The probability of event B, that a person is colorblind, depends on whether or not event A, that the person is a male, has occurred. We say that A and B are dependent events. Tossing Dice

On the other hand, consider tossing a single die two times, and

deﬁne two events: A: Observe a 2 on the ﬁrst toss B: Observe a 2 on the second toss If the die is fair, the probability of event A is P(A) 1/6. Consider the probability of event B. Regardless of whether event A has or has not occurred, the probability of observing a 2 on the second toss is still 1/6. We could write: P(B given that A occurred) 1/6 P(B given that A did not occur) 1/6 Since the probability of event B is not changed by the occurrence of event A, we say that A and B are independent events. The probability of an event A, given that the event B has occurred, is called the conditional probability of A, given that B has occurred, denoted by P(A兩B). The vertical bar is read “given” and the events appearing to the right of the bar are those that you know have occurred. We will use these probabilities to calculate the probability that both A and B occur when the experiment is performed. THE GENERAL MULTIPLICATION RULE The probability that both A and B occur when the experiment is performed is P(A B) P(A)P(B兩A) or P(A B) P(B)P(A兩B)

150

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

EXAMPLE

4.19

In a color preference experiment, eight toys are placed in a container. The toys are identical except for color—two are red, and six are green. A child is asked to choose two toys at random. What is the probability that the child chooses the two red toys? You can visualize the experiment using a tree diagram as shown in Figure 4.13. Deﬁne the following events:

Solution

R: Red toy is chosen G: Green toy is chosen

FI GU R E 4 .1 3

Tree diagram for Example 4.19

●

First choice

Second choice Red (1/7)

Simple event

Red (2/8) Green (6/7) Red (2/7)

RG GR

Green (6/8) Green (5/7)

The event A (both toys are red) can be constructed as the intersection of two events: A (R on first choice) (R on second choice) Since there are only two red toys in the container, the probability of choosing red on the first choise is 2/8. However, once this red toy has been chosen, the probability of red on the second choice is dependent on the outcome of the first choice (see Figure 4.13). If the first choice was red, the probability of choosing a second red toy is only 1/7 because there is only one red toy among the seven remaining. If the first choice was green, the probability of choosing red on the second choice is 2/7 because there are two red toys among the seven remaining. Using this information and the Multiplication Rule, you can find the probability of event A. P(A) P(R on first choice R on second choice) P(R on first choice) P(R on second choice)兩R on first)

冢冣冢冣

2 1 2 1 8 7 56 28 Sometimes you may need to use the Multiplication Rule in a slightly different form, so that you can calculate the conditional probability, P(A円B). Just rearrange the terms in the Multiplication Rule.

4.6 INDEPENDENCE, CONDITIONAL PROBABILITY, AND THE MULTIPLICATION RULE

❍

151

CONDITIONAL PROBABILITIES The conditional probability of event A, given that event B has occurred is P(A B) P(A兩B) if P(B) 0 P(B) The conditional probability of event B, given that event A has occurred is P(A B) P(B兩A) P(A)

P(A) 0

Suppose that in the general population, there are 51% men and 49% women, and that the proportions of colorblind men and women are shown in the probability table below:

Colorblindness, continued

Men(B)

Women (BC)

Total

Colorblind (A) Not Colorblind (AC)

.04 .47

.002 .488

.042 .958

Total

.51

.49

1.00

If a person is drawn at random from this population and is found to be a man (event B), what is the probability that the man is colorblind (event A)? If we know that the event B has occurred, we must restrict our focus to only the 51% of the population that is male. The probability of being colorblind, given that the person is male, is 4% of the 51%, or P(A B) .04 P(A兩B) .078 P(B) .51 What is the probability of being colorblind, given that the person is female? Now we are restricted to only the 49% of the population that is female, and P(A BC) .002 P(A兩BC) .004 P(BC) .49 Notice that the probability of event A changed, depending on whether event B occured. This indicates that these two events are dependent. When two events are independent—that is, if the probability of event B is the same, whether or not event A has occurred, then event A does not affect event B and P(B兩A) P(B) The Multiplication Rule can now be simpliﬁed. THE MULTIPLICATION RULE FOR INDEPENDENT EVENTS If two events A and B are independent, the probability that both A and B occur is P(A B) P(A)P(B) Similarly, if A, B, and C are mutually independent events (all pairs of events are independent), then the probability that A, B, and C all occur is P(A B C) P(A)P(B)P(C)

152

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

Coin Tosses at Football Games A football team is involved in two overtime periods during a given game, so that there are three coins tosses. If the coin is fair, what is the probability that they lose all three tosses? Solution

If the coin is fair, the event can be described in three steps:

A: lose the ﬁrst toss B: lose the second toss C: lose the third toss Since the tosses are independent, and since P(win) P(lose) .5 for any of the three tosses, P(A B C) P(A)P(B)P(C) (.5)(.5)(.5) .125 How can you check to see if two events are independent or dependent? The easiest solution is to redeﬁne the concept of independence in a more formal way. CHECKING FOR INDEPENDENCE Two events A and B are said to be independent if and only if either P(A B) P(A)P(B) or P(B兩A) P(B) Otherwise, the events are said to be dependent. EXAMPLE

4.20

Toss two coins and observe the outcome. Define these events: A: Head on the first coin B: Tail on the second coin Are events A and B independent? From previous examples, you know that S {HH, HT, TH, TT}. Use these four simple events to find

Solution

Remember, independence ⇔ P(A B) P (A)P(B).

1 1 1 P(A) , P(B) , and P(A B) . 2 2 4

冢冣冢冣

1 1 1 1 Since P(A)P(B) and P(A B) , we have P(A)P(B) P(A B) 2 2 4 4 and the two events must be independent.

EXAMPLE

4.21

Refer to the probability table in Example 4.18, which is reproduced below.

Child in College (D) No Child in College (E )

Too High (A)

Right Amount (B)

Too Little (C )

.35 .25

.08 .20

.01 .11

Are events D and A independent? Explain.

4.6 INDEPENDENCE, CONDITIONAL PROBABILITY, AND THE MULTIPLICATION RULE

❍

153

Solution

1. Use the probability table to ﬁnd P(A D) .35, P(A) .60, and P(D) .44. Then P(A)P(D) (.60)(.44) .264 and P(A D) .35 Since these two probabilities are not the same, events A and D are dependent. 2. Alternately, calculate P(A D) .35 P(A兩 D) .80 P(D) .44 Since P(A兩 D) .80 and P(A) .60, we are again led to the conclusion that events A and D are dependent.

What’s the Difference between Mutually Exclusive and Independent Events? Many students ﬁnd it hard to tell the difference between mutually exclusive and independent events. • When two events are mutually exclusive or disjoint, they cannot both happen when the experiment is performed. Once the event B has occurred, event A cannot occur, so that P(A兩B) 0, or vice versa. The occurrence of event B certainly affects the probability that event A can occur. •

Therefore, mutually exclusive events must be dependent.

•

When two events are mutually exclusive or disjoint, P(A B) 0 and P(A B) P(A) P(B).

•

When two events are independent, P(A B) P(A)P(B), and P(A B) P(A) P(B) P(A)P(B).

Exercise Reps Use the relationships above to ﬁll in the blanks in the table below. P(A)

P(B)

Conditions for Events A and B Mutually exclusive

Independent

P(A 8 B)

P(A 9 B)

P(A円B)

.6 .10

Answers are located on the perforated card at the back of this book.

Using probability rules to calculate the probability of an event requires some experience and ingenuity. You need to express the event of interest as a union or intersection (or the combination of both) of two or more events whose probabilities are known or easily calculated. Often you can do this in different ways; the key is to ﬁnd the right combination.

154

❍

EXAMPLE

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

4.22

Two cards are drawn from a deck of 52 cards. Calculate the probability that the draw includes an ace and a ten. Solution

Consider the event of interest: A: Draw an ace and a ten

Then A B C, where B: Draw the ace on the first draw and the ten on the second C: Draw the ten on the first draw and the ace on the second Events B and C were chosen to be mutually exclusive and also to be intersections of events with known probabilities; that is, B B1 B2 and C C1 C2 where B1: Draw an ace on the first draw B2: Draw a ten on the second draw C1: Draw a ten on the first draw C2: Draw an ace on the second draw Applying the Multiplication Rule, you get P(B1 B2) P(B1)P(B2兩B1)

冢冣冢冣

4 4 52 51 and

冢冣冢冣

4 4 P(C1 C2) 52 51 Then, applying the Addition Rule, P(A) P(B) P(C)

冢冣冢冣冢冣冢冣

4 4 4 4 8 52 51 52 51 663 Check each composition carefully to be certain that it is actually equal to the event of interest.

4.6

EXERCISES EXERCISE REPS These exercises refer to the MyPersonal Trainer section on page 153. 4.40 Use event relationships to ﬁll in the blanks in the table below. P(A)

P(B)

Conditions for Events A and B

Mutually exclusive

Independent

P(A 8 B)

P(A 9 B)

.12 .7

P(A円B)

4.6 INDEPENDENCE, CONDITIONAL PROBABILITY, AND THE MULTIPLICATION RULE

❍

155

4.41 Use event relationships to ﬁll in the blanks in the table below. P(A)

P(B)

Conditions for Events A and B

Mutually exclusive

Independent

P(A 8 B)

P(A 9 B)

P(A円B)

.1 0

BASIC TECHNIQUES

A: Observe a number less than 4

4.42 An experiment can result in one of ﬁve equally

B: Observe a number less than or equal to 2 C: Observe a number greater than 3

likely simple events, E1, E2, . . . , E5. Events A, B, and C are deﬁned as follows: A: E1, E3

P(A) .4

B: E1, E2, E4, E5 C: E3, E4

P(B) .8 P(C) .4

Find the probabilities associated with these compound events by listing the simple events in each. b. A B c. B C a. Ac d. A B e. B兩C f. A兩B g. A B C h. (A B)c 4.43 Refer to Exercise 4.42. Use the deﬁnition of a

complementary event to ﬁnd these probabilities: a. P(Ac ) b. P((A B)c ) Do the results agree with those obtained in Exercise 4.42? 4.44 Refer to Exercise 4.42. Use the deﬁnition of

conditional probability to ﬁnd these probabilities: a. P(A兩B) b. P(B兩C) Do the results agree with those obtained in Exercise 4.42? 4.45 Refer to Exercise 4.42. Use the Addition and

Multiplication Rules to ﬁnd these probabilities: a. P(A B) b. P(A B) c. P(B C) Do the results agree with those obtained in Exercise 4.42?

Find the probabilities associated with the events below using either the simple event approach or the rules and deﬁnitions from this section. a. S b. A兩B c. B d. A B C e. A B f. A C g. B C h. A C i. B C 4.48 Refer to Exercise 4.47.

a. Are events A and B independent? Mutually exclusive? b. Are events A and C independent? Mutually exclusive? 4.49 Suppose that P(A) .4 and P(B) .2. If events

A and B are independent, ﬁnd these probabilities: a. P(A B) b. P(A B) 4.50 Suppose that P(A) .3 and P(B) .5. If

events A and B are mutually exclusive, ﬁnd these probabilities: a. P(A B) b. P(A B) 4.51 Suppose that P(A) .4 and P(A B) .12.

a. Find P(B兩A). b. Are events A and B mutually exclusive? c. If P(B) .3, are events A and B independent? 4.52 An experiment can result in one or both of

events A and B with the probabilities shown in this probability table:

4.46 Refer to Exercise 4.42.

a. Are events A and B independent? b. Are events A and B mutually exclusive? 4.47 Dice An experiment consists of tossing a single

die and observing the number of dots that show on the upper face. Events A, B, and C are deﬁned as follows:

B Bc

.34 .15

.46 .05

Find the following probabilities: a. P(A) d. P(A B)

b. P(B) e. P(A兩B)

c. P(A B) f. P(B兩A)

156

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

4.53 Refer to Exercise 4.52.

a. Are events A and B mutually exclusive? Explain. b. Are events A and B independent? Explain.

A: The offender has 10 or more years of education B: The offender is convicted within 2 years after completion of treatment

APPLICATIONS 4.54 Drug Testing Many companies are testing

prospective employees for drug use, with the intent of improving efficiency and reducing absenteeism, accidents, and theft. Opponents claim that this procedure is creating a class of unhirables and that some persons may be placed in this class because the tests themselves are not 100% reliable. Suppose a company uses a test that is 98% accurate—that is, it correctly identiﬁes a person as a drug user or nonuser with probability .98—and to reduce the chance of error, each job applicant is required to take two tests. If the outcomes of the two tests on the same person are independent events, what are the probabilities of these events? a. A nonuser fails both tests. b. A drug user is detected (i.e., he or she fails at least one test). c. A drug user passes both tests. 4.55 Grant Funding Whether a grant proposal is

funded quite often depends on the reviewers. Suppose a group of research proposals was evaluated by a group of experts as to whether the proposals were worthy of funding. When these same proposals were submitted to a second independent group of experts, the decision to fund was reversed in 30% of the cases. If the probability that a proposal is judged worthy of funding by the ﬁrst peer review group is .2, what are the probabilities of these events? a. A worthy proposal is approved by both groups. b. A worthy proposal is disapproved by both groups. c. A worthy proposal is approved by one group. 4.56 Drug Offenders A study of the behavior of a

large number of drug offenders after treatment for drug abuse suggests that the likelihood of conviction within a 2-year period after treatment may depend on the offender’s education. The proportions of the total number of cases that fall into four education/conviction categories are shown in the table below. Status Within 2 Years After Treatment Convicted

Not Convicted

Totals

10 Years or More 9 Years or Less

.10 .27

.30 .33

.40 .60

Totals

.37

.63

1.00

Education

Suppose a single offender is selected from the treatment program. Here are the events of interest:

Find the appropriate probabilities for these events: a. A b. B c. A B d. A B e. Ac f. (A B)c g. (A B)c h. A given that B has occurred i. B given that A has occurred 4.57 Use the probabilities of Exercise 4.56 to show

that these equalities are true: a. P(A B) P(A)P(B兩A) b. P(A B) P(B)P(A兩B) c. P(A B) P(A) P(B) P(A B) 4.58 The Birthday Problem Two people enter a

room and their birthdays (ignoring years) are recorded. a. Identify the nature of the simple events in S. b. What is the probability that the two people have a speciﬁc pair of birthdates? c. Identify the simple events in event A: Both people have the same birthday. d. Find P(A). e. Find P(Ac). 4.59 The Birthday Problem, continued If n peo-

ple enter a room, ﬁnd these probabilities: A: None of the people have the same birthday B: At least two of the people have the same birthday Solve for a. n 3

b. n 4

[NOTE: Surprisingly, P(B) increases rapidly as n increases. For example, for n 20, P(B) .411; for n 40, P(B) .891.] 4.60 Starbucks or Peet’s®? A college student fre-

quents one of two coffee houses on campus, choosing Starbucks 70% of the time and Peet’s 30% of the time. Regardless of where she goes, she buys a cafe mocha on 60% of her visits. a. The next time she goes into a coffee house on campus, what is the probability that she goes to Starbucks and orders a cafe mocha? b. Are the two events in part a independent? Explain. c. If she goes into a coffee house and orders a cafe mocha, what is the probability that she is at Peet’s?

4.6 INDEPENDENCE, CONDITIONAL PROBABILITY, AND THE MULTIPLICATION RULE

d. What is the probability that she goes to Starbucks or orders a cafe mocha or both? 4.61 Inspection Lines A certain manufactured item is visually inspected by two different inspectors. When a defective item comes through the line, the probability that it gets by the ﬁrst inspector is .1. Of those that get past the ﬁrst inspector, the second inspector will “miss” 5 out of 10. What fraction of the defective items get by both inspectors? 4.62 Smoking and Cancer A survey of people in a given region showed that 20% were smokers. The probability of death due to lung cancer, given that a person smoked, was roughly 10 times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to lung cancer in the region is .006, what is the probability of death due to lung cancer given that a person is a smoker? 4.63 Smoke Detectors A smoke-detector system

uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is .95; by device B, .98; and by both devices, .94. a. If smoke is present, find the probability that the smoke will be detected by device A or device B or both devices. b. Find the probability that the smoke will not be detected. 4.64 Plant Genetics Gregor Mendel was a monk who suggested in 1865 a theory of inheritance based on the science of genetics. He identified heterozygous individuals for flower color that had two alleles (one r recessive white color allele and one R dominant red color allele). When these individuals were mated, 3/4 of the offspring were observed to have red flowers and 1/4 had white flowers. The table summarizes this mating; each parent gives one of its alleles to form the gene of the offspring. Parent 2 Parent 1

r R

rr Rr

rR RR

We assume that each parent is equally likely to give either of the two alleles and that, if either one or two of the alleles in a pair is dominant (R), the offspring will have red ﬂowers. a. What is the probability that an offspring in this mating has at least one dominant allele? b. What is the probability that an offspring has at least one recessive allele?

❍

157

c. What is the probability that an offspring has one recessive allele, given that the offspring has red ﬂowers? 4.65 Soccer Injuries During the inaugural season of

Major League Soccer in the United States, the medical teams documented 256 injuries that caused a loss of participation time to the player. The results of this investigation, reported in The American Journal of Sports Medicine, is shown in the table.3 Severity Minor (A) Moderate (B) Major (C) Total

Practice (P)

Game (G)

Total

66 23 12

88 44 23

154 67 35

101

155

256

If one individual is drawn at random from this group of 256 soccer players, ﬁnd the following probabilities: a. P(A) b. P(G) c. P(A G) d. P(G兩A) e. P(G兩B) f. P(G兩C) g. P(C兩P) h. P(Bc ) 4.66 Choosing a Mate Men and women often disagree on how they think about selecting a mate. Suppose that a poll of 1000 individuals in their twenties gave the following responses to the question of whether it is more important for their future mate to be able to communicate their feelings (F ) than it is for that person to make a good living (G). Feelings (F )

Good Living (G)

Totals

Men (M) Women (W )

.35 .36

.20 .09

.55 .45

Totals

.71

.29

1.00

If an individual is selected at random from this group of 1000 individuals, calculate the following probabilities: a. P(F) d. P(F兩W)

b. P(G) e. P(M兩F)

c. P(F兩M) f. P(W兩G)

4.67 Jason and Shaq The two stars of the Miami Heat professional basketball team are very different when it comes to making free throws. ESPN.com reports that Jason Williams makes about 80% of his free throws, while Shaquille O’Neal makes only 53% of his free throws.4 Assume that the free throws are independent, and that each player takes two free throws during a particular game. a. What is the probability that Jason makes both of

his free throws?

158 ❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

b. What is the probability that Shaq makes exactly

Player A has probability 1/6 of winning the tournament if player B enters and probability 3/4 of winning if player B does not enter the tournament. If the probability that player B enters is 1/3, ﬁnd the probability that player A wins the tournament.

one of his two free throws? c. What is the probability that Shaq makes both of his free throws and Jason makes neither of his? 4.68 Golﬁng Player A has entered a golf tournament

but it is not certain whether player B will enter.

BAYES’ RULE (OPTIONAL)

4.7

Let us reconsider the experiment involving colorblindness from Section 4.6. Notice that the two events

Colorblindness

B: the person selected is a man BC: the person selected is a woman taken together make up the sample space S, consisting of both men and women. Since colorblind people can be either male or female, the event A, which is that a person is colorblind, consists of both those simple events that are in A and B and those simple events that are in A and BC. Since these two intersections are mutually exclusive, you can write the event A as A (A B) (A BC) and P(A) P(A B) P(A BC) .04 .002 .042 Suppose now that the sample space can be partitioned into k subpopulations, S1, S2, S3, . . . , Sk, that, as in the colorblindness example, are mutually exclusive and exhaustive; that is, taken together they make up the entire sample space. In a similar way, you can express an event A as A (A S1) (A S2 ) (A S3 ) (A Sk ) Then P(A) P(A S1) P(A S2 ) P(A S3 ) P(A Sk ) This is illustrated for k 3 in Figure 4.14. F IGU R E 4 .1 4

Decomposition of event A

●

S A∩S1

A∩S2

A∩S3

4.7 BAYES’ RULE (OPTIONAL)

❍

159

You can go one step further and use the Multiplication Rule to write P(A Si) as P(Si)P(A兩Si), for i 1, 2, . . . , k. The result is known as the Law of Total Probability. LAW OF TOTAL PROBABILITY Given a set of events S1, S2, S3, . . . , Sk that are mutually exclusive and exhaustive and an event A, the probability of the event A can be expressed as P(A) P(S1)P(A兩S1) P(S2)P(A兩S2) P(S3)P(A兩S3) P(Sk)P(A兩Sk) EXAMPLE

TABLE 4.6

4.23

Sneakers are no longer just for the young. In fact, most adults own multiple pairs of sneakers. Table 4.6 gives the fraction of U.S. adults 20 years of age and older who own five or more pairs of wearable sneakers, along with the fraction of the U.S. adult population 20 years or older in each of five age groups.5 Use the Law of Total Probability to determine the unconditional probability of an adult 20 years and older owning five or more pairs of wearable sneakers. ●

Probability Table Groups and Ages

Fraction with 5 Pairs Fraction of U.S. Adults 20 and Older

G1 20–24

G2 25–34

G3 35–49

G4 50–64

G5 65

.26 .09

.20 .20

.13 .31

.18 .23

.14 .17

Let A be the event that a person chosen at random from the U.S. adult population 18 years of age and older owns five or more pairs of wearable sneakers. Let G1, G2, . . . , G5 represent the event that the person selected belongs to each of the five age groups, respectively. Since the five groups are exhaustive, you can write the event A as

Solution

A (A G1) (A G2) (A G3) (A G4) (A G5) Using the Law of Total Probability, you can find the probability of A as P(A) P(A G1) P(A G2) P(A G3) P(A G4) P(A G5) P(G1)P(A兩G1) P(G2)P(A兩G2) P(G3)P(A兩G3) P(G4)P(A兩G4) P(G5)P(A兩G5) From the probabilities in Table 4.6, P(A) (.09)(.26) (.20)(.20) (.31)(.13) (.23)(.18) (.17)(.14) .0234 .0400 .0403 .0414 .0238 .1689 The unconditional probability that a person selected at random from the population of U.S. adults 20 years of age and older owns at least five pairs of wearable sneakers is about .17. Notice that the Law of Total Probability is a weighted average of the probabilities within each group, with weights .09, .20, .31, .23, and .17, which reflect the relative sizes of the groups. Often you need to find the conditional probability of an event B, given that an event A has occurred. One such situation occurs in screening tests, which used to be

160

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

associated primarily with medical diagnostic tests but are now ﬁnding applications in a variety of ﬁelds. Automatic test equipment is routinely used to inspect parts in highvolume production processes. Steroid testing of athletes, home pregnancy tests, and AIDS testing are some other applications. Screening tests are evaluated on the probability of a false negative or a false positive, and both of these are conditional probabilities. A false positive is the event that the test is positive for a given condition, given that the person does not have the condition. A false negative is the event that the test is negative for a given condition, given that the person has the condition. You can evaluate these conditional probabilities using a formula derived by the probabilist Thomas Bayes. The experiment involves selecting a sample from one of k subpopulations that are mutually exclusive and exhaustive. Each of these subpopulations, denoted by S1, S2, . . . , Sk, has a selection probability P(S1), P(S2), P(S3), . . . , P(Sk), called prior probabilities. An event A is observed in the selection. What is the probability that the sample came from subpopulation Si, given that A has occurred? You know from Section 4.6 that P(Si |A) [P(A Si)]/P(A), which can be rewritten as P(Si |A) [P(Si)P(A|Si)]/P(A). Using the Law of Total Probability to rewrite P(A), you have P(Si)P(A|Si) P(Si |A) P(S1)P(A|S1) P(S2)P(A|S2) P(S3)P(A|S3) P(Sk)P(A|Sk) These new probabilities are often referred to as posterior probabilities—that is, probabilities of the subpopulations (also called states of nature) that have been updated after observing the sample information contained in the event A. Bayes suggested that if the prior probabilities are unknown, they can be taken to be 1/k, which implies that each of the events S1 through Sk is equally likely. BAYES’ RULE Let S1, S2, . . . , Sk represent k mutually exclusive and exhaustive subpopulations with prior probabilities P(S1), P(S2), . . . , P(Sk). If an event A occurs, the posterior probability of Si given A is the conditional probability P(Si)P(A|Si) P(Si |A) k 冱 P(Sj)P(A|Sj) j1

for i 1, 2, . . . , k. EXAMPLE

4.24

Refer to Example 4.23. Find the probability that the person selected was 65 years of age or older, given that the person owned at least ﬁve pairs of wearable sneakers. Solution

You need to ﬁnd the conditional probability given by

P(A G5) P(G5|A) P(A) You have already calculated P(A) .1689 using the Law of Total Probability. Therefore,

4.7 BAYES’ RULE (OPTIONAL)

❍

161

P(G5|A) P(G5)P(A|G5) P(G1)P(A|G1) P(G2)P(A|G2) P(G3)P(A|G3) P(G4P(A|G4) P(G5)P(A|G5) (.17)(.14) (.09)(.26) (.20)(.20) (.31)(.13) (.23)(.18) (.17)(.14) .0238 .1409 .1689 In this case, the posterior probability of .14 is somewhat less than the prior probability of .17 (from Table 4.6). This group a priori was the second smallest, and only a small proportion of this segment had ﬁve or more pairs of wearable sneakers. What is the posterior probability for those aged 35 to 49? For this group of adults, we have (.31)(.13) P(G3|A) .2386 (.09)(.26) (.20)(.20) (.31)(.13) (.23)(.18) (.17)(.14) This posterior probability of .24 is substantially less than the prior probability of .31. In effect, this group was a priori the largest segment of the population sampled, but at the same time, the proportion of individuals in this group who had at least ﬁve pairs of wearable sneakers was the smallest of any of the groups. These two facts taken together cause a downward adjustment of almost one-third in the a priori probability of .31.

4.7

EXERCISES

BASIC TECHNIQUES 4.69 Bayes’ Rule A sample is selected from one of two populations, S1 and S2, with probabilities P(S1) .7 and P(S2) .3. If the sample has been selected from S1, the probability of observing an event A is P(A兩S1) .2. Similarly, if the sample has been selected from S2, the probability of observing A is P(A兩S2) .3.

a. If a sample is randomly selected from one of the two populations, what is the probability that event A occurs? b. If the sample is randomly selected and event A is observed, what is the probability that the sample was selected from population S1? From population S2? 4.70 Bayes’ Rule II If an experiment is conducted,

one and only one of three mutually exclusive events S1, S2, and S3 can occur, with these probabilities: P(S1) .2

P(S2) .5

P(S3) .3

The probabilities of a fourth event A occurring, given that event S1, S2, or S3 occurs, are P(A|S1) .2

P(A|S2) .1

P(A|S3) .3

If event A is observed, ﬁnd P(S1|A), P(S2|A), and P(S3|A). 4.71 Law of Total Probability A population can be divided into two subgroups that occur with probabilities 60% and 40%, respectively. An event A occurs 30% of the time in the ﬁrst subgroup and 50% of the time in the second subgroup. What is the unconditional probability of the event A, regardless of which subgroup it comes from?

APPLICATIONS 4.72 Violent Crime City crime records show that

20% of all crimes are violent and 80% are nonviolent, involving theft, forgery, and so on. Ninety percent of violent crimes are reported versus 70% of nonviolent crimes.

162 ❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

a. What is the overall reporting rate for crimes in the city? b. If a crime in progress is reported to the police, what is the probability that the crime is violent? What is the probability that it is nonviolent? c. Refer to part b. If a crime in progress is reported to the police, why is it more likely that it is a nonviolent crime? Wouldn’t violent crimes be more likely to be reported? Can you explain these results? 4.73 Worker Error A worker-operated machine pro-

duces a defective item with probability .01 if the worker follows the machine’s operating instructions exactly, and with probability .03 if he does not. If the worker follows the instructions 90% of the time, what proportion of all items produced by the machine will be defective? 4.74 Airport Security Suppose that, in a particular city, airport A handles 50% of all airline traffic, and airports B and C handle 30% and 20%, respectively. The detection rates for weapons at the three airports are .9, .8, and .85, respectively. If a passenger at one of the airports is found to be carrying a weapon through the boarding gate, what is the probability that the passenger is using airport A? Airport C? 4.75 Football Strategies A particular football team is known to run 30% of its plays to the left and 70% to the right. A linebacker on an opposing team notes that the right guard shifts his stance most of the time (80%) when plays go to the right and that he uses a balanced stance the remainder of the time. When plays go to the left, the guard takes a balanced stance 90% of the time and the shift stance the remaining 10%. On a particular play, the linebacker notes that the guard takes a balanced stance. a. What is the probability that the play will go to the left? b. What is the probability that the play will go to the right? c. If you were the linebacker, which direction would you prepare to defend if you saw the balanced stance? 4.76 No Pass, No Play Many public schools are implementing a “no pass, no play” rule for athletes. Under this system, a student who fails a course is disqualified from participating in extracurricular activities during the next grading period. Suppose the probability that an athlete who has not previously been disqualified will be disqualified is .15 and the

probability that an athlete who has been disqualified will be disqualified again in the next time period is .5. If 30% of the athletes have been disqualified before, what is the unconditional probability that an athlete will be disqualified during the next grading period? 4.77 Medical Diagnostics Medical case histories

indicate that different illnesses may produce identical symptoms. Suppose a particular set of symptoms, which we will denote as event H, occurs only when any one of three illnesses—A, B, or C—occurs. (For the sake of simplicity, we will assume that illnesses A, B, and C are mutually exclusive.) Studies show these probabilities of getting the three illnesses: P(A) .01 P(B) .005 P(C) .02 The probabilities of developing the symptoms H, given a specific illness, are P(H|A) .90 P(H|B) .95 P(H|C) .75 Assuming that an ill person shows the symptoms H, what is the probability that the person has illness A? 4.78 Cheating on Your Taxes? Suppose 5% of all people filing the long income tax form seek deductions that they know are illegal, and an additional 2% incorrectly list deductions because they are unfamiliar with income tax regulations. Of the 5% who are guilty of cheating, 80% will deny knowledge of the error if confronted by an investigator. If the filer of the long form is confronted with an unwarranted deduction and he or she denies the knowledge of the error, what is the probability that he or she is guilty? 4.79 Screening Tests Suppose that a certain disease is present in 10% of the population, and that there is a screening test designed to detect this disease if present. The test does not always work perfectly. Sometimes the test is negative when the disease is present, and sometimes it is positive when the disease is absent. The table below shows the proportion of times that the test produces various results. Test Is Positive (P ) Disease Present (D) Disease Absent (Dc)

Test Is Negative (N )

.08

.22

.05

.85

4.8 DISCRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS

a. Find the following probabilities from the table: P(D), P(Dc ), P(N兩Dc ), P(N兩D). b. Use Bayes’ Rule and the results of part a to find P(D兩N). c. Use the definition of conditional probability to find P(D兩N). (Your answer should be the same as the answer to part b.)

4.8

❍

163

d. Find the probability of a false positive, that the test is positive, given that the person is disease-free. e. Find the probability of a false negative, that the test is negative, given that the person has the disease. f. Are either of the probabilities in parts d or e large enough that you would be concerned about the reliability of this screening method? Explain.

DISCRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS In Chapter 1, variables were deﬁned as characteristics that change or vary over time and/or for different individuals or objects under consideration. Quantitative variables generate numerical data, whereas qualitative variables generate categorical data. However, even qualitative variables can generate numerical data if the categories are numerically coded to form a scale. For example, if you toss a single coin, the qualitative outcome could be recorded as “0” if a head and “1” if a tail.

Random Variables A numerically valued variable x will vary or change depending on the particular outcome of the experiment being measured. For example, suppose you toss a die and measure x, the number observed on the upper face. The variable x can take on any of six values—1, 2, 3, 4, 5, 6—depending on the random outcome of the experiment. For this reason, we refer to the variable x as a random variable. A variable x is a random variable if the value that it assumes, corresponding to the outcome of an experiment, is a chance or random event. Definition

You can think of many examples of random variables: • • •

x Number of defects on a randomly selected piece of furniture x SAT score for a randomly selected college applicant x Number of telephone calls received by a crisis intervention hotline during a randomly selected time period

As in Chapter 1, quantitative random variables are classiﬁed as either discrete or continuous, according to the values that x can assume. It is important to distinguish between discrete and continuous random variables because different techniques are used to describe their distributions. We focus on discrete random variables in the remainder of this chapter; continuous random variables are the subject of Chapter 6.

Probability Distributions In Chapters 1 and 2, you learned how to construct the relative frequency distribution for a set of numerical measurements on a variable x. The distribution gave this information about x:

164

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

• •

What values of x occurred How often each value of x occurred

You also learned how to use the mean and standard deviation to measure the center and variability of this data set. In this chapter, we deﬁned probability as the limiting value of the relative frequency as the experiment is repeated over and over again. Now we deﬁne the probability distribution for a random variable x as the relative frequency distribution constructed for the entire population of measurements. The probability distribution for a discrete random variable is a formula, table, or graph that gives the possible values of x, and the probability p(x) associated with each value of x. Definition

The values of x represent mutually exclusive numerical events. Summing p(x) over all values of x is equivalent to adding the probabilities of all simple events and therefore equals 1. REQUIREMENTS FOR A DISCRETE PROBABILITY DISTRIBUTION • • EXAMPLE

0 p(x) 1 S p(x) 1

Toss two fair coins and let x equal the number of heads observed. Find the probability distribution for x.

4.25

The simple events for this experiment with their respective probabilities are listed in Table 4.7. Since E1 HH results in two heads, this simple event results in the value x 2. Similarly, the value x 1 is assigned to E2, and so on. Solution

TABLE 4.7

●

Simple Events and Probabilities in Tossing Two Coins Simple Event

Coin 1

Coin 2

P(Ei)

E1 E2 E3 E4

H H T T

H T H T

1/4 1/4 1/4 1/4

2 1 1 0

For each value of x, you can calculate p(x) by adding the probabilities of the simple events in that event. For example, when x 0, 1 p(0) P(E4) 4 and when x 1, 1 p(1) P(E2) P(E3) 2

4.8 DISCRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS

❍

165

The values of x and their respective probabilities, p(x), are listed in Table 4.8. Notice that the probabilities add to 1. TABLE 4.8

●

Probability Distribution for x (x Number of Heads) x

Simple Events in x p(x)

0 1 2

E4 E2, E3 E1

1/4 1/2 1/4 S p(x) 1

The probability distribution in Table 4.8 can be graphed using the methods of Section 1.5 to form the probability histogram in Figure 4.15.† The three values of the random variable x are located on the horizontal axis, and the probabilities p(x) are located on the vertical axis (replacing the relative frequencies used in Chapter 1). Since the width of each bar is 1, the area under the bar is the probability of observing the particular value of x and the total area equals 1.

Probability histogram for Example 4.25

● 1/2

p(x)

FI GU R E 4 .1 5

1/4

0 0

1 x

There are two Java applets that will allow you to approximate discrete probability distributions using simulation methods. That is, even though the probabilities p(x) can only be found as the long-run relative frequencies when the experiment is repeated an inﬁnite number of times, we can get close to these probabilities if we repeat the experiment a large number of times. The applets called Flipping Fair Coins and Flipping Weighted Coins are two such simulations. The fastest way to generate the approximate probability distribution for x, the number of heads in n tosses of the coin, is to repeat the experiment “100 at a Time,” using the button at the bottom of the applet. The probability distribution will build up rather quickly. You can approximate the values of p(x) and compare to the actual values calculated using probability rules. We will use these applets for the MyApplet Exercises at the end of the chapter. †

The probability distribution in Table 4.8 can also be presented using a formula, which is given in Section 5.2.

166

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

F I GU R E 4 .1 6

Flipping Fair Coins applet

F IGU R E 4 .1 7

Flipping Weighted Coins applet

●

The Mean and Standard Deviation for a Discrete Random Variable The probability distribution for a discrete random variable looks very similar to the relative frequency distribution discussed in Chapter 1. The difference is that the relative frequency distribution describes a sample of n measurements, whereas the probability distribution is constructed as a model for the entire population of measurements. Just as the mean x苶 and the standard deviation s measured the center and spread of the sample data, you can calculate similar measures to describe the center and spread of the population. The population mean, which measures the average value of x in the population, is also called the expected value of the random variable x. It is the value that you would expect to observe on average if the experiment is repeated over and over again. The formula for calculating the population mean is easier to understand by example. Toss those two fair coins again, and let x be the number of heads observed. We constructed this probability distribution for x: x

p(x)

1/4

1/2

1/4

4.8 DISCRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS

❍

167

Suppose the experiment is repeated a large number of times—say, n 4,000,000 times. Intuitively, you would expect to observe approximately 1 million zeros, 2 million ones, and 1 million twos. Then the average value of x would equal Sum of measurements 1,000,000(0) 2,000,000(1) 1,000,000(2) n 4,000,000

冢冣

1 1 1 (0) (1) (2) 4 2 4 Note that the first term in this sum is (0)p(0), the second is equal to (1)p(1), and the third is (2)p(2). The average value of x, then, is 1 2 Sxp(x) 0 1 2 4 This result provides some intuitive justification for the definition of the expected value of a discrete random variable x. Let x be a discrete random variable with probability distribution p(x). The mean or expected value of x is given as

Definition

m E(x) S xp(x) where the elements are summed over all values of the random variable x. We could use a similar argument to justify the formulas for the population variance s 2 and the population standard deviation s. These numerical measures describe the spread or variability of the random variable using the “average” or “expected value” of the squared deviations of the x-values from their mean m. Let x be a discrete random variable with probability distribution p(x) and mean m. The variance of x is

Definition

s 2 E[(x m)2] S(x m)2p(x) where the summation is over all values of the random variable x.† The standard deviation s of a random variable x is equal to the positive square root of its variance. Definition

EXAMPLE

4.26

An electronics store sells a particular model of computer notebook. There are only four notebooks in stock, and the manager wonders what today’s demand for this particular model will be. She learns from the marketing department that the probability distribution for x, the daily demand for the laptop, is as shown in the table. Find the

It can be shown (proof omitted) that s 2 S(x m)2p(x) Sx 2p(x) m2. This result is analogous to the computing formula for the sum of squares of deviations given in Chapter 2.

†

168

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

mean, variance, and standard deviation of x. Is it likely that ﬁve or more customers will want to buy a laptop today? x

p(x)

.10

.40

.20

.15

.10

.05

Table 4.9 shows the values of x and p(x), along with the individual terms used in the formulas for m and s 2. The sum of the values in the third column is

Solution

m S xp(x) (0)(.10) (1)(.40) (5)(.05) 1.90 while the sum of the values in the ﬁfth column is s 2 S(x m)2p(x) (0 1.9)2(.10) (1 1.9)2(.40) (5 1.9)2(.05) 1.79 and 苶2 兹1.79 苶 1.34 s 兹s TABLE 4.9

●

Calculations for Example 4.26 x

p(x)

xp(x)

(x m)2

(x m)2 p(x)

0 1 2 3 4 5

.10 .40 .20 .15 .10 .05

.00 .40 .40 .45 .40 .25

3.61 .81 .01 1.21 4.41 9.61

.361 .324 .002 .1815 .441 .4805

Totals

1.00

m 1.90

s 2 1.79

The graph of the probability distribution is shown in Figure 4.18. Since the distribution is approximately mound-shaped, approximately 95% of all measurements should lie within two standard deviations of the mean—that is, m 2s ⇒ 1.90 2(1.34)

or .78 to 4.58

Since x 5 lies outside this interval, you can say it is unlikely that ﬁve or more customers will want to buy a laptop today. In fact, P(x 5) is exactly .05, or 1 time in 20.

Probability distribution for Example 4.26

● .4

p(x)

FI GU R E 4 .1 8

0 0

3 x

4.8 DISCRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS

EXAMPLE

4.27

❍

169

In a lottery conducted to beneﬁt a local charity, 8000 tickets are to be sold at $10 each. The prize is a $24,000 automobile. If you purchase two tickets, what is your expected gain? Your gain x may take one of two values. You will either lose $20 (i.e., your “gain” will be $20) or win $23,980, with probabilities 7998/8000 and 2/8000, respectively. The probability distribution for the gain x is shown in the table:

Solution

p(x)

$20 $23,980

7998/8000 2/8000

The expected gain will be m S xp(x)

冢

冣

冢

冣

7998 2 ($20) ($23,980) $14 8000 8000 Recall that the expected value of x is the average of the theoretical population that would result if the lottery were repeated an inﬁnitely large number of times. If this were done, your average or expected gain per lottery ticket would be a loss of $14.

EXAMPLE

4.28

Determine the yearly premium for a $10,000 insurance policy covering an event that, over a long period of time, has occurred at the rate of 2 times in 100. Let x equal the yearly financial gain to the insurance company resulting from the sale of the policy, and let C equal the unknown yearly premium. Calculate the value of C such that the expected gain E(x) will equal zero. Then C is the premium required to break even. To this, the company would add administrative costs and profit. The first step in the solution is to determine the values that the gain x may take and then to determine p(x). If the event does not occur during the year, the insurance company will gain the premium of x C dollars. If the event does occur, the gain will be negative; that is, the company will lose $10,000 less the premium of C dollars already collected. Then x (10,000 C) dollars. The probabilities associated with these two values of x are 98/100 and 2/100, respectively. The probability distribution for the gain is shown in the table:

Solution

x Gain

p(x)

C (10,000 C )

98/100 2/100

Since the company wants the insurance premium C such that, in the long run (for many similar policies), the mean gain will equal zero, you can set the expected value of x equal to zero and solve for C. Then m E(x) Sxp(x)

冢冣

98 2 C [10,000 C)] 0 100 100 or

98 2 C C 200 0 100 100 Solving this equation for C, you obtain C $200. Therefore, if the insurance company charged a yearly premium of $200, the average gain calculated for a large number of

170

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

similar policies would equal zero. The actual premium would equal $200 plus administrative costs and proﬁt. The method for calculating the expected value of x for a continuous random variable is similar to what you have done, but in practice it involves the use of calculus. Nevertheless, the basic results concerning expectations are the same for continuous and discrete random variables. For example, regardless of whether x is continuous or discrete, m E(x) and s 2 E[(x m)2].

4.8

EXERCISES

BASIC TECHNIQUES

4.83 Probability Distribution II A random variable

4.80 Discrete or Continuous? Identify the follow-

x can assume ﬁve values: 0, 1, 2, 3, 4. A portion of the probability distribution is shown here:

ing as discrete or continuous random variables: a. Total number of points scored in a football game b. Shelf life of a particular drug c. Height of the ocean’s tide at a given location d. Length of a 2-year-old black bass e. Number of aircraft near-collisions in a year 4.81 Discrete or Continuous? II Identify the following as discrete or continuous random variables: a. Increase in length of life attained by a cancer patient as a result of surgery b. Tensile breaking strength (in pounds per square inch) of 1-inch-diameter steel cable c. Number of deer killed per year in a state wildlife preserve d. Number of overdue accounts in a department store at a particular time e. Your blood pressure 4.82 Probability Distribution I A random variable

x has this probability distribution: x

p(x)

.05

a. b. c. d.

Find p (4). Construct a probability histogram to describe p(x). Find m, s 2, and s. Locate the interval m 2s on the x-axis of the histogram. What is the probability that x will fall into this interval? e. If you were to select a very large number of values of x from the population, would most fall into the interval m 2s? Explain.

p(x)

a. Find p(3). b. Construct a probability histogram for p (x). c. Calculate the population mean, variance, and standard deviation. d. What is the probability that x is greater than 2? e. What is the probability that x is 3 or less? 4.84 Dice Let x equal the number observed on the throw of a single balanced die. a. Find and graph the probability distribution for x. b. What is the average or expected value of x? c. What is the standard deviation of x? d. Locate the interval m 2s on the x-axis of the graph in part a. What proportion of all the measurements would fall into this range? 4.85 Grocery Visits Let x represent the number of times a customer visits a grocery store in a 1-week period. Assume this is the probability distribution of x: x

p(x)

Find the expected value of x, the average number of times a customer visits the store. APPLICATIONS 4.86 Letterman or Leno? Who is the king of late

night TV? An Internet survey estimates that, when given a choice between David Letterman and Jay Leno, 52% of the population prefers to watch Jay

4.8 DISCRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS

Leno. Suppose that you randomly select three late night TV watchers and ask them which of the two talk show hosts they prefer. a. Find the probability distribution for x, the number of people in the sample of three who would prefer Jay Leno. b. Construct the probability histogram for p(x). c. What is the probability that exactly one of the three would prefer Jay Leno? d. What are the population mean and standard deviation for the random variable x? 4.87 Which Key Fits? A key ring contains four office

keys that are identical in appearance, but only one will open your office door. Suppose you randomly select one key and try it. If it does not fit, you randomly select one of the three remaining keys. If it does not fit, you randomly select one of the last two. Each different sequence that could occur in selecting the keys represents one of a set of equiprobable simple events. a. List the simple events in S and assign probabilities to the simple events. b. Let x equal the number of keys that you try before you find the one that opens the door (x 1, 2, 3, 4). Then assign the appropriate value of x to each simple event. c. Calculate the values of p(x) and display them in a table. d. Construct a probability histogram for p(x). 4.88 Roulette Exercise 4.10 described the game of roulette. Suppose you bet $5 on a single number—say, the number 18. The payoff on this type of bet is usually 35 to 1. What is your expected gain? 4.89 Gender Bias? A company has five applicants

for two positions: two women and three men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let x equal the number of women chosen to fill the two positions. a. Find p (x). b. Construct a probability histogram for x. 4.90 Defective Equipment A piece of electronic

equipment contains six computer chips, two of which are defective. Three chips are selected at random, removed from the piece of equipment, and inspected. Let x equal the number of defectives observed, where x 0, 1, or 2. Find the probability distribution for x. Express the results graphically as a probability histogram.

❍

171

4.91 Drilling Oil Wells Past experience has shown that, on the average, only 1 in 10 wells drilled hits oil. Let x be the number of drillings until the ﬁrst success (oil is struck). Assume that the drillings represent independent events.

a. Find p(1), p(2), and p(3). b. Give a formula for p(x). c. Graph p(x). 4.92 Tennis, Anyone? Two tennis professionals, A

and B, are scheduled to play a match; the winner is the ﬁrst player to win three sets in a total that cannot exceed ﬁve sets. The event that A wins any one set is independent of the event that A wins any other, and the probability that A wins any one set is equal to .6. Let x equal the total number of sets in the match; that is, x 3, 4, or 5. Find p(x). 4.93 Tennis, again The probability that tennis

player A can win a set against tennis player B is one measure of the comparative abilities of the two players. In Exercise 4.92 you found the probability distribution for x, the number of sets required to play a best-of-ﬁve-sets match, given that the probability that A wins any one set—call this P(A)—is .6. a. Find the expected number of sets required to complete the match for P(A) .6. b. Find the expected number of sets required to complete the match when the players are of equal ability—that is, P(A) .5. c. Find the expected number of sets required to complete the match when the players differ greatly in ability—that is, say, P(A) .9. 4.94 The PGA One professional golfer plays best on

short-distance holes. Experience has shown that the numbers x of shots required for 3-, 4-, and 5-par holes have the probability distributions shown in the table: Par-3 Holes

Par-4 Holes

Par-5 Holes

p(x)

2 3 4 5

.12 .80 .06 .02

3 4 5 6

.14 .80 .04 .02

4 5 6 7

.04 .80 .12 .04

What is the golfer’s expected score on these holes? a. A par-3 hole b. A par-4 hole c. A par-5 hole 4.95 Insuring Your Diamonds You can insure a $50,000 diamond for its total value by paying a

172

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

premium of D dollars. If the probability of theft in a given year is estimated to be .01, what premium should the insurance company charge if it wants the expected gain to equal $1000? 4.96 FDA Testing The maximum patent life for a new drug is 17 years. Subtracting the length of time required by the FDA for testing and approval of the drug provides the actual patent life of the drug—that is, the length of time that a company has to recover research and development costs and make a proﬁt. Suppose the distribution of the lengths of patent life for new drugs is as shown here: Years, x

p(x)

.03

.05

.07

.10

.14

.20

Years, x

p(x)

.18

.12

.07

.03

.01

a. Find the expected number of years of patent life for a new drug. b. Find the standard deviation of x. c. Find the probability that x falls into the interval m 2s. 4.97 Coffee Breaks Are you a coffee drinker? If so, how many coffee breaks do you take when you are at work or at school? Most coffee drinkers take a little time for their favorite beverage, and many take more than one coffee break every day. The table below, adapted from a Snapshot in USA Today shows the probability distribution for x, the number of daily coffee breaks taken per day by coffee drinkers.6 x

p(x)

.28

.37

.17

.12

.05

.01

a. What is the probability that a randomly selected coffee drinker would take no coffee breaks during the day? b. What is the probability that a randomly selected coffee drinker would take more than two coffee breaks during the day? c. Calculate the mean and standard deviation for the random variable x. d. Find the probability that x falls into the interval m 2s. 4.98 Shipping Charges From experience, a shipping company knows that the cost of delivering a small package within 24 hours is $14.80. The company charges $15.50 for shipment but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only 2% of its packages within the 24-hour period, what is the expected gain per package? 4.99 Actuaries A manufacturing representative is considering taking out an insurance policy to cover possible losses incurred by marketing a new product. If the product is a complete failure, the representative feels that a loss of $800,000 would be incurred; if it is only moderately successful, a loss of $250,000 would be incurred. Insurance actuaries have determined from market surveys and other available information that the probabilities that the product will be a failure or only moderately successful are .01 and .05, respectively. Assuming that the manufacturing representative is willing to ignore all other possible losses, what premium should the insurance company charge for a policy in order to break even?

CHAPTER REVIEW Key Concepts and Formulas I.

Experiments and the Sample Space

1. Experiments, events, mutually exclusive events, simple events 2. The sample space 3. Venn diagrams, tree diagrams, probability tables II. Probabilities

1. Relative frequency deﬁnition of probability

2. Properties of probabilities a. Each probability lies between 0 and 1. b. Sum of all simple-event probabilities equals 1. 3. P(A), the sum of the probabilities for all simple events in A III. Counting Rules

1. mn Rule; extended mn Rule

MY MINITAB

n! 2. Permutations: P nr (n r)!

❍

173

6. Multiplication Rule: P(A B) P(A)P(B|A) 7. Law of Total Probability

n! 3. Combinations: C nr r!(n r)!

8. Bayes’ Rule

IV. Event Relations

V. Discrete Random Variables and Probability Distributions

1. Unions and intersections 2. Events

1. Random variables, discrete and continuous 2. Properties of probability distributions

a. Disjoint or mutually exclusive: P(A B) 0 b. Complementary: P(A) 1 P(Ac)

a. 0 p(x) 1 b. Sp(x) 1

P(A B) 3. Conditional probability: P(A|B) P(B) 4. Independent and dependent events 5. Addition Rule: P(A B) P(A) P(B) P(A B)

3. Mean or expected value of a discrete random variable: m Sxp(x) 4. Variance and standard deviation of a discrete random variable: s 2 S(x m)2p(x) and s 兹苶 s2

Discrete Probability Distributions Although MINITAB cannot help you solve the types of general probability problems presented in this chapter, it is useful for graphing the probability distribution p (x) for a general discrete random variable x when the probabilities are known, and for calculating the mean, variance, and standard deviation of the random variable x. In Chapters 5 and 6, we will use MINITAB to calculate exact probabilities for three special cases: the binomial, the Poisson, and the normal random variables. Suppose you have this general probability distribution: x

p(x)

.25

.35

.25

.15

Enter the values of x and p(x) into columns C1 and C2 of a new MINITAB worksheet. In the gray boxes just below C3, C4, and C5, respectively, type the names “Mean,” “Variance,” and “Std Dev.” You can now use the Calc 씮 Calculator command to calculate m, s 2, and s and to store the results in columns C3–C5 of the worksheet. Use the same approach for the three parameters. In the Calculator dialog box, select “Mean” as the column in which to store m. In the Expression box, use the Functions list, the calculator keys, and the variables list on the left to highlight, select, and create the expression for the mean (see Figure 4.19): SUM(‘x’*‘p(x)’)

174

❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

FI GU R E 4 .1 9

●

MINITAB will multiply each row element in C1 times the corresponding row element in C2, sum the resulting products, and store the result in C3! You can check the result by hand if you like. The formulas for the variance and standard deviation are selected in a similar way: Variance: SUM((‘x’ ‘Mean’)**2*‘p(x)’) Std Dev: SQRT(‘Variance’) To see the tabular form of the probability distribution and the three parameters, use Data 씮 Display Data and select all ﬁve columns. Click OK and the results will be displayed in the Session window, as shown in Figure 4.20. The probability histogram can be plotted using the MINITAB command Graph 씮 Scatterplot 씮 Simple 씮 OK. In the Scatterplot dialog box (Figure 4.21), select ‘p(x)’ for Y variables and ‘x’ for X variables. To display the discrete probability bars, click on Data View, uncheck the box marked “Symbols,” and check the box marked “Project Lines.” Click OK twice to see the plot. You will see a single straight line projected at each of the four values of x. If you want the plot to look more like the discrete probability histograms in Section 4.8, position your cursor on one of the lines, right-click the mouse and choose “Edit Project Lines.” Under the “Attributes” tab, select Custom and change the line size to 75. Click OK. If the bar width is not satisfactory, you can readjust the line size. Finally, right-click on the X-axis, choose “Edit X Scale” and select .5 and 5.5 for the minimum and maximum Scale Ranges. Click OK. The probability histogram is shown in Figure 4.22. Locate the mean on the graph. Is it at the center of the distribution? If you mark off two standard deviations on either side of the mean, do most of the possible values of x fall into this interval?

MY MINITAB

FIGU R E 4 .2 0

●

F IGU R E 4 .2 1

●

F IGU R E 4 .2 2

●

❍

175

176 ❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

Supplementary Exercises Starred (*) exercises are optional. 4.100 Playing the Slots A slot machine has three slots; each will show a cherry, a lemon, a star, or a bar when spun. The player wins if all three slots show the same three items. If each of the four items is equally likely to appear on a given spin, what is your probability of winning? 4.101 Whistle Blowers “Whistle blowers” is the name given to employees who report corporate fraud, theft, or other unethical and perhaps criminal activities by fellow employees or by their employer. Although there is legal protection for whistle blowers, it has been reported that approximately 23% of those who reported fraud suffered reprisals such as demotion or poor performance ratings. Suppose the probability that an employee will fail to report a case of fraud is .69. Find the probability that a worker who observes a case of fraud will report it and will subsequently suffer some form of reprisal. 4.102 Aspirin Two cold tablets are accidentally

placed in a box containing two aspirin tablets. The four tablets are identical in appearance. One tablet is selected at random from the box and is swallowed by the first patient. A tablet is then selected at random from the three remaining tablets and is swallowed by the second patient. Define the following events as specific collections of simple events: a. The sample space S b. The event A that the first patient obtained a cold tablet c. The event B that exactly one of the two patients obtained a cold tablet d. The event C that neither patient obtained a cold tablet 4.103 Refer to Exercise 4.102. By summing the probabilities of simple events, find P(A), P(B), P(A B), P(A B), P(C), P(A C), and P(A C). 4.104 DVRs A retailer sells two styles of highpriced digital video recorders (DVR) that experience indicates are in equal demand. (Fifty percent of all potential customers prefer style 1, and 50% favor style 2.) If the retailer stocks four of each, what is the probability that the first four customers seeking a DVR all purchase the same style?

to the purchaser, three are defective. Two of the seven are selected for thorough testing and are then classiﬁed as defective or nondefective. What is the probability that no defectives are found? 4.106 Heavy Equipment A heavy-equipment salesman can contact either one or two customers per day with probabilities 1/3 and 2/3, respectively. Each contact will result in either no sale or a $50,000 sale with probabilities 9/10 and 1/10, respectively. What is the expected value of his daily sales? 4.107 Fire Insurance A county containing a large

number of rural homes is thought to have 60% of those homes insured against fire. Four rural homeowners are chosen at random from the entire population, and x are found to be insured against fire. Find the probability distribution for x. What is the probability that at least three of the four will be insured? 4.108 Fire Alarms A fire-detection device uses three temperature-sensitive cells acting independently of one another in such a manner that any one or more can activate the alarm. Each cell has a probability p .8 of activating the alarm when the temperature reaches 100°F or higher. Let x equal the number of cells activating the alarm when the temperature reaches 100°F. a. Find the probability distribution of x. b. Find the probability that the alarm will function when the temperature reaches 100°F. c. Find the expected value and the variance for the random variable x. 4.109 Catching a Cold Is your chance of getting a cold influenced by the number of social contacts you have? A study by Sheldon Cohen, a psychology professor at Carnegie Mellon University, seems to show that the more social relationships you have, the less susceptible you are to colds. A group of 276 healthy men and women were grouped according to their number of relationships (such as parent, friend, church member, neighbor). They were then exposed to a virus that causes colds. An adaptation of the results is shown in the table:7 Number of Relationships Three or Fewer

Four or Five

Six or More

4.105 Interstate Commerce A shipping container

Cold No Cold

49 31

43 57

34 62

contains seven complex electronic systems. Unknown

Total

100

SUPPLEMENTARY EXERCISES

a. If one person is selected at random from the 276 people in the study, what is the probability that the person got a cold? b. If two people are randomly selected, what is the probability that one has four or ﬁve relationships and the other has six or more relationships? c. If a single person is randomly selected and has a cold, what is the probability that he or she has three or fewer relationships? 4.110 Plant Genetics Refer to the experiment

conducted by Gregor Mendel in Exercise 4.64. Suppose you are interested in following two independent traits in snap peas—seed texture (S smooth, s wrinkled) and seed color (Y yellow, y green)—in a second-generation cross of heterozygous parents. Remember that the capital letter represents the dominant trait. Complete the table with the gene pairs for both traits. All possible pairings are equally likely. Seed Color Seed Texture

(ss yy)

(ss yY)

❍

177

4.112 Racial Bias? Four union men, two from a

minority group, are assigned to four distinctly different one-man jobs, which can be ranked in order of desirability. a. Deﬁne the experiment. b. List the simple events in S. c. If the assignment to the jobs is unbiased—that is, if any one ordering of assignments is as probable as any other—what is the probability that the two men from the minority group are assigned to the least desirable jobs? 4.113 A Reticent Salesman A salesperson ﬁgures

that the probability of her consummating a sale during the ﬁrst contact with a client is .4 but improves to .55 on the second contact if the client did not buy during the ﬁrst contact. Suppose this salesperson makes one and only one callback to any client. If she contacts a client, calculate the probabilities for these events: a. The client will buy. b. The client will not buy.

sS Ss SS

a. What proportion of the offspring from this cross will have smooth yellow peas? b. What proportion of the offspring will have smooth green peas? c. What proportion of the offspring will have wrinkled yellow peas? d. What proportion of the offspring will have wrinkled green peas? e. Given that an offspring has smooth yellow peas, what is the probability that this offspring carries one s allele? One s allele and one y allele? 4.111 Profitable Stocks An investor has the option of investing in three of five recommended stocks. Unknown to her, only two will show a substantial profit within the next 5 years. If she selects the three stocks at random (giving every combination of three stocks an equal chance of selection), what is the probability that she selects the two profitable stocks? What is the probability that she selects only one of the two profitable stocks?

4.114 Bus or Subway A man takes either a bus or

the subway to work with probabilities .3 and .7, respectively. When he takes the bus, he is late 30% of the days. When he takes the subway, he is late 20% of the days. If the man is late for work on a particular day, what is the probability that he took the bus? 4.115 Guided Missiles The failure rate for a guided missile control system is 1 in 1000. Suppose that a duplicate, but completely independent, control system is installed in each missile so that, if the ﬁrst fails, the second can take over. The reliability of a missile is the probability that it does not fail. What is the reliability of the modiﬁed missile? 4.116 Rental Trucks A rental truck agency services its vehicles on a regular basis, routinely checking for mechanical problems. Suppose that the agency has six moving vans, two of which need to have new brakes. During a routine check, the vans are tested one at a time.

a. What is the probability that the last van with brake problems is the fourth van tested? b. What is the probability that no more than four vans need to be tested before both brake problems are detected? c. Given that one van with bad brakes is detected in the ﬁrst two tests, what is the probability that the remaining van is found on the third or fourth test?

178 ❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

4.117 Pennsylvania Lottery Probability played a role in the rigging of the April 24, 1980, Pennsylvania state lottery. To determine each digit of the three-digit winning number, each of the numbers 0, 1, 2, . . . , 9 is written on a Ping-Pong ball, the 10 balls are blown into a compartment, and the number selected for the digit is the one on the ball that ﬂoats to the top of the machine. To alter the odds, the conspirators injected a liquid into all balls used in the game except those numbered 4 and 6, making it almost certain that the lighter balls would be selected and determine the digits in the winning number. They then proceeded to buy lottery tickets bearing the potential winning numbers. How many potential winning numbers were there (666 was the eventual winner)? *4.118 Lottery, continued Refer to Exercise 4.117.

Hours after the rigging of the Pennsylvania state lottery was announced on September 19, 1980, Connecticut state lottery officials were stunned to learn that their winning number for the day was 666. a. All evidence indicates that the Connecticut selection of 666 was pure chance. What is the probability that a 666 would be drawn in Connecticut, given that a 666 had been selected in the April 24, 1980, Pennsylvania lottery? b. What is the probability of drawing a 666 in the April 24, 1980, Pennsylvania lottery (remember, this drawing was rigged) and a 666 on the September 19, 1980, Connecticut lottery? *4.119 ACL/MCL Tears The American Journal of

Sports Medicine published a study of 810 women collegiate rugby players who have a history of knee injuries. For these athletes, the two common knee injuries investigated were medial cruciate ligament (MCL) sprains and anterior cruciate ligament (ACL) tears.8 For backﬁeld players, it was found that 39% had MCL sprains and 61% had ACL tears. For forwards, it was found that 33% had MCL sprains and 67% had ACL tears. Since a rugby team consists of eight forwards and seven backs, you can assume that 47% of the players with knee injuries are backs and 53% are forwards. a. Find the unconditional probability that a rugby player selected at random from this group of players has experienced an MCL sprain. b. Given that you have selected a player who has an MCL sprain, what is the probability that the player is a forward?

c. Given that you have selected a player who has an ACL tear, what is the probability that the player is a back? 4.120 MRIs Magnetic resonance imaging (MRI) is an accepted noninvasive test to evaluate changes in the cartilage in joints. An article in The American Journal of Sports Medicine compared the results of MRI evaluation with arthroscopic surgical evaluation of cartilage tears at two sites in the knees of 35 patients. The 2 35 70 examinations produced the classiﬁcations shown in the table.9 Actual tears were conﬁrmed by arthroscopic surgical examination. Tears

No Tears

Total

MRI Positive MRI Negative

27 4

0 39

27 43

Total

a. What is the probability that a site selected at random has a tear and has been identified as a tear by MRI? b. What is the probability that a site selected at random has no tear and has been identified as having a tear? c. What is the probability that a site selected at random has a tear and has not been identified by MRI? d. What is the probability of a positive MRI, given that there is a tear? e. What is the probability of a false negative—that is, a negative MRI, given that there is a tear? 4.121 The Match Game Two men each toss a coin.

They obtain a “match” if either both coins are heads or both are tails. Suppose the tossing is repeated three times. a. What is the probability of three matches? b. What is the probability that all six tosses (three for each man) result in tails? c. Coin tossing provides a model for many practical experiments. Suppose that the coin tosses represent the answers given by two students for three speciﬁc true–false questions on an examination. If the two students gave three matches for answers, would the low probability found in part a suggest collusion? 4.122 Contract Negotiations Experience has

shown that, 50% of the time, a particular union– management contract negotiation led to a contract

SUPPLEMENTARY EXERCISES

settlement within a 2-week period, 60% of the time the union strike fund was adequate to support a strike, and 30% of the time both conditions were satisﬁed. What is the probability of a contract settlement given that the union strike fund is adequate to support a strike? Is settlement of a contract within a 2-week period dependent on whether the union strike fund is adequate to support a strike? 4.123 Work Tenure Suppose the probability of remaining with a particular company 10 years or longer is 1/6. A man and a woman start work at the company on the same day.

a. What is the probability that the man will work there less than 10 years? b. What is the probability that both the man and the woman will work there less than 10 years? (Assume they are unrelated and their lengths of service are independent of each other.) c. What is the probability that one or the other or both will work 10 years or longer? 4.124 Accident Insurance Accident records collected by an automobile insurance company give the following information: The probability that an insured driver has an automobile accident is .15; if an accident has occurred, the damage to the vehicle amounts to 20% of its market value with probability .80, 60% of its market value with probability .12, and a total loss with probability .08. What premium should the company charge on a $22,000 car so that the expected gain by the company is zero? 4.125 Waiting Times Suppose that at a particular

supermarket the probability of waiting 5 minutes or longer for checkout at the cashier’s counter is .2. On a given day, a man and his wife decide to shop individually at the market, each checking out at different cashier counters. They both reach cashier counters at the same time. a. What is the probability that the man will wait less than 5 minutes for checkout? b. What is probability that both the man and his wife will be checked out in less than 5 minutes? (Assume that the checkout times for the two are independent events.) c. What is the probability that one or the other or both will wait 5 minutes or longer? 4.126 Quality Control A quality-control plan calls

for accepting a large lot of crankshaft bearings if a

❍

179

sample of seven is drawn and none are defective. What is the probability of accepting the lot if none in the lot are defective? If 1/10 are defective? If 1/2 are defective? 4.127 Mass Transit Only 40% of all people in a

community favor the development of a mass transit system. If four citizens are selected at random from the community, what is the probability that all four favor the mass transit system? That none favors the mass transit system? 4.128 Blood Pressure Meds A research physician compared the effectiveness of two blood pressure drugs A and B by administering the two drugs to each of four pairs of identical twins. Drug A was given to one member of a pair; drug B to the other. If, in fact, there is no difference in the effects of the drugs, what is the probability that the drop in the blood pressure reading for drug A exceeds the corresponding drop in the reading for drug B for all four pairs of twins? Suppose drug B created a greater drop in blood pressure than drug A for each of the four pairs of twins. Do you think this provides sufficient evidence to indicate that drug B is more effective in lowering blood pressure than drug A? 4.129 Blood Tests To reduce the cost of detecting a disease, blood tests are conducted on a pooled sample of blood collected from a group of n people. If no indication of the disease is present in the pooled blood sample (as is usually the case), none have the disease. If analysis of the pooled blood sample indicates that the disease is present, each individual must submit to a blood test. The individual tests are conducted in sequence. If, among a group of five people, one person has the disease, what is the probability that six blood tests (including the pooled test) are required to detect the single diseased person? If two people have the disease, what is the probability that six tests are required to locate both diseased people? 4.130 Tossing a Coin How many times should a coin be tossed to obtain a probability equal to or greater than .9 of observing at least one head? 4.131 Flextime The number of companies offering flexible work schedules has increased as companies try to help employees cope with the demands of home and work. One flextime schedule is to work four 10-hour shifts. However, a big obstacle to flextime schedules for workers paid hourly is state legislation on overtime. A survey provided the following information for 220 firms located in two cities in California.

180 ❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

Flextime Schedule City

Available

Not Available

Total

A B

39 25

75 81

114 106

Totals

156

220

A company is selected at random from this pool of 220 companies. a. What is the probability that the company is located in city A? b. What is the probability that the company is located in city B and offers ﬂextime work schedules?

4.133 Pepsi™ or Coke™? A taste-testing experiment is conducted at a local supermarket, where passing shoppers are asked to taste two soft-drink samples—one Pepsi and one Coke—and state their preference. Suppose that four shoppers are chosen at random and asked to participate in the experiment, and that there is actually no difference in the taste of the two brands. a. What is the probability that all four shoppers choose Pepsi? b. What is the probability that exactly one of the four shoppers chooses Pepsi?

c. What is the probability that the company does not have ﬂextime schedules? d. What is the probability that the company is located in city B, given that the company has ﬂextime schedules available?

4.134 Viruses A certain virus afflicted the families

4.132 A Color Recognition Experiment An

4.135 Orchestra Politics The board of directors of a major symphony orchestra has voted to create a players’ committee for the purpose of handling employee complaints. The council will consist of the president and vice president of the symphony board and two orchestra representatives. The two orchestra representatives will be randomly selected from a list of six volunteers, consisting of four men and two women. a. Find the probability distribution for x, the number of women chosen to be orchestra representatives. b. Find the mean and variance for the random variable x.

experiment is run as follows—the colors red, yellow, and blue are each flashed on a screen for a short period of time. A subject views the colors and is asked to choose the one he feels was flashed for the longest time. The experiment is repeated three times with the same subject. a. If all the colors were flashed for the same length of time, find the probability distribution for x, the number of times that the subject chose the color red. Assume that his three choices are independent. b. Construct the probability histogram for the random variable x.

in 3 adjacent houses in a row of 12 houses. If three houses were randomly chosen from a row of 12 houses, what is the probability that the 3 houses would be adjacent? Is there reason to believe that this virus is contagious?

c. What is the probability that both orchestra representatives will be women?

Exercises 4.136 Two fair dice are tossed. Use the Tossing Dice

applet to answer the following questions. a. What is the probability that the sum of the number of dots shown on the upper faces is equal to 7? To 11? b. What is the probability that you roll “doubles”— that is, both dice have the same number on the upper face? c. What is the probability that both dice show an odd number?

4.137 If you toss a pair of dice, the sum T of the number of dots appearing on the upper faces of the dice can assume the value of an integer in the interval 2 T 12. a. Use the Tossing Dice applet to ﬁnd the probability distribution for T. Display this probability distribution in a table. b. Construct a probability histogram for p(T). How would you describe the shape of this distribution?

CASE STUDY

4.138 Access the Flipping Fair Coins applet. The experiment consists of tossing three fair coins and recording x, the number of heads. a. Use the laws of probability to write down the simple events in this experiment. b. Find the probability distribution for x. Display the distribution in a table and in a probability histogram. c. Use the Flipping Fair Coins applet to simulate the probability distribution—that is, repeat the cointossing experiment a large number of times until the relative frequency histogram is very close to the actual probability distribution. Start by performing the experiment once (click ) to see what is happening. Then speed up the process by clicking . Generate at least 2000 values of x. Sketch the histogram that you have generated. d. Compare the histograms in parts b and c. Does the simulation conﬁrm your answer from part b? 4.139 Refer to Exercise 4.138.

CASE STUDY

❍

181

a. If you were to toss only one coin, what would the probability distribution for x look like? b. Perform a simulation using the Flipping Fair Coins applet with n 1, and compare your results with part a. 4.140 Refer to Exercise 4.138. Access the Flipping Weighted Coins applet. The experiment consists of tossing three coins that are not fair, and recording x, the number of heads.

a. Perform a simulation of the experiment using the Flipping Weighted Coins applet. Is the distribution symmetric or skewed? Which is more likely, heads or tails? b. Suppose that we do not know the probability of getting a head, P(H). Write a formula for calculating the probability of no heads in three tosses. c. Use the approximate probability P(x 0) from your simulation and the results of part b to approximate the value of P(T). What is the probability of getting a head?

Probability and Decision Making in the Congo In his exciting novel Congo, Michael Crichton describes a search by Earth Resources Technology Service (ERTS), a geological survey company, for deposits of boroncoated blue diamonds, diamonds that ERTS believes to be the key to a new generation of optical computers.10 In the novel, ERTS is racing against an international consortium to find the Lost City of Zinj, a city that thrived on diamond mining and existed several thousand years ago (according to African fable), deep in the rain forests of eastern Zaire. After the mysterious destruction of its first expedition, ERTS launches a second expedition under the leadership of Karen Ross, a 24-year-old computer genius who is accompanied by Professor Peter Elliot, an anthropologist; Amy, a talking gorilla; and the famed mercenary and expedition leader, “Captain” Charles Munro. Ross’s efforts to find the city are blocked by the consortium’s offensive actions, by the deadly rain forest, and by hordes of “talking” killer gorillas whose perceived mission is to defend the diamond mines. Ross overcomes these obstacles by using space-age computers to evaluate the probabilities of success for all possible circumstances and all possible actions that the expedition might take. At each stage of the expedition, she is able to quickly evaluate the chances of success. At one stage in the expedition, Ross is informed by her Houston headquarters that their computers estimate that she is 18 hours and 20 minutes behind the competing Euro-Japanese team, instead of 40 hours ahead. She changes plans and decides to

182 ❍

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS

have the 12 members of her team—Ross, Elliot, Munro, Amy, and eight native porters—parachute into a volcanic region near the estimated location of Zinj. As Crichton relates, “Ross had double-checked outcome probabilities from the Houston computer, and the results were unequivocal. The probability of a successful jump was .7980, meaning that there was approximately one chance in ﬁve that someone would be badly hurt. However, given a successful jump, the probability of expedition success was .9943, making it virtually certain that they would beat the consortium to the site.” Keeping in mind that this is an excerpt from a novel, let us examine the probability, .7980, of a successful jump. If you were one of the 12-member team, what is the probability that you would successfully complete your jump? In other words, if the probability of a successful jump by all 12 team members is .7980, what is the probability that a single member could successfully complete the jump?

Several Useful Discrete Distributions GENERAL OBJECTIVES Discrete random variables are used in many practical applications. Three important discrete random variables—the binomial, the Poisson, and the hypergeometric—are presented in this chapter. These random variables are often used to describe the number of occurrences of a specified event in a fixed number of trials or a fixed unit of time or space.

CHAPTER INDEX ● The binomial probability distribution (5.2) ● The hypergeometric probability distribution (5.4) ● The mean and variance for the binomial random variable (5.2) ● The Poisson probability distribution (5.3)

How Do I Use Table 1 to Calculate Binomial Probabilities? How Do I Calculate Poisson Probabilities Using the Formula? How Do I Use Table 2 to Calculate Poisson Probabilities?

A Mystery: Cancers Near a Reactor Is the Pilgrim I nuclear reactor responsible for an increase in cancer cases in the surrounding area? A political controversy was set off when the Massachusetts Department of Public Health found an unusually large number of cases in a 4-mile-wide coastal strip just north of the nuclear reactor in Plymouth, Massachusetts. The case study at the end of this chapter examines how this question can be answered using one of the discrete probability distributions presented here.

183

184 ❍

CHAPTER 5 SEVERAL USEFUL DISCRETE DISTRIBUTIONS

5.1

INTRODUCTION Examples of discrete random variables can be found in a variety of everyday situations and across most academic disciplines. However, there are three discrete probability distributions that serve as models for a large number of these applications. In this chapter we study the binomial, the Poisson, and the hypergeometric probability distributions and discuss their usefulness in different physical situations.

5.2

THE BINOMIAL PROBABILITY DISTRIBUTION A coin-tossing experiment is a simple example of an important discrete random variable called the binomial random variable. Many practical experiments result in data similar to the head or tail outcomes of the coin toss. For example, consider the political polls used to predict voter preferences in elections. Each sampled voter can be compared to a coin because the voter may be in favor of our candidate— a “head”—or not—a “tail.” In most cases, the proportion of voters who favor our candidate does not equal 1/2; that is, the coin is not fair. In fact, the proportion of voters who favor our candidate is exactly what the poll is designed to measure! Here are some other situations that are similar to the coin-tossing experiment: • • •

A sociologist is interested in the proportion of elementary school teachers who are men. A soft-drink marketer is interested in the proportion of cola drinkers who prefer her brand. A geneticist is interested in the proportion of the population who possess a gene linked to Alzheimer’s disease.

Each sampled person is analogous to tossing a coin, but the probability of a “head” is not necessarily equal to 1/2. Although these situations have different practical objectives, they all exhibit the common characteristics of the binomial experiment. Definition

A binomial experiment is one that has these ﬁve characteristics:

1. The experiment consists of n identical trials. 2. Each trial results in one of two outcomes. For lack of a better name, the one outcome is called a success, S, and the other a failure, F. 3. The probability of success on a single trial is equal to p and remains the same from trial to trial. The probability of failure is equal to (1 p) q. 4. The trials are independent. 5. We are interested in x, the number of successes observed during the n trials, for x 0, 1, 2, . . . , n. EXAMPLE

5.1

Suppose there are approximately 1,000,000 adults in a county and an unknown proportion p favor term limits for politicians. A sample of 1000 adults will be chosen in such a way that every one of the 1,000,000 adults has an equal chance of being selected, and each adult is asked whether he or she favors term limits. (The ultimate objective of this survey is to estimate the unknown proportion p, a problem that we will discuss in Chapter 8.) Is this a binomial experiment?

5.2 THE BINOMIAL PROBABILITY DISTRIBUTION

Solution

❍

185

Does the experiment have the ﬁve binomial characteristics?

1. A “trial” is the choice of a single adult from the 1,000,000 adults in the county. This sample consists of n 1000 identical trials. 2. Since each adult will either favor or not favor term limits, there are two outcomes that represent the “successes” and “failures” in the binomial experiment.† 3. The probability of success, p, is the probability that an adult favors term limits. Does this probability remain the same for each adult in the sample? For all practical purposes, the answer is yes. For example, if 500,000 adults in the population favor term limits, then the probability of a “success” when the first adult is chosen is 500,000/1,000,000 1/2. When the second adult is chosen, the probability p changes slightly, depending on the first choice. That is, there will be either 499,999 or 500,000 successes left among the 999,999 adults. In either case, p is still approximately equal to 1/2. 4. The independence of the trials is guaranteed because of the large group of adults from which the sample is chosen. The probability of an adult favoring term limits does not change depending on the responses of previously chosen people. 5. The random variable x is the number of adults in the sample who favor term limits. Because the survey satisfies the five characteristics reasonably well, for all practical purposes it can be viewed as a binomial experiment.

EXAMPLE

5.2

A patient ﬁlls a prescription for a 10-day regimen of 2 pills daily. Unknown to the pharmacist and the patient, the 20 tablets consist of 18 pills of the prescribed medication and 2 pills that are the generic equivalent of the prescribed medication. The patient selects two pills at random for the ﬁrst day’s dosage. If we check the selection and record the number of pills that are generic, is this a binomial experiment? Again, check the sampling procedure for the characteristics of a binomial experiment.

Solution

1. A “trial” is the selection of a pill from the 20 in the prescription. This experiment consists of n 2 trials. 2. Each trial results in one of two outcomes. Either the pill is generic (call this a “success”) or not (a “failure”). 3. Since the pills in a prescription bottle can be considered randomly “mixed,” the unconditional probability of drawing a generic pill on a given trial would be 2/20. 4. The condition of independence between trials is not satisfied, because the probability of drawing a generic pill on the second trial is dependent on the first trial. For example, if the first pill drawn is generic, then there is only 1 generic pill in the remaining 19. Therefore, P(generic on trial 2兩generic on trial 1) 1/19 †

Although it is traditional to call the two possible outcomes of a trial “success” and “failure,” they could have been called “head” and “tail,” “red” and “white,” or any other pair of words. Consequently, the outcome called a “success” does not need to be viewed as a success in the ordinary use of the word.

186 ❍

CHAPTER 5 SEVERAL USEFUL DISCRETE DISTRIBUTIONS

If the ﬁrst selection does not result in a generic pill, then there are still 2 generic pills in the remaining 19, and the probability of a “success” (a generic pill) changes to P(generic on trial 2兩no generic on trial 1) 2/19 Therefore the trials are dependent and the sampling does not represent a binomial experiment. Think about the difference between these two examples. When the sample (the n identical trials) came from a large population, the probability of success p stayed about the same from trial to trial. When the population size N was small, the probability of success p changed quite dramatically from trial to trial, and the experiment was not binomial. RULE OF THUMB If the sample size is large relative to the population size—in particular, if n/N .05—then the resulting experiment is not binomial. In Chapter 4, we tossed two fair coins and constructed the probability distribution for x, the number of heads—a binomial experiment with n 2 and p .5. The general binomial probability distribution is constructed in the same way, but the procedure gets complicated as n gets large. Fortunately, the probabilities p(x) follow a general pattern. This allows us to use a single formula to ﬁnd p(x) for any given value of x. THE BINOMIAL PROBABILITY DISTRIBUTION A binomial experiment consists of n identical trials with probability of success p on each trial. The probability of k successes in n trials is n! P(x k) C nk p kq nk p kq nk k!(n k)! for values of k 0, 1, 2, . . . , n. The symbol C nk equals n! k!(n k)! where n! n(n 1)(n 2) (2)(1) and 0! ⬅ 1. The general formulas for m, s 2, and s given in Chapter 4 can be used to derive the following simpler formulas for the binomial mean and standard deviation. MEAN AND STANDARD DEVIATION FOR THE BINOMIAL RANDOM VARIABLE The random variable x, the number of successes in n trials, has a probability distribution with this center and spread: Mean: m np Variance: s 2 npq 苶 Standard deviation: s 兹npq

5.2 THE BINOMIAL PROBABILITY DISTRIBUTION

EXAMPLE

❍

187

Find P(x 2) for a binomial random variable with n 10 and p .1.

5.3

P(x 2) is the probability of observing 2 successes and 8 failures in a sequence of 10 trials. You might observe the 2 successes ﬁrst, followed by 8 consecutive failures:

Solution

S, S, F, F, F, F, F, F, F, F Since p is the probability of success and q is the probability of failure, this particular sequence has probability

n! n(n 1)(n 2) . . . (2)(1) For example, 5! 5(4)(3)(2)(1) 120

ppqqqqqqqq p 2q8 However, many other sequences also result in x 2 successes. The binomial formula uses C 10 2 to count the number of sequences and gives the exact probability when you use the binomial formula with k 2:

and 0! ⬅ 1.

2 102 P(x 2) C 10 2 (.1) (.9) 10(9) 10! (.1)2(.9)8 (.01)(.430467) .1937 2(1) 2!(10 2)!

You could repeat the procedure in Example 5.3 for each value of x—0, 1, 2, . . . , 10—and find all the values of p(x) necessary to construct a probability histogram for x. This would be a long and tedious job, but the resulting graph would look like Figure 5.1(a). You can check the height of the bar for x 2 and find p(2) P(x 2) .1937. The graph is skewed right; that is, most of the time you will observe small values of x. The mean or “balancing point” is around x 1; in fact, you can use the formula to find the exact mean: m np 10(.1) 1 Figures 5.1(b) and 5.1(c) show two other binomial distributions with n 10 but with different values of p. Look at the shapes of these distributions. When p .5, the distribution is exactly symmetric about the mean, m np 10(.5) 5. When p .9, the distribution is the “mirror image” of the distribution for p .1 and is skewed to the left. FIGU R E 5 .1

Binomial probability distributions

●

p(x)

.40

.25 n = 10, p = .5 µ=5 σ = 1.58

.20 n = 10, p = .1 µ=1 σ = .95

.30 .20

.15 .10 .05 0

.10

(b) 0

6 (a)

p(x) .40 n = 10, p = .9 µ=9 σ = .95

.30 .20 .10 0

6 (c)

188 ❍

EXAMPLE

CHAPTER 5 SEVERAL USEFUL DISCRETE DISTRIBUTIONS

5.4

Over a long period of time, it has been observed that a professional basketball player can make a free throw on a given trial with probability equal to .8. Suppose he shoots four free throws. 1. What is the probability that he will make exactly two free throws? 2. What is the probability that he will make at least one free throw? A “trial” is a single free throw, and you can deﬁne a “success” as a basket and a “failure” as a miss, so that n 4 and p .8. If you assume that the player’s chance of making the free throw does not change from shot to shot, then the number x of times that he makes the free throw is a binomial random variable.

Solution

1. P(x 2) C 42(.8)2(.2)2 4(3)(2)(1) 4! (.64)(.04) (.64)(.04) .1536 2(1)(2)(1) 2!2! The probability is .1536 that he will make exactly two free throws. 2. P(at least one) P(x 1) p(1) p(2) p(3) p(4) 1 p(0) 1 C 40(.8)0(.2)4 1 .0016 .9984. Although you could calculate P(x 1), P(x 2), P(x 3) and P(x 4) to ﬁnd this probability, using the complement of the event makes your job easier; that is, P(x 1) 1 P(x 1) 1 P(x 0). Can you think of any reason your assumption of independent trials might be wrong? If the player learns from his previous attempt (that is, he adjusts his shooting according to his last attempt), then his probability p of making the free throw may change, possibly increase, from shot to shot. The trials would not be independent and the experiment would not be binomial.

Use Table 1 in Appendix I rather than the binomial formula whenever possible. This is an easier way!

Calculating binomial probabilities can become tedious even for relatively small values of n. As n gets larger, it becomes almost impossible without the help of a calculator or computer. Fortunately, both of these tools are available to us. Computergenerated tables of cumulative binomial probabilities are given in Table 1 of Appendix I for values of n ranging from 2 to 25 and for selected values of p. These probabilities can also be generated using MINITAB or the Java applets on the Premium Website. Cumulative binomial probabilities differ from the individual binomial probabilities that you calculated with the binomial formula. Once you ﬁnd the column of probabilities for the correct values of n and p in Table 1, the row marked k gives the sum of all the binomial probabilities from x 0 to x k. Table 5.1 shows part of Table 1 for n 5 and p .6. If you look in the row marked k 3, you will ﬁnd P(x 3) p(0) p(1) p(2) p(3) .663

5.2 THE BINOMIAL PROBABILITY DISTRIBUTION

TABLE 5.1

●

❍

189

Portion of Table 1 in Appendix I for n 5 p k

.01

.05

.10

.20

.30

.40

.50

—

.60

.70

.80

.90

.95

.99

.010

—

.087

—

.317

—

.663

—

.922

—

1.000

—

If the probability you need to calculate is not in this form, you will need to think of a way to rewrite your probability to make use of the tables!

EXAMPLE

5.5

Use the cumulative binomial table for n 5 and p .6 to ﬁnd the probabilities of these events: 1. Exactly three successes 2. Three or more successes Solution

1. If you find k 3 in Table 5.1, the tabled value is P(x 3) p(0) p(1) p(2) p(3) Since you want only P(x 3) p(3), you must subtract out the unwanted probability: P(x 2) p(0) p(1) p(2) which is found in Table 5.1 with k 2. Then P(x 3) P(x 3) P(x 2) .663 .317 .346 2. To find P(three or more successes) P(x 3) using Table 5.1, you must use the complement of the event of interest. Write P(x 3) 1 P(x 3) 1 P(x 2) You can find P(x 2) in Table 5.1 with k 2. Then P(x 3) 1 P(x 2) 1 .317 .683

190 ❍

CHAPTER 5 SEVERAL USEFUL DISCRETE DISTRIBUTIONS

How Do I Use Table 1 to Calculate Binomial Probabilities? 1. Find the necessary values of n and p. Isolate the appropriate column in Table 1. 2. Table 1 gives P(x k) in the row marked k. Rewrite the probability you need so that it is in this form. •

List the values of x in your event.

•

From the list, write the event as either the difference of two probabilities:

P(x a) P(x b) for a b or the complement of the event: 1 P(x a) or just the event itself: P(x a) or P(x a) P(x a 1) Exercise Reps A. Consider a binomial random variable with n 5 and p .6. Isolate the appropriate column in Table 1 and ﬁll in the probabilities below. One of the probabilities, P(x 3) is ﬁlled in for you. k

P(x k)

.663

B. Fill in the blanks in the table below. The second problem is done for you.

The Problem

List the Values of x

Write the Probability

Rewrite the Probability (if needed)

Find the Probability

4 or less 4 or more

4, 5

P(x

1 P(x 3)

1 .663 .337

More than 4 Fewer than 4 Between 2 and 4 (inclusive) Exactly 4

Progress Report •

Still having trouble? Try again using the Exercise Reps at the end of this section.

•

Mastered binomial probabilities? You can skip the Exercise Reps at the end of this section!

Answers are located on the perforated card at the back of this book.

The Java applet called Calculating Binomial Probabilities gives a visual display of the binomial distribution for values of n 100 and any p that you choose. You can use this applet to calculate binomial probabilities for any value of x or for any interval a x b. To reproduce the results of Example 5.5, enter 5 in the box

5.2 THE BINOMIAL PROBABILITY DISTRIBUTION

❍

191

labeled “n” and 0.6 in the box labeled “p,” pressing the “Enter” key after each entry. Next enter the beginning and ending values for x (if you need to calculate an individual probability, both entries will be the same). The probability will be calculated and shaded in red on your monitor (light blue in Figure 5.2) when you press “Enter.” What is the probability of three or more successes from Figure 5.2? Does this conﬁrm our answer in Example 5.5? You will use this applet again for the MyApplet Exercises section at the end of the chapter. FIGU R E 5 .2

Calculating Binomial Probabilities applet

EXAMPLE

5.6

●

A regimen consisting of a daily dose of vitamin C was tested to determine its effectiveness in preventing the common cold. Ten people who were following the prescribed regimen were observed for a period of 1 year. Eight survived the winter without a cold. Suppose the probability of surviving the winter without a cold is .5 when the vitamin C regimen is not followed. What is the probability of observing eight or more survivors, given that the regimen is ineffective in increasing resistance to colds? If you assume that the vitamin C regimen is ineffective, then the probability p of surviving the winter without a cold is .5. The probability distribution for x, the number of survivors, is Solution

x 10x p (x) C 10 x (.5) (.5)

You have learned four ways to ﬁnd P(8 or more survivors) P(x 8). You will get the same results with any of the four; choose the most convenient method for your particular problem. 1. The binomial formula: P(8 or more) p(8) p(9) p(10) 10 10 10 C 10 C10 C 10 8 (.5) 9 (.5) 10 (.5)

.055 2. The cumulative binomial tables: Find the column corresponding to p .5 in the table for n 10: P(8 or more) P(x 8) 1 P(x 7) 1 .945 .055

192 ❍

CHAPTER 5 SEVERAL USEFUL DISCRETE DISTRIBUTIONS

3. The Calculating Binomial Probabilities applet: Enter n 10, p .5 and calculate the probability that x is between 8 and 10. The probability, P(x 8) .0547, is shaded in red on your monitor (light blue in Figure 5.3). FIGU R E 5 .3

Java applet for Example 5.6

●

4. Output from MINITAB: The output shown in Figure 5.4 gives the cumulative distribution function, which gives the same probabilities you found in the cumulative binomial tables. The probability density function gives the individual binomial probabilities, which you found using the binomial formula. FI GU R E 5 .4

MINITAB output for Example 5.6

● Cumulative Distribution Function

Probability Density Function

Binomial with n = 10 and p = 0.5

x 0 1 2 3 4 5 6 7 8 9 10

P( X 1.96)

9.4 A LARGE-SAMPLE TEST OF HYPOTHESIS FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS

•

❍

365

The p-value approach: Calculate the p-value, the probability that z is greater than z 1.84 plus the probability that z is less than z 1.84, as shown in Figure 9.11:

p-value P(z 1.84) P(z 1.84) (1 .9671) .0329 .0658 The p-value lies between .10 and .05, so you can reject H0 at the .10 level but not at the .05 level of significance. Since the p-value of .0658 exceeds the specified significance level a .05, H0 cannot be rejected. Again, you should not be willing to accept H0 until b is evaluated for some meaningful values of (m1 m2 ).

Hypothesis Testing and Confidence Intervals Whether you use the critical value or the p-value approach for testing hypotheses about (m1 m2 ), you will always reach the same conclusion because the calculated value of the test statistic and the critical value are related exactly in the same way that the p-value and the significance level a are related. You might remember that the confidence intervals constructed in Chapter 8 could also be used to answer questions about the difference between two population means. In fact, for a two-tailed test, the (1 a)100% confidence interval for the parameter of interest can be used to test its value, just as you did informally in Chapter 8. The value of a indicated by the confidence coefficient in the confidence interval is equivalent to the significance level a in the statistical test. For a one-tailed test, the equivalent confidence interval approach would use the one-sided confidence bounds in Section 8.8 with confidence coefficient a. In addition, by using the confidence interval approach, you gain a range of possible values for the parameter of interest, regardless of the outcome of the test of hypothesis. •

•

EXAMPLE

9.10

If the confidence interval you construct contains the value of the parameter specified by H0, then that value is one of the likely or possible values of the parameter and H0 should not be rejected. If the hypothesized value lies outside of the confidence limits, the null hypothesis is rejected at the a level of significance.

Construct a 95% confidence interval for the difference in average academic achievements between car owners and non-owners. Using the confidence interval, can you conclude that there is a difference in the population means for the two groups of students? For the large-sample statistics discussed in Chapter 8, the 95% confidence interval is given as

Solution

Point estimator 1.96 (Standard error of the estimator) For the difference in two population means, the conﬁdence interval is approximated as

冪莦莦

(x苶1 苶x2) 1.96

s2 s2 1 2 n1 n2

冪莦莦

.36 .40 (2.70 2.54) 1.96 100 100 .16 .17

366 ❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

or .01 (m1 m2 ) .33. This interval gives you a range of possible values for the difference in the population means. Since the hypothesized difference, (m1 m2 ) 0, is contained in the conﬁdence interval, you should not reject H0. Look at the signs of the possible values in the conﬁdence interval. You cannot tell from the interval whether the difference in the means is negative (), positive (), or zero (0)—the latter of the three would indicate that the two means are the same. Hence, you can really reach no conclusion in terms of the question posed. There is not enough evidence to indicate that there is a difference in the average achievements for car owners versus non-owners. The conclusion is the same one reached in Example 9.9.

9.4

EXERCISES

BASIC TECHNIQUES 9.18 Independent random samples of 80 measure-

ments were drawn from two quantitative populations, 1 and 2. Here is a summary of the sample data: Sample Size Sample Mean Sample Variance

Sample 1

Sample 2

80 11.6 27.9

80 9.7 38.4

a. If your research objective is to show that m1 is larger than m 2, state the alternative and the null hypotheses that you would choose for a statistical test. b. Is the test in part a one- or two-tailed? c. Calculate the test statistic that you would use for the test in part a. Based on your knowledge of the standard normal distribution, is this a likely or unlikely observation, assuming that H0 is true and the two population means are the same? d. p-value approach: Find the p-value for the test. Test for a signiﬁcant difference in the population means at the 1% signiﬁcance level. e. Critical value approach: Find the rejection region when a .01. Do the data provide sufficient evidence to indicate a difference in the population means? 9.19 Independent random samples of 36 and 45

observations are drawn from two quantitative populations, 1 and 2, respectively. The sample data summary is shown here: Sample Size Sample Mean Sample Variance

Sample 1

Sample 2

36 1.24 .0560

45 1.31 .0540

Do the data present sufficient evidence to indicate that the mean for population 1 is smaller than the mean for population 2? Use one of the two methods of testing presented in this section, and explain your conclusions. 9.20 Suppose you wish to detect a difference between

m1 and m 2 (either m1 m 2 or m1 m 2 ) and, instead of running a two-tailed test using a .05, you use the following test procedure. You wait until you have collected the sample data and have calculated x苶1 and 苶x2. If 苶x1 is larger than x苶2, you choose the alternative hypothesis Ha : m1 m 2 and run a one-tailed test placing a1 .05 in the upper tail of the z distribution. If, on the other hand, 苶x2 is larger than x苶1, you reverse the procedure and run a one-tailed test, placing a2 .05 in the lower tail of the z distribution. If you use this procedure and if m1 actually equals m 2, what is the probability a that you will conclude that m1 is not equal to m 2 (i.e., what is the probability a that you will incorrectly reject H0 when H0 is true)? This exercise demonstrates why statistical tests should be formulated prior to observing the data. APPLICATIONS 9.21 Cure for the Common Cold? An experiment

was planned to compare the mean time (in days) required to recover from a common cold for persons given a daily dose of 4 milligrams (mg) of vitamin C versus those who were not given a vitamin supplement. Suppose that 35 adults were randomly selected for each treatment category and that the mean recovery times and standard deviations for the two groups were as follows:

9.4 A LARGE-SAMPLE TEST OF HYPOTHESIS FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS

No Vitamin Supplement

4 mg Vitamin C

35 6.9 2.9

35 5.8 1.2

Sample Size Sample Mean Sample Standard Deviation

a. Suppose your research objective is to show that the use of vitamin C reduces the mean time required to recover from a common cold and its complications. Give the null and alternative hypotheses for the test. Is this a one- or a two-tailed test? b. Conduct the statistical test of the null hypothesis in part a and state your conclusion. Test using a .05. 9.22 Healthy Eating Americans are becoming more

conscious about the importance of good nutrition, and some researchers believe we may be altering our diets to include less red meat and more fruits and vegetables. To test the theory that the consumption of red meat has decreased over the last 10 years, a researcher decides to select hospital nutrition records for 400 subjects surveyed 10 years ago and to compare their average amount of beef consumed per year to amounts consumed by an equal number of subjects interviewed this year. The data are given in the table. Ten Years Ago

This Year

73 25

63 28

Sample Mean Sample Standard Deviation

a. Do the data present sufficient evidence to indicate that per-capita beef consumption has decreased in the last 10 years? Test at the 1% level of significance. b. Find a 99% lower confidence bound for the difference in the average per-capita beef consumptions for the two groups. (This calculation was done as part of Exercise 8.76.) Does your confidence bound confirm your conclusions in part a? Explain. What additional information does the confidence bound give you? 9.23 Lead Levels in Drinking Water Analyses of

drinking water samples for 100 homes in each of two different sections of a city gave the following means and standard deviations of lead levels (in parts per million): Sample Size Mean Standard Deviation

Section 1

Section 2

100 34.1 5.9

100 36.0 6.0

a. Calculate the test statistic and its p-value (observed signiﬁcance level) to test for a difference in the two

❍

367

population means. Use the p-value to evaluate the statistical significance of the results at the 5% level. b. Use a 95% confidence interval to estimate the difference in the mean lead levels for the two sections of the city. c. Suppose that the city environmental engineers will be concerned only if they detect a difference of more than 5 parts per million in the two sections of the city. Based on your confidence interval in part b, is the statistical significance in part a of practical significance to the city engineers? Explain. 9.24 Starting Salaries, again In an attempt to

compare the starting salaries for college graduates who majored in chemical engineering and computer science (see Exercise 8.45), random samples of 50 recent college graduates in each major were selected and the following information obtained. Major

Mean

Chemical Engineering Computer Science

$53,659 51,042

2225 2375

a. Do the data provide sufficient evidence to indicate a difference in average starting salaries for college graduates who majored in chemical engineering and computer science? Test using a .05. b. Compare your conclusions in part a with the results of part b in Exercise 8.45. Are they the same? Explain. 9.25 Hotel Costs In Exercise 8.18, we explored the average cost of lodging at three different hotel chains.6 We randomly select 50 billing statements from the computer databases of the Marriott, Radisson, and Wyndham hotel chains, and record the nightly room rates. A portion of the sample data is shown in the table. Marriott Sample Average Sample Standard Deviation

$170 17.5

Radisson $145 10

a. Before looking at the data, would you have any preconceived idea about the direction of the difference between the average room rates for these two hotels? If not, what null and alternative hypotheses should you test? b. Use the critical value approach to determine if there is a signiﬁcant difference in the average room rates for the Marriott and the Radisson hotel chains. Use a .01. c. Find the p-value for this test. Does this p-value conﬁrm the results of part b?

368 ❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

9.26 Hotel Costs II Refer to Exercise 9.25. The table below shows the sample data collected to compare the average room rates at the Wyndham and Radisson hotel chains.6 Wyndham

Radisson

$150 16.5

$145 10

Sample Average Sample Standard Deviation

a. Do the data provide sufficient evidence to indicate a difference in the average room rates for the Wyndham and the Radisson hotel chains? Use a .05. b. Construct a 95% confidence interval for the difference in the average room rates for the two chains. Does this interval confirm your conclusions in part a? 9.27 MMT in Gasoline The addition of MMT, a compound containing manganese (Mn), to gasoline as an octane enhancer has caused concern about human exposure to Mn because high intakes have been linked to serious health effects. In a study of ambient air concentrations of fine Mn, Wallace and Slonecker (Journal of the Air and Waste Management Association) presented the accompanying summary information about the amounts of fine Mn (in nanograms per cubic meter) in mostly rural national park sites and in mostly urban California sites.7

Mean Standard Deviation Number of Sites

National Parks

California

.94 1.2 36

2.8 2.8 26

a. Is there sufficient evidence to indicate that the mean concentrations differ for these two types of sites at the a .05 level of significance? Use the largesample z-test. What is the p-value of this test? b. Construct a 95% confidence interval for (m1 m2 ). Does this interval confirm your conclusions in part a?

9.5

9.28 Noise and Stress In Exercise 8.48, you compared the effect of stress in the form of noise on the ability to perform a simple task. Seventy subjects were divided into two groups; the ﬁrst group of 30 subjects acted as a control, while the second group of 40 was the experimental group. Although each subject performed the task in the same control room, each of the experimental group subjects had to perform the task while loud rock music was played. The time to ﬁnish the task was recorded for each subject and the following summary was obtained:

n x苶 s

Control

Experimental

30 15 minutes 4 minutes

40 23 minutes 10 minutes

a. Is there sufficient evidence to indicate that the average time to complete the task was longer for the experimental “rock music” group? Test at the 1% level of signiﬁcance. b. Construct a 99% one-sided upper bound for the difference (control experimental) in average times for the two groups. Does this interval conﬁrm your conclusions in part a? 9.29 What’s Normal II Of the 130 people in Exer-

cise 9.16, 65 were female and 65 were male.3 The means and standard deviations of their temperatures are shown below. Sample Mean Standard Deviation

Men

Women

98.11 0.70

98.39 0.74

a. Use the p-value approach to test for a signiﬁcant difference in the average temperatures for males versus females. b. Are the results signiﬁcant at the 5% level? At the 1% level?

A LARGE-SAMPLE TEST OF HYPOTHESIS FOR A BINOMIAL PROPORTION When a random sample of n identical trials is drawn from a binomial population, the sample proportion pˆ has an approximately normal distribution when n is large, with mean p and standard error SE

冪莦n pq

9.5 A LARGE-SAMPLE TEST OF HYPOTHESIS FOR A BINOMIAL PROPORTION

❍

369

When you test a hypothesis about p, the proportion in the population possessing a certain attribute, the test follows the same general form as the large-sample tests in Sections 9.3 and 9.4. To test a hypothesis of the form H0 : p p0 versus a one- or two-tailed alternative Ha : p p0

Ha : p p0

the test statistic is constructed using pˆ, the best estimator of the true population proportion p. The sample proportion pˆ is standardized, using the hypothesized mean and standard error, to form a test statistic z, which has a standard normal distribution if H0 is true. This large-sample test is summarized next. LARGE-SAMPLE STATISTICAL TEST FOR p 1. Null hypothesis: H0 : p p0 2. Alternative hypothesis: One-Tailed Test

Ha : p p0 (or, Ha : p p0)

Two-Tailed Test

Ha : p p0

pˆ p0 pˆ p 3. Test statistic: z 0 p q SE 0 0 n

冪莦

with

x pˆ n

where x is the number of successes in n binomial trials.† 4. Rejection region: Reject H0 when One-Tailed Test

Two-Tailed Test

z za (or z za when the alternative hypothesis is Ha : p p0)

z za/2

or when p-value a

α 0

zα

α/2

α/2 –z α/2

zα/2

Assumption: The sampling satisﬁes the assumptions of a binomial experiment (see Section 5.2), and n is large enough so that the sampling distribution of pˆ can be approximated by a normal distribution (np0 5 and nq0 5).

†

An equivalent test statistic can be found by multiplying the numerator and denominator by z by n to obtain x np0 z np0q0 兹苶

370 ❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

EXAMPLE

9.11

Regardless of age, about 20% of American adults participate in fitness activities at least twice a week. However, these fitness activities change as the people get older, and occasionally participants become nonparticipants as they age. In a local survey of n 100 adults over 40 years old, a total of 15 people indicated that they participated in a fitness activity at least twice a week. Do these data indicate that the participation rate for adults over 40 years of age is significantly less than the 20% figure? Calculate the p-value and use it to draw the appropriate conclusions. Assuming that the sampling procedure satisfies the requirements of a binomial experiment, you can answer the question posed using a one-tailed test of hypothesis:

Solution

H0 : p .2

versus Ha : p .2

Begin by assuming that H0 is true—that is, the true value of p is p0 .2. Then pˆ x/n will have an approximate normal distribution with mean p0 and standard error 兹p苶. 0q0/n (NOTE: This is different from the estimation procedure in which the 苶.) ˆqˆ/n The observed value of pˆ is 15/100 unknown standard error is estimated by 兹p .15 and the test statistic is p-value a ⇔ reject H0 p-value a ⇔ do not reject H0

.15 .20 pˆ p0 z 1.25 (.20)(.80) p q 00 100 n

冪莦冪莦莦

The p-value associated with this test is found as the area under the standard normal curve to the left of z 1.25 as shown in Figure 9.12. Therefore, p-value P(z 1.25) .1056 FI GU R E 9 .1 2

p-value for Example 9.11

● f(z)

p-value = .1056

–1.25

If you use the guidelines for evaluating p-values, then .1056 is greater than .10, and you would not reject H0. There is insufficient evidence to conclude that the percentage of adults over age 40 who participate in ﬁtness activities twice a week is less than 20%.

Statistical Significance and Practical Importance It is important to understand the difference between results that are “signiﬁcant” and results that are practically “important.” In statistical language, the word signiﬁcant does not necessarily mean “important,” but only that the results could not have occurred by chance. For example, suppose that in Example 9.11, the researcher had used

9.5 A LARGE-SAMPLE TEST OF HYPOTHESIS FOR A BINOMIAL PROPORTION

❍

371

n 400 adults in her experiment and had observed the same sample proportion. The test statistic is now .15 .20 pˆ p0 z 2.50 (.20)(.80) p q 00 400 n

冪莦冪莦莦

with p-value P(z 2.50) .0062 Now the results are highly significant: H0 is rejected, and there is sufficient evidence to indicate that the percentage of adults over age 40 who participate in physical fitness activities is less than 20%. However, is this drop in activity really important? Suppose that physicians would be concerned only about a drop in physical activity of more than 10%. If there had been a drop of more than 10% in physical activity, this would imply that the true value of p was less than .10. What is the largest possible value of p? Using a 95% upper one-sided confidence bound, you have

冪莦

pˆ qˆ pˆ 1.645 n

冪莦莦

(.15)(.85) .15 1.645 400 .15 .029 or p .179. The physical activity for adults aged 40 and older has dropped from 20%, but you cannot say that it has dropped below 10%. So, the results, although statistically signiﬁcant, are not practically important. In this book, you will learn how to determine whether results are statistically signiﬁcant. When you use these procedures in a practical situation, however, you must also make sure the results are practically important.

9.5

EXERCISES

BASIC TECHNIQUES 9.30 A random sample of n 1000 observations from a binomial population produced x 279. a. If your research hypothesis is that p is less than .3, what should you choose for your alternative hypothesis? Your null hypothesis? b. What is the critical value that determines the rejection region for your test with a .05? c. Do the data provide sufficient evidence to indicate that p is less than .3? Use a 5% signiﬁcance level. 9.31 A random sample of n 1400 observations from a binomial population produced x 529. a. If your research hypothesis is that p differs from .4, what hypotheses should you test?

b. Calculate the test statistic and its p-value. Use the p-value to evaluate the statistical signiﬁcance of the results at the 1% level. c. Do the data provide sufficient evidence to indicate that p is different from .4? 9.32 A random sample of 120 observations was

selected from a binomial population, and 72 successes were observed. Do the data provide sufficient evidence to indicate that p is greater than .5? Use one of the two methods of testing presented in this section, and explain your conclusions. APPLICATIONS 9.33 Childhood Obesity According to PARADE magazine’s “What America Eats” survey involving

372

❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

n 1015 adults, almost half of parents say their children’s weight is ﬁne.8 Only 9% of parents describe their children as overweight. However, the American Obesity Association says the number of overweight children and teens is at least 15%. Suppose that the number of parents in the sample is n 750 and the number of parents who describe their children as overweight is x 68.

a. How would you proceed to test the hypothesis that the proportion of parents who describe their children as overweight is less than the actual proportion reported by the American Obesity Association? b. What conclusion are you able to draw from these data at the a .05 level of signiﬁcance? c. What is the p-value associated with this test? 9.34 Plant Genetics A peony plant with red

petals was crossed with another plant having streaky petals. A geneticist states that 75% of the offspring resulting from this cross will have red ﬂowers. To test this claim, 100 seeds from this cross were collected and germinated, and 58 plants had red petals. a. What hypothesis should you use to test the geneticist’s claim? b. Calculate the test statistic and its p-value. Use the p-value to evaluate the statistical signiﬁcance of the results at the 1% level. 9.35 Early Detection of Breast Cancer Of those women who are diagnosed to have early-stage breast cancer, one-third eventually die of the disease. Suppose a community public health department instituted a screening program to provide for the early detection of breast cancer and to increase the survival rate p of those diagnosed to have the disease. A random sample of 200 women was selected from among those who were periodically screened by the program and who were diagnosed to have the disease. Let x represent the number of those in the sample who survive the disease. a. If you wish to detect whether the community screening program has been effective, state the null hypothesis that should be tested. b. State the alternative hypothesis. c. If 164 women in the sample of 200 survive the disease, can you conclude that the community screening program was effective? Test using a .05 and explain the practical conclusions from your test. d. Find the p-value for the test and interpret it.

9.36 Sweet Potato Whitefly Suppose that 10% of the fields in a given agricultural area are infested with the sweet potato whitefly. One hundred fields in this area are randomly selected, and 25 are found to be infested with whitefly.

a. Assuming that the experiment satisfies the conditions of the binomial experiment, do the data indicate that the proportion of infested fields is greater than expected? Use the p-value approach, and test using a 5% significance level. b. If the proportion of infested fields is found to be significantly greater than .10, why is this of practical significance to the agronomist? What practical conclusions might she draw from the results? 9.37 Brown or Blue? An article in the Washington Post stated that nearly 45% of the U.S. population is born with brown eyes, although they don’t necessarily stay that way.9 To test the newspaper’s claim, a random sample of 80 people was selected, and 32 had brown eyes. Is there sufficient evidence to dispute the newspaper’s claim regarding the proportion of browneyed people in the United States? Use a .01. 9.38 Colored Contacts Refer to Exercise 9.37.

Contact lenses, worn by about 26 million Americans, come in many styles and colors. Most Americans wear soft lenses, with the most popular colors being the blue varieties (25%), followed by greens (24%), and then hazel or brown. A random sample of 80 tinted contact lens wearers was checked for the color of their lenses. Of these people, 22 wore blue lenses and only 15 wore green lenses.9 a. Do the sample data provide sufficient evidence to indicate that the proportion of tinted contact lens wearers who wear blue lenses is different from 25%? Use a .05. b. Do the sample data provide sufficient evidence to indicate that the proportion of tinted contact lens wearers who wear green lenses is different from 24%? Use a .05. c. Is there any reason to conduct a one-tailed test for either part a or b? Explain. 9.39 A Cure for Insomnia An experimenter has

prepared a drug-dose level that he claims will induce sleep for at least 80% of people suffering from insomnia. After examining the dosage we feel that his claims regarding the effectiveness of his dosage are inﬂated. In an attempt to disprove his claim, we administer his prescribed dosage to 50 insomniacs and observe that

9.6 A LARGE-SAMPLE TEST OF HYPOTHESIS FOR THE DIFFERENCE BETWEEN TWO BINOMIAL PROPORTIONS

❍

373

37 of them have had sleep induced by the drug dose. Is there enough evidence to refute his claim at the 5% level of signiﬁcance?

percentage of adults who say that they always vote is different from the percentage reported in Time? Test using a .01.

9.40 Who Votes? About three-fourths of voting age Americans are registered to vote, but many do not bother to vote on Election Day. Only 64% voted in 1992, and 60% in 2000, but turnout in off-year elections is even lower. An article in Time stated that 35% of adult Americans are registered voters who always vote.10 To test this claim, a random sample of n 300 adult Americans was selected and x 123 were registered regular voters who always voted. Does this sample provide sufficient evidence to indicate that the

9.41 Man’s Best Friend The Humane Society

9.6

reports that there are approximately 65 million dogs owned in the United States and that approximately 40% of all U.S. households own at least one dog.11 In a random sample of 300 households, 114 households said that they owned at least one dog. Does this data provide sufficient evidence to indicate that the proportion of households with at least one dog is different from that reported by the Humane Society? Test using a .05.

A LARGE-SAMPLE TEST OF HYPOTHESIS FOR THE DIFFERENCE BETWEEN TWO BINOMIAL PROPORTIONS When random and independent samples are selected from two binomial populations, the focus of the experiment may be the difference ( p1 p2 ) in the proportions of individuals or items possessing a speciﬁed characteristic in the two populations. In this situation, you can use the difference in the sample proportions ( pˆ1 pˆ 2 ) along with its standard error, SE

莦冪莦 n n p1q1

p2q2

in the form of a z-statistic to test for a signiﬁcant difference in the two population proportions. The null hypothesis to be tested is usually of the form H0 : p1 p2

Remember: Each trial results in one of two outcomes (S or F).

H0 : ( p1 p2) 0

versus either a one- or two-tailed alternative hypothesis. The formal test of hypothesis is summarized in the next display. In estimating the standard error for the z-statistic, you should use the fact that when H0 is true, the two population proportions are equal to some common value—say, p. To obtain the best estimate of this common value, the sample data are “pooled” and the estimate of p is Total number of successes x1 x2 pˆ Total number of trials n1 n2 Remember that, in order for the difference in the sample proportions to have an approximately normal distribution, the sample sizes must be large and the proportions should not be too close to 0 or 1.

374 ❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

LARGE-SAMPLE STATISTICAL TEST FOR (p1 p2) 1. Null hypothesis: H0 : ( p1 p2 ) 0 or equivalently H0 : p1 p2 2. Alternative hypothesis: One-Tailed Test

Two-Tailed Test

Ha : ( p1 p2 ) 0 [or Ha : ( p1 p2 ) 0]

Ha : ( p1 p2 ) 0

pˆ1 pˆ 2 pˆ1 pˆ 2 ( pˆ1 pˆ 2) 0 3. Test statistic: z SE pq pq pq pq 11 22 n1 n2 n1 n2

冪莦莦冪莦莦

where pˆ1 x1/n1 and pˆ 2 x2/n2. Since the common value of p1 p2 p (used in the standard error) is unknown, it is estimated by x1 x2 pˆ n1 n2 and the test statistic is ( pˆ1 pˆ2) 0 z pˆ qˆ pˆ qˆ n1 n2

冪莦莦

pˆ1 pˆ 2 z 1 1 pˆ qˆ n1 n2

冢莦冣冪莦

4. Rejection region: Reject H0 when One-Tailed Test

Two-Tailed Test

z za [or z za when the alternative hypothesis is Ha : ( p1 p2 ) 0]

z za/2

or when p-value a

α 0

zα

α/2

α/2 –z α/2

zα/2

Assumptions: Samples are selected in a random and independent manner from two binomial populations, and n1 and n2 are large enough so that the sampling distribution of ( pˆ1 pˆ2 ) can be approximated by a normal distribution. That is, n1 pˆ1, n1qˆ1, n2 pˆ 2, and n2qˆ2 should all be greater than 5. EXAMPLE

9.12

The records of a hospital show that 52 men in a sample of 1000 men versus 23 women in a sample of 1000 women were admitted because of heart disease. Do these data present sufficient evidence to indicate a higher rate of heart disease among men admitted to the hospital? Use a .05. Assume that the number of patients admitted for heart disease has an approximate binomial probability distribution for both men and women with parameters

Solution

9.6 A LARGE-SAMPLE TEST OF HYPOTHESIS FOR THE DIFFERENCE BETWEEN TWO BINOMIAL PROPORTIONS

❍

375

p1 and p2, respectively. Then, since you wish to determine whether p1 p2, you will test the null hypothesis p1 p2—that is, H0 : ( p1 p2) 0—against the alternative hypothesis Ha : p1 p2 or, equivalently, Ha : ( p1 p2) 0. To conduct this test, use the z-test statistic and approximate the standard error using the pooled estimate of p. Since Ha implies a one-tailed test, you can reject H0 only for large values of z. Thus, for a .05, you can reject H0 if z 1.645 (see Figure 9.13). The pooled estimate of p required for the standard error is x1 x2 52 23 .0375 pˆ n1 n2 1000 1000 FIGU R E 9 .1 3

Location of the rejection region in Example 9.12

●

f(z)

α = .05 0

1.645

Rejection region

and the test statistic is .052 .023 pˆ1 pˆ2 z 3.41 1 1 1 1 (.0375)(.9625) pˆqˆ 1000 1000 n1 n2

冢莦莦冣冢莦冣冪莦莦莦冪莦

Since the computed value of z falls in the rejection region, you can reject the hypothesis that p1 p2. The data present sufficient evidence to indicate that the percentage of men entering the hospital because of heart disease is higher than that of women. (NOTE: This does not imply that the incidence of heart disease is higher in men. Perhaps fewer women enter the hospital when afflicted with the disease!) How much higher is the proportion of men than women entering the hospital with heart disease? A 95% lower one-sided conﬁdence bound will help you ﬁnd the lowest likely value for the difference.

冪莦莦

pˆ qˆ pˆ qˆ ( pˆ1 pˆ 2 ) 1.645 11 2 2 n1 n2

冪莦莦莦莦

.052(.948) .023(.977) (.052 .023) 1.645 1000 1000 .029 .014 or ( p1 p2) .015. The proportion of men is roughly 1.5% higher than women. Is this of practical importance? This is a question for the researcher to answer. In some situations, you may need to test for a difference D0 (other than 0) between two binomial proportions. If this is the case, the test statistic is modiﬁed for testing

376 ❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

H0 : ( p1 p2) D0, and a pooled estimate for a common p is no longer used in the standard error. The modiﬁed test statistic is ( pˆ1 pˆ 2 ) D0 z pˆ qˆ pˆ qˆ 11 22 n1 n2

冪莦莦

Although this test statistic is not used often, the procedure is no different from other large-sample tests you have already mastered!

9.6

EXERCISES

BASIC TECHNIQUES

9.44 Independent random samples of 280 and 350

9.42 Independent random samples of n1 140 and

observations were selected from binomial populations 1 and 2, respectively. Sample 1 had 132 successes, and sample 2 had 178 successes. Do the data present sufficient evidence to indicate that the proportion of successes in population 1 is smaller than the proportion in population 2? Use one of the two methods of testing presented in this section, and explain your conclusions.

n2 140 observations were randomly selected from binomial populations 1 and 2, respectively. Sample 1 had 74 successes, and sample 2 had 81 successes. a. Suppose you have no preconceived idea as to which parameter, p1 or p2, is the larger, but you want to detect only a difference between the two parameters if one exists. What should you choose as the alternative hypothesis for a statistical test? The null hypothesis? b. Calculate the standard error of the difference in the two sample proportions, ( pˆ1 pˆ2 ). Make sure to use the pooled estimate for the common value of p. c. Calculate the test statistic that you would use for the test in part a. Based on your knowledge of the standard normal distribution, is this a likely or unlikely observation, assuming that H0 is true and the two population proportions are the same? d. p-value approach: Find the p-value for the test. Test for a signiﬁcant difference in the population proportions at the 1% signiﬁcance level. e. Critical value approach: Find the rejection region when a .01. Do the data provide sufficient evidence to indicate a difference in the population proportions? 9.43 Refer to Exercise 9.42. Suppose, for practical

reasons, you know that p1 cannot be larger than p2. a. Given this knowledge, what should you choose as the alternative hypothesis for your statistical test? The null hypothesis? b. Does your alternative hypothesis in part a imply a one- or two-tailed test? Explain. c. Conduct the test and state your conclusions. Test using a .05.

APPLICATIONS 9.45 Treatment versus Control An experiment

was conducted to test the effect of a new drug on a viral infection. The infection was induced in 100 mice, and the mice were randomly split into two groups of 50. The ﬁrst group, the control group, received no treatment for the infection. The second group received the drug. After a 30-day period, the proportions of survivors, pˆ1 and pˆ 2, in the two groups were found to be .36 and .60, respectively. a. Is there sufficient evidence to indicate that the drug is effective in treating the viral infection? Use a .05. b. Use a 95% conﬁdence interval to estimate the actual difference in the cure rates for the treated versus the control groups. 9.46 Movie Marketing Marketing to targeted age

groups has become a standard method of advertising, even in movie theater advertising. Advertisers use computer software to track the demographics of moviegoers and then decide on the type of products to advertise before a particular movie.12 One statistic that might be of interest is how frequently adults with children under 18 attend movies as compared to those without children. Suppose that a theater database is used to randomly select 1000 adult ticket purchasers. These adults are then surveyed and asked whether they

9.6 A LARGE-SAMPLE TEST OF HYPOTHESIS FOR THE DIFFERENCE BETWEEN TWO BINOMIAL PROPORTIONS

were frequent moviegoers—that is, do they attend movies 12 or more times a year? The results are shown in the table:

Sample Size Number Who Attend 12 Times per Year

With Children under 18

Without Children

440 123

560 145

a. Is there a signiﬁcant difference in the population proportions of frequent moviegoers in these two demographic groups? Use a .01. b. Why would a statistically signiﬁcant difference in these population proportions be of practical importance to the advertiser? 9.47 M&M’S In Exercise 8.53, you investigated

whether Mars, Inc., uses the same proportion of red M&M’S in its plain and peanut varieties. Random samples of plain and peanut M&M’S provide the following sample data for the experiment: Sample Size Number of Red M&M’S

Plain

Peanut

56 12

32 8

Use a test of hypothesis to determine whether there is a signiﬁcant difference in the proportions of red candies for the two types of M&M’S. Let a .05 and compare your results with those of Exercise 8.53. 9.48 Hormone Therapy and Alzheimer’s Disease

In the last few years, many research studies have shown that the purported beneﬁts of hormone replacement therapy (HRT) do not exist, and in fact, that hormone replacement therapy actually increases the risk of several serious diseases. A four-year experiment involving 4532 women, reported in The Press Enterprise, was conducted at 39 medical centers. Half of the women took placebos and half took Prempro, a widely prescribed type of hormone replacement therapy. There were 40 cases of dementia in the hormone group and 21 in the placebo group.13 Is there sufficient evidence to indicate that the risk of dementia is higher for patients using Prempro? Test at the 1% level of signiﬁcance. 9.49 HRT, continued Refer to Exercise 9.48. Cal-

culate a 99% lower one-sided conﬁdence bound for the difference in the risk of dementia for women using hormone replacement therapy versus those who do not. Would this difference be of practical importance to a woman considering HRT? Explain.

❍

377

9.50 Clopidogrel and Aspirin A large study was conducted to test the effectiveness of clopidogrel in combination with aspirin in warding off heart attacks and strokes.14 The trial involved more than 15,500 people 45 years of age or older from 32 countries, including the United States, who had been diagnosed with cardiovascular disease or had multiple risk factors. The subjects were randomly assigned to one of two groups. After two years, there was no difference in the risk of heart attack, stroke, or dying from heart disease between those who took clopidogrel and low-dose aspirin daily and those who took low-dose aspirin plus a dummy pill. The two-drug combination actually increased the risk of dying (5.4% versus 3.8%) or dying specifically from cardiovascular disease (3.9% versus 2.2%). a. The subjects were randomly assigned to one of the two groups. Explain how you could use the random number table to make these assignments. b. No sample sizes were given in the article: however, let us assume that the sample sizes for each group were n1 7720 and n2 7780. Determine whether the risk of dying was significantly different for the two groups. c. What do the results of the study mean in terms of practical significance? 9.51 Baby’s Sleeping Position Does a baby’s

sleeping position affect the development of motor skills? In one study, published in the Archives of Pediatric Adolescent Medicine, 343 full-term infants were examined at their 4-month checkups for various developmental milestones, such as rolling over, grasping a rattle, reaching for an object, and so on.15 The baby’s predominant sleep position—either prone (on the stomach) or supine (on the back) or side—was determined by a telephone interview with the parent. The sample results for 320 of the 343 infants for whom information was received are shown here:

Number of Infants Number That Roll Over

Prone

Supine or Side

121 93

199 119

The researcher reported that infants who slept in the side or supine position were less likely to roll over at the 4-month checkup than infants who slept primarily in the prone position (P .001). Use a large-sample test of hypothesis to conﬁrm or refute the researcher’s conclusion.

378 ❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

9.7

SOME COMMENTS ON TESTING HYPOTHESES A statistical test of hypothesis is a fairly clear-cut procedure that enables an experimenter to either reject or accept the null hypothesis H0, with measured risks a and b. The experimenter can control the risk of falsely rejecting H0 by selecting an appropriate value of a. On the other hand, the value of b depends on the sample size and the values of the parameter under test that are of practical importance to the experimenter. When this information is not available, an experimenter may decide to select an affordable sample size, in the hope that the sample will contain sufficient information to reject the null hypothesis. The chance that this decision is in error is given by a, whose value has been set in advance. If the sample does not provide sufficient evidence to reject H0, the experimenter may wish to state the results of the test as “The data do not support the rejection of H0” rather than accepting H0 without knowing the chance of error b. Some experimenters prefer to use the observed p-value of the test to evaluate the strength of the sample information in deciding to reject H0. These values can usually be generated by computer and are often used in reports of statistical results: •

If the p-value is greater than .05, the results are reported as NS—not signiﬁcant at the 5% level.

•

If the p-value lies between .05 and .01, the results are reported as P .05— signiﬁcant at the 5% level.

•

If the p-value lies between .01 and .001, the results are reported as P .01— “highly signiﬁcant” or signiﬁcant at the 1% level.

•

If the p-value is less than .001, the results are reported as P .001—“very highly signiﬁcant” or signiﬁcant at the .1% level.

Still other researchers prefer to construct a confidence interval for a parameter and perform a test informally. If the value of the parameter specified by H0 is included within the upper and lower limits of the confidence interval, then “H0 is not rejected.” If the value of the parameter specified by H0 is not contained within the interval, then “H0 is rejected.” These results will agree with a two-tailed test; one-sided confidence bounds are used for one-tailed alternatives. Finally, consider the choice between a one- and two-tailed test. In general, experimenters wish to know whether a treatment causes what could be a beneficial increase in a parameter or what might be a harmful decrease in a parameter. Therefore, most tests are two-tailed unless a one-tailed test is strongly dictated by practical considerations. For example, assume you will sustain a large financial loss if the mean m is greater than m0 but not if it is less. You will then want to detect values larger than m0 with a high probability and thereby use a right-tailed test. In the same vein, if pollution levels higher than m0 cause critical health risks, then you will certainly wish to detect levels higher than m0 with a right-tailed test of hypothesis. In any case, the choice of a one- or two-tailed test should be dictated by the practical consequences that result from a decision to reject or not reject H0 in favor of the alternative.

CHAPTER REVIEW

❍

379

CHAPTER REVIEW 4. In a Type II error, b is the probability of accepting H0 when it is in fact false. The power of the test is (1 b), the probability of rejecting H0 when it is false.

Key Concepts and Formulas I.

Parts of a Statistical Test

1. Null hypothesis: a contradiction of the alternative hypothesis 2. Alternative hypothesis: the hypothesis the researcher wants to support

III. Large-Sample Test Statistics Using the z Distribution

To test one of the four population parameters when the sample sizes are large, use the following test statistics:

3. Test statistic and its p-value: sample evidence calculated from the sample data 4. Rejection region—critical values and signiﬁcance levels: values that separate rejection and nonrejection of the null hypothesis 5. Conclusion: Reject or do not reject the null hypothesis, stating the practical signiﬁcance of your conclusion

Parameter

Test Statistic

x苶 m0 z s/兹苶n

II. Errors and Statistical Significance

1. The signiﬁcance level a is the probability of rejecting H0 when it is in fact true. 2. The p-value is the probability of observing a test statistic as extreme as or more extreme than the one observed; also, the smallest value of a for which H0 can be rejected. 3. When the p-value is less than the signiﬁcance level a, the null hypothesis is rejected. This happens when the test statistic exceeds the critical value.

Supplementary Exercises Starred (*) exercises are optional. 9.52 a. Deﬁne a and b for a statistical test of hypothesis. b. For a ﬁxed sample size n, if the value of a is decreased, what is the effect on b? c. In order to decrease both a and b for a particular alternative value of m, how must the sample size change? 9.53 What is the p-value for a test of hypothesis?

How is it calculated for a large-sample test? 9.54 What conditions must be met so that the z test can be used to test a hypothesis concerning a population mean m?

m1 m2

p1 p2

pˆ p0 z pq 00 n

冪莦

(x苶1 x苶2) D0 z s2 s2 1 2 n1 n2

冪莦莦

pˆ1 pˆ 2 z 1 1 pˆ qˆ n1 n2

冪莦冢莦冣

(pˆ1 pˆ 2) D0 z pˆ qˆ pˆ qˆ 11 22 n1 n2

冪莦莦

9.55 Deﬁne the power of a statistical test. As the

alternative value of m gets farther from m0, how is the power affected? 9.56 Acidity in Rainfall Refer to Exercise 8.31 and the collection of water samples to estimate the mean acidity (in pH) of rainfalls in the northeastern United States. As noted, the pH for pure rain falling through clean air is approximately 5.7. The sample of n 40 rainfalls produced pH readings with x苶 3.7 and s .5. Do the data provide sufficient evidence to indicate that the mean pH for rainfalls is more acidic (Ha : m 5.7 pH) than pure rainwater? Test using a .05. Note that this inference is appropriate only for the area in which the rainwater specimens were collected.

380

❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

9.57 Washing Machine Colors A manufacturer of automatic washers provides a particular model in one of three colors. Of the first 1000 washers sold, it is noted that 400 were of the first color. Can you conclude that more than one-third of all customers have a preference for the first color? a. Find the p-value for the test.

b. If you plan to conduct your test using a .05, what will be your test conclusions? 9.58 Generation Next Born between 1980 and

1990, Generation Next was the topic of Exercise 8.60.16 In a survey of 500 female and 500 male students in Generation Next, 345 of the females and 365 of the males reported that they decided to attend college in order to make more money. a. Is there a signiﬁcant difference in the population proportions of female and male students who decided to attend college in order to make more money? Use a .01. b. Can you think of any reason why a statistically signiﬁcant difference in these population proportions might be of practical importance? To whom might this difference be important? 9.59 Bass Fishing The pH factor is a measure of the

acidity or alkalinity of water. A reading of 7.0 is neutral; values in excess of 7.0 indicate alkalinity; those below 7.0 imply acidity. Loren Hill states that the best chance of catching bass occurs when the pH of the water is in the range 7.5 to 7.9.17 Suppose you suspect that acid rain is lowering the pH of your favorite ﬁshing spot and you wish to determine whether the pH is less than 7.5. a. State the alternative and null hypotheses that you would choose for a statistical test. b. Does the alternative hypothesis in part a imply a one- or a two-tailed test? Explain.

9.60 Pennsylvania Lottery A central Pennsylvania

attorney reported that the Northumberland County district attorney’s (DA) office trial record showed only 6 convictions in 27 trials from January to mid-July 1997. Four central Pennsylvania county DAs responded, “Don’t judge us by statistics!”18 a. If the attorney’s information is correct, would you reject a claim by the DA of a 50% or greater conviction rate? b. The actual records show that there have been 455 guilty pleas and 48 cases that have gone to trial. Even assuming that the 455 guilty pleas are the only convictions of the 503 cases reported, what is the 95% confidence interval for p, the true proportion of convictions by this district attorney? c. Using the results of part b, are you willing to reject a figure of 50% or greater for the true conviction rate? Explain. 9.61 White-Tailed Deer In an article entitled “A Strategy for Big Bucks,” Charles Dickey discusses studies of the habits of white-tailed deer that indicate that they live and feed within very limited ranges— approximately 150 to 205 acres.19 To determine whether there was a difference between the ranges of deer located in two different geographic areas, 40 deer were caught, tagged, and fitted with small radio transmitters. Several months later, the deer were tracked and identified, and the distance x from the release point was recorded. The mean and standard deviation of the distances from the release point were as follows:

Sample Size Sample Mean Sample Standard Deviation

Location 1

Location 2

40 2980 ft 1140 ft

40 3205 ft 963 ft

a. If you have no preconceived reason for believing one population mean is larger than another, what would you choose for your alternative hypothesis? Your null hypothesis?

c. Suppose that a random sample of 30 water specimens gave pH readings with x苶 7.3 and s .2. Just glancing at the data, do you think that the difference x苶 7.5 .2 is large enough to indicate that the mean pH of the water samples is less than 7.5? (Do not conduct the test.)

b. Does your alternative hypothesis in part a imply a one- or a two-tailed test? Explain. c. Do the data provide sufficient evidence to indicate that the mean distances differ for the two geographic locations? Test using a .05.

d. Now conduct a statistical test of the hypotheses in part a and state your conclusions. Test using a .05. Compare your statistically based decision with your intuitive decision in part c.

9.62 Female Models In a study to assess various

effects of using a female model in automobile advertising, 100 men were shown photographs of two automobiles matched for price, color, and size, but of

SUPPLEMENTARY EXERCISES

different makes. One of the automobiles was shown with a female model to 50 of the men (group A), and both automobiles were shown without the model to the other 50 men (group B). In group A, the automobile shown with the model was judged as more expensive by 37 men; in group B, the same automobile was judged as the more expensive by 23 men. Do these results indicate that using a female model inﬂuences the perceived cost of an automobile? Use a one-tailed test with a .05. 9.63 Bolts Random samples of 200 bolts manufactured by a type A machine and 200 bolts manufactured by a type B machine showed 16 and 8 defective bolts, respectively. Do these data present sufficient evidence to suggest a difference in the performance of the machine types? Use a .05. 9.64 Biomass Exercise 7.63 reported that the biomass for tropical woodlands, thought to be about 35 kilograms per square meter (kg/m2), may in fact be too high and that tropical biomass values vary regionally—from about 5 to 55 kg/m2.20 Suppose you measure the tropical biomass in 400 randomly selected square-meter plots and obtain 苶x 31.75 and s 10.5. Do the data present sufficient evidence to indicate that scientists are overestimating the mean biomass for tropical woodlands and that the mean is in fact lower than estimated? a. State the null and alternative hypotheses to be tested. b. Locate the rejection region for the test with a .01. c. Conduct the test and state your conclusions. 9.65 Adolescents and Social Stress In a study to compare ethnic differences in adolescents’ social stress, researchers recruited subjects from three middle schools in Houston, Texas.21 Social stress among four ethnic groups was measured using the Social Attitudinal Familial and Environment Scale for Children (SAFE-C). In addition, demographic information about the 316 students was collected using self-administered questionnaires. A tabulation of student responses to a question regarding their socioeconomic status (SES) compared with other families in which the students chose one of ﬁve responses (much worse off, somewhat worse off, about the same, better off, or much better off ) resulted in the tabulation that follows. European African Hispanic American American American Sample Size About the Same

144 68

66 42

77 48

Asian American 19 8

❍

381

a. Do these data support the hypothesis that the proportion of adolescent African Americans who state that their SES is “about the same” exceeds that for adolescent Hispanic Americans? b. Find the p-value for the test. c. If you plan to test using a .05, what is your conclusion? 9.66* Adolescents and Social Stress, continued

Refer to Exercise 9.65. Some thought should have been given to designing a test for which b is tolerably low when p1 exceeds p2 by an important amount. For example, ﬁnd a common sample size n for a test with a .05 and b .20 when in fact p1 exceeds p2 by 0.1. (HINT: The maximum value of p(1 p) .25.) 9.67 Losing Weight In a comparison of the mean 1-month weight losses for women aged 20–30 years, these sample data were obtained for each of two diets: Sample Size n Sample Mean x苶 Sample Variance s 2

Diet I

Diet II

40 10 lb 4.3

40 8 lb 5.7

Do the data provide sufficient evidence to indicate that diet I produces a greater mean weight loss than diet II? Use a .05. 9.68 Increased Yield An agronomist has shown experimentally that a new irrigation/fertilization regimen produces an increase of 2 bushels per quadrat (signiﬁcant at the 1% level) when compared with the regimen currently in use. The cost of implementing and using the new regimen will not be a factor if the increase in yield exceeds 3 bushels per quadrat. Is statistical signiﬁcance the same as practical importance in this situation? Explain. 9.69 Breaking Strengths of Cables A test of the breaking strengths of two different types of cables was conducted using samples of n1 n2 100 pieces of each type of cable. Cable I

Cable II

x苶1 1925 s1 40

苶x2 1905 s2 30

Do the data provide sufficient evidence to indicate a difference between the mean breaking strengths of the two cables? Use a .05. 9.70 Put on the Brakes The braking ability was

compared for two 2008 automobile models. Random samples of 64 automobiles were tested for each type. The recorded measurement was the distance (in feet)

382 ❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

required to stop when the brakes were applied at 40 miles per hour. These are the computed sample means and variances:

2005 were randomly selected and their SAT scores recorded in the following table:

Model I

Model II

x苶1 118 s 12 102

苶x22 109 s 2 87

Sample Average Sample Standard Deviation

Do the data provide sufficient evidence to indicate a difference between the mean stopping distances for the two models? 9.71 Spraying Fruit Trees A fruit grower wants to test a new spray that a manufacturer claims will reduce the loss due to insect damage. To test the claim, the grower sprays 200 trees with the new spray and 200 other trees with the standard spray. The following data were recorded: Mean Yield per Tree 苶x (lb) Variance s 2

New Spray

Standard Spray

240 980

227 820

a. Do the data provide sufficient evidence to conclude that the mean yield per tree treated with the new spray exceeds that for trees treated with the standard spray? Use a .05. b. Construct a 95% conﬁdence interval for the difference between the mean yields for the two sprays. 9.72 Actinomycin D A biologist hypothesizes that

high concentrations of actinomycin D inhibit RNA synthesis in cells and hence the production of proteins as well. An experiment conducted to test this theory compared the RNA synthesis in cells treated with two concentrations of actinomycin D: .6 and .7 microgram per milliliter. Cells treated with the lower concentration (.6) of actinomycin D showed that 55 out of 70 developed normally, whereas only 23 out of 70 appeared to develop normally for the higher concentration (.7). Do these data provide sufficient evidence to indicate a difference between the rates of normal RNA synthesis for cells exposed to the two different concentrations of actinomycin D? a. Find the p-value for the test. b. If you plan to conduct your test using a .05, what will be your test conclusions? 9.73 SAT Scores How do California high school

students compare to students nationwide in their college readiness, as measured by their SAT scores? The national average scores for the class of 2005 were 508 on the verbal portion and 520 on the math portion.22 Suppose that 100 California students from the class of

Verbal

Math

499 98

516 96

a. Do the data provide sufficient evidence to indicate that the average verbal score for all California students in the class of 2005 is different from the national average? Test using a .05. b. Do the data provide sufficient evidence to indicate that the average math score for all California students in the class of 2005 is different from the national average? Test using a .05. c. Could you use this data to determine if there is a difference between the average math and verbal scores for all California students in the class of 2005? Explain your answer. 9.74 A Maze Experiment In a maze running study, a rat is run in a T maze and the result of each run recorded. A reward in the form of food is always placed at the right exit. If learning is taking place, the rat will choose the right exit more often than the left. If no learning is taking place, the rat should randomly choose either exit. Suppose that the rat is given n 100 runs in the maze and that he chooses the right exit x 64 times. Would you conclude that learning is taking place? Use the p-value approach, and make a decision based on this p-value. 9.75 PCBs Polychlorinated biphenyls (PCBs) have been found to be dangerously high in some game birds found along the marshlands of the southeastern coast of the United States. The Federal Drug Administration (FDA) considers a concentration of PCBs higher than 5 parts per million (ppm) in these game birds to be dangerous for human consumption. A sample of 38 game birds produced an average of 7.2 ppm with a standard deviation of 6.2 ppm. Is there sufficient evidence to indicate that the mean ppm of PCBs in the population of game birds exceeds the FDA’s recommended limit of 5 ppm? Use a .01. 9.76* PCBs, continued Refer to Exercise 9.75. a. Calculate b and 1 b if the true mean ppm of PCBs is 6 ppm. b. Calculate b and 1 b if the true mean ppm of PCBs is 7 ppm.

c. Find the power, 1 b, when m 8, 9, 10, and 12. Use these values to construct a power curve for the test in Exercise 9.75.

❍

MYAPPLET EXERCISES

d. For what values of m does this test have power greater than or equal to .90? 9.77 9/11 Conspiracy Some Americans believe that the entire 9/11 catastrophe was planned and executed by federal officials in order to provide the United States with a pretext for going to war in the Middle East and as a means of consolidating and extending the power of the then-current administration. This group of Americans is larger than you think. A Scripps-Howard poll of n 1010 adults in August of 2006 found that 36% of American consider such a scenario very or somewhat likely!23 In a follow-up poll, a random sample of n 100 adult Americans found that 26 of those sampled agreed that the conspiracy theory was either likely or somewhat likely. Does this sample contradict the reported 36% ﬁgure? Test at the a .05 level of signiﬁcance. 9.78 Heights and Gender It is a well-accepted fact that males are taller on the average than females. But how much taller? The genders of 105 biomedical students (Exercise 1.54) were also recorded and the data are summarized below:

Sample Size Sample Mean Sample Standard Deviation

Males

Females

48 69.58 2.62

77 64.43 2.58

a. Perform a test of hypothesis to either confirm or refute our initial claim that males are taller on the average than females? Use a .01. b. If the results of part a show that our claim was correct, construct a 99% confidence one-sided lower confidence bound for the average difference in heights between male and female college students. How much taller are males than females? 9.79 English as a Second Language The state of

California is working very hard to make sure that all elementary-aged students whose native language is not English become proﬁcient in English by the sixth

383

grade. Their progress is monitored each year using the California English Language Development Test.24 The results for two school districts in southern California for a recent school year are shown below. District Number of Students Tested Percentage Fluent

Riverside

Palm Springs

6124 40

5512 37

Does this data provide sufficient statistical evidence to indicate that the percentage of students who are ﬂuent in English differs for these two districts? Test using a .01. 9.80 Breaststroke Swimmers How much training time does it take to become a world-class breaststroke swimmer? A survey published in The American Journal of Sports Medicine reported the number of meters per week swum by two groups of swimmers—those who competed only in breaststroke and those who competed in the individual medley (which includes breaststroke). The number of meters per week practicing the breaststroke swim was recorded and the summary statistics are shown below.25

Sample Size Sample Mean Sample Standard Deviation

Breaststroke

Individual Medley

130 9017 7162

80 5853 1961

Is there sufficient evidence to indicate a difference in the average number of meters swum by these two groups of swimmers? Test using a .01. 9.81 Breaststroke, continued Refer to Exer-

cise 9.80. a. Construct a 99% conﬁdence interval for the difference in the average number of meters swum by breaststroke versus individual medley swimmers. b. How much longer do pure breaststroke swimmers practice that stroke than individual medley swimmers? What is the practical reason for this difference?

Exercises 9.82 School Workers In Exercise 8.109, the aver-

age hourly wage for public school cafeteria workers was given as $10.33.26 If n 40 randomly selected public school cafeteria workers within one school district are found to have an average hourly wage of 苶x $9.75 with a standard deviation of s $1.65, would

this information contradict the reported average of $10.33? a. What are the null and alternative hypotheses to be tested? b. Use the Large-Sample Test of a Population Mean applet to ﬁnd the observed value of the test statistic.

384 ❍

CHAPTER 9 LARGE-SAMPLE TESTS OF HYPOTHESES

c. Use the Large-Sample Test of a Population Mean applet to ﬁnd the p-value of this test. d. Based on your results from part c, what conclusions can you draw about the average hourly wage of $10.33? 9.83 Daily Wages The daily wages in a particular

industry are normally distributed with a mean of $94 and a standard deviation of $11.88. Suppose a company in this industry employs 40 workers and pays them $91.50 per week on the average. Can these workers be viewed as a random sample from among all workers in the industry? a. What are the null and alternative hypotheses to be tested? b. Use the Large-Sample Test of a Population Mean applet to ﬁnd the observed value of the test statistic. c. Use the Large-Sample Test of a Population Mean applet to ﬁnd the p-value for this test. d. If you planned to conduct your test using a .01, what would be your test conclusions? e. Was it necessary to know that the daily wages are normally distributed? Explain your answer. 9.84 Refer to Example 9.8. Use the Power of a

z-Test applet to verify the power of the test of H0: m 880

CASE STUDY

for values of m equal to 870, 875, 880, 885 and 890. Check your answers against the values shown in Table 9.2. 9.85 Refer to Example 9.8. a. Use the method given in Example 9.8 to calculate the power of the test of H0: m 880

versus Ha: m 880

when n 30 and the true value of m is 870 tons. b. Repeat part a using n 70 and m 870 tons. c. Use the Power of a z-Test applet to verify your hand-calculated results in parts a and b. d. What is the effect of increasing the sample size on the power of the test? 9.86 Use the appropriate slider on the Power of a

z-Test applet to answer the following questions. Write a sentence for each part, describing what you see using the applet. a. What effect does increasing the sample size have on the power of the test? b. What effect does increasing the distance between the true value of m and the hypothesized value, m 880, have on the power of the test? c. What effect does decreasing the signiﬁcance level a have on the power of the test?

versus Ha: m 880

An Aspirin a Day . . . ? On Wednesday, January 27, 1988, the front page of the New York Times read, “Heart attack risk found to be cut by taking aspirin: Lifesaving effects seen.” A very large study of U.S. physicians showed that a single aspirin tablet taken every other day reduced by one-half the risk of heart attack in men.27 Three days later, a headline in the Times read, “Value of daily aspirin disputed in British study of heart attacks.” How could two seemingly similar studies, both involving doctors as participants, reach such opposite conclusions? The U.S. physicians’ health study consisted of two randomized clinical trials in one. The ﬁrst tested the hypothesis that 325 milligrams (mg) of aspirin taken every other day reduces mortality from cardiovascular disease. The second tested whether 50 mg of b-carotene taken on alternate days decreases the incidence of cancer. From names on an American Medical Association computer tape, 261,248 male physicians between the ages of 40 and 84 were invited to participate in the trial. Of those who responded, 59,285 were willing to participate. After the exclusion of those physicians who had a history of medical disorders, or who were currently taking aspirin or had negative reactions to aspirin, 22,071 physicians were randomized into one of four treatment groups: (1) buffered aspirin and b-carotene, (2) buffered aspirin and a

CASE STUDY

❍

385

b-carotene placebo, (3) aspirin placebo and b-carotene, and (4) aspirin placebo and b-carotene placebo. Thus, half were assigned to receive aspirin and half to receive b-carotene. The study was conducted as a double-blind study, in which neither the participants nor the investigators responsible for following the participants knew to which group a participant belonged. The results of the American study concerning myocardial infarctions (the technical name for heart attacks) are given in the following table: American Study Aspirin (n 11,037) Myocardial Infarction Fatal Nonfatal Total

Placebo (n 11,034)

5 99

18 171

104

189

The objective of the British study was to determine whether 500 mg of aspirin taken daily would reduce the incidence of and mortality from cardiovascular disease. In 1978 all male physicians in the United Kingdom were invited to participate. After the usual exclusions, 5139 doctors were randomly allocated to take aspirin, unless some problem developed, and one-third were randomly allocated to avoid aspirin. Placebo tablets were not used, so the study was not blind! The results of the British study are given here: British Study

Myocardial Infarction Fatal Nonfatal Total

Aspirin (n 3429)

Control (n 1710)

89 (47.3) 80 (42.5)

47 (49.6) 41 (43.3)

169 (89.8)

88 (92.9)

To account for unequal sample sizes, the British study reported rates per 10,000 subject-years alive (given in parentheses). 1. Test whether the American study does in fact indicate that the rate of heart attacks for physicians taking 325 mg of aspirin every other day is significantly different from the rate for those on the placebo. Is the American claim justified? 2. Repeat the analysis using the data from the British study in which one group took 500 mg of aspirin every day and the control group took none. Based on their data, is the British claim justified? 3. Can you think of some possible reasons the results of these two studies, which were alike in some respects, produced such different conclusions?

Inference from Small Samples © CORBIS SYGMA

GENERAL OBJECTIVE The basic concepts of large-sample statistical estimation and hypothesis testing for practical situations involving population means and proportions were introduced in Chapters 8 and 9. Because all of these techniques rely on the Central Limit Theorem to justify the normality of the estimators and test statistics, they apply only when the samples are large. This chapter supplements the large-sample techniques by presenting small-sample tests and conﬁdence intervals for population means and variances. Unlike their large-sample counterparts, these small-sample techniques require the sampled populations to be normal, or approximately so.

CHAPTER INDEX ● Comparing two population variances (10.7) ● Inferences concerning a population variance (10.6) ● Paired-difference test: Dependent samples (10.5) ● Small-sample assumptions (10.8) ● Small-sample inferences concerning the difference in two means: Independent random samples (10.4)

Would You Like a Four-Day Workweek? Will a flexible workweek schedule result in positive benefits for both employer and employee? Four obvious benefits are (1) less time traveling from field positions to the office, (2) fewer employees parked in the parking lot, (3) reduced travel expenses, and (4) allowance for employees to have another day off. But does the flexible workweek make employees more efficient and cause them to take fewer sick and personal days? The answers to some of these questions are posed in the case study at the end of this chapter.

● Small-sample inferences concerning a population mean (10.3) ● Student’s t distribution (10.2)

How Do I Decide Which Test to Use?

386

10.2 STUDENT’S t DISTRIBUTION

10.1

❍

387

INTRODUCTION Suppose you need to run an experiment to estimate a population mean or the difference between two means. The process of collecting the data may be very expensive or very time-consuming. If you cannot collect a large sample, the estimation and test procedures of Chapters 8 and 9 are of no use to you. This chapter introduces some equivalent statistical procedures that can be used when the sample size is small. The estimation and testing procedures involve these familiar parameters: • • • •

A single population mean, m The difference between two population means, (m1 m2) A single population variance, s 2 The comparison of two population variances, s 21 and s 22

Small-sample tests and conﬁdence intervals for binomial proportions will be omitted from our discussion.†

10.2

STUDENT’S t DISTRIBUTION In conducting an experiment to evaluate a new but very costly process for producing synthetic diamonds, you are able to study only six diamonds generated by the process. How can you use these six measurements to make inferences about the average weight m of diamonds from this process? In discussing the sampling distribution of x苶 in Chapter 7, we made these points: • •

When n 30, the Central Limit Theorem will not guarantee that x苶 m 苶 s/兹n

When the original sampled population is normal, 苶x and z ( x苶 m)/(s/兹n苶) both have normal distributions, for any sample size. 苶), When the original sampled population is not normal, 苶x, z ( x苶 m)/(s/兹n 苶) all have approximately normal distributions, if the and z ⬇ ( x苶 m)/(s/兹n sample size is large.

苶) does not Unfortunately, when the sample size n is small, the statistic (x苶 m)/(s/兹n have a normal distribution. Therefore, all the critical values of z that you used in Chapters 8 and 9 are no longer correct. For example, you cannot say that 苶x will lie within 1.96 standard errors of m 95% of the time. This problem is not new; it was studied by statisticians and experimenters in the early 1900s. To ﬁnd the sampling distribution of this statistic, there are two ways to proceed:

is approximately normal.

•

†

Use an empirical approach. Draw repeated samples and compute ( 苶x m)/(s/兹n苶) for each sample. The relative frequency distribution that you construct using these values will approximate the shape and location of the sampling distribution. Use a mathematical approach to derive the actual density function or curve that describes the sampling distribution.

A small-sample test for the binomial parameter p will be presented in Chapter 15.

388 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

This second approach was used by an Englishman named W.S. Gosset in 1908. He derived a complicated formula for the density function of x苶 m t s/兹n苶 for random samples of size n from a normal population, and he published his results under the pen name “Student.” Ever since, the statistic has been known as Student’s t. It has the following characteristics: • •

•

FI GU R E 1 0 . 1

Standard normal z and the t distribution with 5 degrees of freedom

It is mound-shaped and symmetric about t 0, just like z. It is more variable than z, with “heavier tails”; that is, the t curve does not approach the horizontal axis as quickly as z does. This is because the t statistic involves two random quantities, 苶x and s, whereas the z statistic involves only the sample mean, x苶. You can see this phenomenon in Figure 10.1. The shape of the t distribution depends on the sample size n. As n increases, the variability of t decreases because the estimate s of s is based on more and more information. Eventually, when n is inﬁnitely large, the t and z distributions are identical!

●

Normal distribution

t distribution

For a one-sample t, df n 1.

FI GU R E 1 0 . 2

Tabulated values of Student’s t

The divisor (n 1) in the formula for the sample variance s2 is called the number of degrees of freedom (df ) associated with s2. It determines the shape of the t distribution. The origin of the term degrees of freedom is theoretical and refers to the number of independent squared deviations in s2 that are available for estimating s 2. These degrees of freedom may change for different applications and, since they specify the correct t distribution to use, you need to remember to calculate the correct degrees of freedom for each application. The table of probabilities for the standard normal z distribution is no longer useful in calculating critical values or p-values for the t statistic. Instead, you will use Table 4 in Appendix I, which is partially reproduced in Table 10.1. When you index a particular number of degrees of freedom, the table records ta, a value of t that has tail area a to its right, as shown in Figure 10.2. ● f(t)

a 0

10.2 STUDENT’S t DISTRIBUTION

TABLE 10.1

EXAMPLE

●

10.1

❍

389

Format of the Student’s t Table from Table 4 in Appendix I df

t.100

t.050

t.025

t.010

t.005

1 2 3 4 5 6 7 8 9 . . . 26 27 28 29 inf.

3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 . . . 1.315 1.314 1.313 1.311 1.282

6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 . . . 1.706 1.703 1.701 1.699 1.645

12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 . . . 2.056 2.052 2.048 2.045 1.960

31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 . . . 2.479 2.473 2.467 2.462 2.326

63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 . . . 2.779 2.771 2.763 2.756 2.576

1 2 3 4 5 6 7 8 9 . . . 26 27 28 29 inf.

For a t distribution with 5 degrees of freedom, the value of t that has area .05 to its right is found in row 5 in the column marked t.050. For this particular t distribution, the area to the right of t 2.015 is .05; only 5% of all values of the t statistic will exceed this value.

You can use the Student’s t Probabilities applet to find the t-value described in Example 10.1. The first applet, shown in Figure 10.3, provides t-values and their two-tailed probabilities, while the second applet provides t-values and one-tailed probabilities. Use the slider on the right side of the applet to select the proper degrees of freedom. For Example 10.1, you should choose df 5 and type .10 in the box marked “prob:” at the bottom of the first applet. The applet will provide the value of t that puts .05 in one tail of the t distribution. The second applet will show the identical t for a one-tailed area of .05. The applet in Figure 10.3 shows t 2.02 which is correct to two decimal places. We will use this applet for the MyApplet Exercises at the end of the chapter. FIGU R E 1 0 . 3

Student’s t Probabilities applet

●

390 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

EXAMPLE

Suppose you have a sample of size n 10 from a normal distribution. Find a value of t such that only 1% of all values of t will be smaller.

10.2

Solution The degrees of freedom that specify the correct t distribution are df n 1 9, and the necessary t-value must be in the lower portion of the distribution, with area .01 to its left, as shown in Figure 10.4. Since the t distribution is symmetric about 0, this value is simply the negative of the value on the right-hand side with area .01 to its right, or t.01 2.821. FI GU R E 1 0 . 4

t Distribution for Example 10.2

● f(t)

.01 –2.821

Comparing the t and z Distributions Look at one of the columns in Table 10.1. As the degrees of freedom increase, the critical value of t decreases until, when df inf., the critical t-value is the same as the critical z-value for the same tail area. You can use the Comparing t and z applet to visualize this concept. Look at the three applets in Figure 10.5, which show the critical values for t.025 compared with z.025 for df 8, 29 and 100. (The slider on the right side of the applet allows you to change the df.) The red curve (black in Figure 10.5) is the standard normal distribution, with z.025 1.96. FI GU R E 1 0 . 5

Comparing t and z applet

●

10.3 SMALL-SAMPLE INFERENCES CONCERNING A POPULATION MEAN

❍

391

The blue curve is the t distribution. With 8 df, you can clearly see a difference in the t and z curves, especially in the critical values that cut off an area of .025 in the tails. As the degrees of freedom increase, the difference in the shapes of t and z becomes very similar, as do their critical values, until at df 100, there is almost no difference. This helps to explain why we use n 30 as the somewhat arbitrary dividing line between large and small samples. When n 30 (df 29), the critical values of t are quite close to their normal counterparts. Rather than produce a t table with rows for many more degrees of freedom, the critical values of z are sufficient when the sample size reaches n 30.

Assumptions behind Student’s t Distribution

Assumptions for one-sample t: • Random sample • Normal distribution

The critical values of t allow you to make reliable inferences only if you follow all the rules; that is, your sample must meet these requirements speciﬁed by the t distribution: • •

The sample must be randomly selected. The population from which you are sampling must be normally distributed.

These requirements may seem quite restrictive. How can you possibly know the shape of the probability distribution for the entire population if you have only a sample? If this were a serious problem, however, the t statistic could be used in only very limited situations. Fortunately, the shape of the t distribution is not affected very much as long as the sampled population has an approximately mound-shaped distribution. Statisticians say that the t statistic is robust, meaning that the distribution of the statistic does not change signiﬁcantly when the normality assumption is violated. How can you tell whether your sample is from a normal population? Although there are statistical procedures designed for this purpose, the easiest and quickest way to check for normality is to use the graphical techniques of Chapter 2: Draw a dotplot or construct a stem and leaf plot. As long as your plot tends to “mound up” in the center, you can be fairly safe in using the t statistic for making inferences. The random sampling requirement, on the other hand, is quite critical if you want to produce reliable inferences. If the sample is not random, or if it does not at least behave as a random sample, then your sample results may be affected by some unknown factor and your conclusions may be incorrect. When you design an experiment or read about experiments conducted by others, look critically at the way the data have been collected!

10.3

SMALL-SAMPLE INFERENCES CONCERNING A POPULATION MEAN As with large-sample inference, small-sample inference can involve either estimation or hypothesis testing, depending on the preference of the experimenter. We explained the basics of these two types of inference in the earlier chapters, and we use them again 苶), and a different sampling distrinow, with a different sample statistic, t ( 苶x m)/(s/兹n bution, the Student’s t, with (n 1) degrees of freedom.

392 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

SMALL-SAMPLE HYPOTHESIS TEST FOR m 1. Null hypothesis: H0 : m m0 2. Alternative hypothesis: One-Tailed Test

Two-Tailed Test

Ha : m m0 (or, Ha : m m0)

Ha : m m0

x苶 m0 3. Test statistic: t s/兹苶n 4. Rejection region: Reject H0 when One-Tailed Test

Two-Tailed Test

t ta (or t ta when the alternative hypothesis is Ha : m m0)

t ta/2

–t

t ta/2

or when p-value a

α 0

tα

α/2

α/2 α/2

α/2

The critical values of t, ta, and ta/2 are based on (n 1) degrees of freedom. These tabulated values can be found using Table 4 of Appendix I or the Student’s t Probabilities applet. Assumption: The sample is randomly selected from a normally distributed population. SMALL-SAMPLE (1 a)100% CONFIDENCE INTERVAL FOR m s 苶x ta/2 兹苶n where s/兹n苶 is the estimated standard error of x苶, often referred to as the standard error of the mean. EXAMPLE

10.3

A new process for producing synthetic diamonds can be operated at a proﬁtable level only if the average weight of the diamonds is greater than .5 karat. To evaluate the proﬁtability of the process, six diamonds are generated, with recorded weights .46, .61, .52, .48, .57, and .54 karat. Do the six measurements present sufficient evidence to indicate that the average weight of the diamonds produced by the process is in excess of .5 karat? The population of diamond weights produced by this new process has mean m, and you can set out the formal test of hypothesis in steps, as you did in Chapter 9:

Solution

10.3 SMALL-SAMPLE INFERENCES CONCERNING A POPULATION MEAN

1–2

393

Null and alternative hypotheses: H0: m .5

❍

versus Ha: m .5

Test statistic: You can use your calculator to verify that the mean and standard deviation for the six diamond weights are .53 and .0559, respectively. The test statistic is a t statistic, calculated as x苶 m0 .53 .5 t 1.32 s/兹n苶 .0559/兹苶6 As with the large-sample tests, the test statistic provides evidence for either rejecting or accepting H0 depending on how far from the center of the t distribution it lies.

Rejection region: If you choose a 5% level of signiﬁcance (a .05), the righttailed rejection region is found using the critical values of t from Table 4 of Appendix I. With df n 1 5, you can reject H0 if t t.05 2.015, as shown in Figure 10.6.

Conclusion: Since the calculated value of the test statistic, 1.32, does not fall in the rejection region, you cannot reject H0. The data do not present sufficient evidence to indicate that the mean diamond weight exceeds .5 karat.

FIGU R E 1 0 . 6

Rejection region for Example 10.3

● f(t)

.05 0 A 95% conﬁdence interval tells you that, if you were to construct many of these intervals (all of which would have slightly different endpoints), 95% of them would enclose the population mean.

1.32 2.015 Reject H0

As in Chapter 9, the conclusion to accept H0 would require the difficult calculation of b, the probability of a Type II error. To avoid this problem, we choose to not reject H0. We can then calculate the lower bound for m using a small-sample lower onesided confidence bound. This bound is similar to the large-sample one-sided confidence bound, except that the critical za is replaced by a critical ta from Table 4. For this example, a 95% lower one-sided confidence bound for m is: s x苶 ta 兹苶n .0559 .53 2.015 兹6苶 .53 .046 The 95% lower bound for m is m .484. The range of possible values includes mean diamond weights both smaller and greater than .5; this confirms the failure of our test to show that m exceeds .5.

394 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

Remember from Chapter 9 that there are two ways to conduct a test of hypothesis: •

•

The critical value approach: Set up a rejection region based on the critical values of the statistic’s sampling distribution. If the test statistic falls in the rejection region, you can reject H0. The p-value approach: Calculate the p-value based on the observed value of the test statistic. If the p-value is smaller than the significance level, a, you can reject H0. If there is no preset significance level, use the guidelines in Section 9.3 to judge the statistical significance of your sample results.

We used the ﬁrst approach in the solution to Example 10.3. We use the second approach to solve Example 10.4. EXAMPLE

10.4

Labels on 1-gallon cans of paint usually indicate the drying time and the area that can be covered in one coat. Most brands of paint indicate that, in one coat, a gallon will cover between 250 and 500 square feet, depending on the texture of the surface to be painted. One manufacturer, however, claims that a gallon of its paint will cover 400 square feet of surface area. To test this claim, a random sample of ten 1-gallon cans of white paint were used to paint 10 identical areas using the same kind of equipment. The actual areas (in square feet) covered by these 10 gallons of paint are given here: 310 376

311 303

412 410

368 365

447 350

Do the data present sufficient evidence to indicate that the average coverage differs from 400 square feet? Find the p-value for the test, and use it to evaluate the statistical signiﬁcance of the results. Solution Remember from Chapter 2 how to calculate 苶x and s using the data entry method on your calculator.

To test the claim, the hypotheses to be tested are

H0 : m 400

versus Ha : m 400

The sample mean and standard deviation for the recorded data are s 48.417 苶x 365.2 and the test statistic is x苶 m0 365.2 400 t 2.27 48.417/兹苶 10 s/兹n苶 The p-value for this test is the probability of observing a value of the t statistic as contradictory to the null hypothesis as the one observed for this set of data—namely, t 2.27. Since this is a two-tailed test, the p-value is the probability that either t 2.27 or t 2.27. Unlike the z-table, the table for t gives the values of t corresponding to upper-tail areas equal to .100, .050, .025, .010, and .005. Consequently, you can only approximate the upper-tail area that corresponds to the probability that t 2.27. Since the t statistic for this test is based on 9 df, we refer to the row corresponding to df 9 in Table 4. The ﬁve critical values for various tail areas are shown in Figure 10.7, an enlargement of the tail of the t distribution with 9 degrees of freedom. The value t 2.27 falls between t.025 2.262 and t.010 2.821. Therefore, the right-tail area corresponding to the probability that t 2.27 lies between .01 and .025. Since this area represents only half of the p-value, you can write 1 .01 (p-value) .025 2

.02 p-value .05

10.3 SMALL-SAMPLE INFERENCES CONCERNING A POPULATION MEAN

FIGU R E 1 0 . 7

Calculating the p-value for Example 10.4 (shaded area 12 p-value)

❍

395

● f(t)

.100 .050 .025 .010 .005 2.262 1.383 1.833

2.821 3.250 2.27

What does this tell you about the significance of the statistical results? For you to reject H0, the p-value must be less than the specified significance level, a. Hence, you could reject H0 at the 5% level, but not at the 2% or 1% level. Therefore, the p-value for this test would typically be reported by the experimenter as p-value .05

(or sometimes P .05)

For this test of hypothesis, H0 is rejected at the 5% signiﬁcance level. There is sufficient evidence to indicate that the average coverage differs from 400 square feet. Within what limits does this average coverage really fall? A 95% conﬁdence interval gives the upper and lower limits for m as s x苶 ta/2 兹苶n

冢冣

冢

48.417 365.2 2.262 10 兹苶

冣

365.2 34.63 Thus, you can estimate that the average area covered by 1 gallon of this brand of paint lies in the interval 330.6 to 399.8. A more precise interval estimate (a shorter interval) can generally be obtained by increasing the sample size. Notice that the upper limit of this interval is very close to the value of 400 square feet, the coverage claimed on the label. This coincides with the fact that the observed value of t 2.27 is just slightly less than the left-tail critical value of t.025 2.262, making the p-value just slightly less than .05. Most statistical computing packages contain programs that will implement the Student’s t-test or construct a confidence interval for m when the data are properly entered into the computer’s database. Most of these programs will calculate and report the exact p-value of the test, allowing you to quickly and accurately draw conclusions about the statistical significance of the results. The results of the MINITAB one-sample t-test and confidence interval procedures are given in Figure 10.8. Besides the observed value of t 2.27 and the confidence interval (330.6, 399.8), the output gives the sample mean, the sample standard deviation, the standard error of the mean (SE Mean s/兹n苶), and the exact p-value of the test (P .049). This is consistent with the range for the p-value that we found using Table 4 in Appendix I: .02 p-value .05

396 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

FI GU R E 1 0 . 8

MINITAB output for Example 10.4

● One-Sample T: Area Test of mu = 400 Variable Area Variable Area

N 10

not = 400

Mean 365.2

95% CI (330.6, 399.8)

StDev 48.4

SE Mean 15.3 T -2.27

P 0.049

You can use the Small Sample Test of a Population Mean applet to visualize the p-values for either one- or two-tailed tests of the population mean m. The procedure follows the same pattern as with previous applets. You enter the values of 苶x, n, and s and press “Enter” after each entry; the applet will calculate t and give you the option of choosing one- or two-tailed p-values (Area to Left, Area to Right, or Two Tails), as well as a Middle area that you will not need. FI GU R E 1 0 . 9

Small Sample Test of a Population Mean applet

●

For the data of Example 10.4, the p-value is the two-tailed area to the right of t 2.273 and to the left of t 2.273. Can you ﬁnd this same p-value in the MINITAB printout shown in Figure 10.9?

You can see the value of using the computer output or the Java applet to evaluate statistical results: • •

The exact p-value eliminates the need for tables and critical values. All of the numerical calculations are done for you.

The most important job—which is left for the experimenter—is to interpret the results in terms of their practical signiﬁcance!

10.3 SMALL-SAMPLE INFERENCES CONCERNING A POPULATION MEAN

10.3

APPLICATIONS

10.1 Find the following t-values in Table 4 of

Appendix I: a. t.05 for 5 df c. t.10 for 18 df

10.6 Tuna Fish Is there a difference in the

b. t.025 for 8 df d. t.025 for 30 df

rejection region in these situations: A two-tailed test with a .01 and 12 df A right-tailed test with a .05 and 16 df A two-tailed test with a .05 and 25 df A left-tailed test with a .01 and 7 df

10.3 Use Table 4 in Appendix I to approximate the

p-value for the t statistic in each situation: a. A two-tailed test with t 2.43 and 12 df b. A right-tailed test with t 3.21 and 16 df c. A two-tailed test with t 1.19 and 25 df d. A left-tailed test with t 8.77 and 7 df 10.4 Test Scores The test scores on a

100-point test were recorded for 20 students:

EX1004

93 86 89 68

91 82 67 65

86 76 62 75

75 57 72 84

a. Can you reasonably assume that these test scores have been selected from a normal population? Use a stem and leaf plot to justify your answer. b. Calculate the mean and standard deviation of the scores. c. If these students can be considered a random sample from the population of all students, ﬁnd a 95% conﬁdence interval for the average test score in the population. 10.5 The following n 10 observations are a sample

from a normal population: 7.4

7.1

6.5

prices of tuna, depending on the method of packaging? Consumer Reports gives the estimated average price for a 6-ounce can or a 7.06-ounce pouch of tuna, based on prices paid nationally in supermarkets.1 These prices are recorded for a variety of different brands of tuna.

EX1006

10.2 Find the critical value(s) of t that specify the

71 73 84 77

397

EXERCISES

BASIC TECHNIQUES

a. b. c. d.

❍

7.5

7.6

6.3

6.9

7.7

6.5

7.0

a. Find the mean and standard deviation of these data. b. Find a 99% upper one-sided conﬁdence bound for the population mean m. c. Test H0 : m 7.5 versus Ha : m 7.5. Use a .01. d. Do the results of part b support your conclusion in part c?

Light Tuna in Water

White Tuna in Oil

White Tuna in Water

Light Tuna in Oil

.99 1.92 1.23 .85 .65 .69 .60

1.27 1.22 1.19 1.22

1.49 1.29 1.27 1.35 1.29 1.00 1.27 1.28

2.56 1.92 1.30 1.79 1.23

.53 1.41 1.12 .63 .67 .60 .66

.62 .66 62 .65 .60 .67

Source: Case Study “Pricing of Tuna” Copyright 2001 by Consumers Union of U.S., Inc., Yonkers, NY 10703-1057, a nonproﬁt organization. Reprinted with permission from the June 2001 issue of Consumer Reports® for educational purposes only. No commercial use or reproduction permitted. www.ConsumerReports.org®.

Assume that the tuna brands included in this survey represent a random sample of all tuna brands available in the United States. a. Find a 95% confidence interval for the average price for light tuna in water. Interpret this interval. That is, what does the “95%” refer to? b. Find a 95% confidence interval for the average price for white tuna in oil. How does the width of this interval compare to the width of the interval in part a? Can you explain why? c. Find 95% confidence intervals for the other two samples (white tuna in water and light tuna in oil). Plot the four treatment means and their standard errors in a two-dimensional plot similar to Figure 8.5. What kind of broad comparisons can you make about the four treatments? (We will discuss the procedure for comparing more than two population means in Chapter 11.) 10.7 Dissolved O2 Content Industrial wastes and

sewage dumped into our rivers and streams absorb oxygen and thereby reduce the amount of dissolved oxygen available for ﬁsh and other forms of aquatic life. One state agency requires a minimum of 5 parts per million (ppm) of dissolved oxygen in order for the oxygen content to be sufficient to support aquatic life. Six water specimens taken from a river at a speciﬁc location during the low-water season (July) gave

398 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

readings of 4.9, 5.1, 4.9, 5.0, 5.0, and 4.7 ppm of dissolved oxygen. Do the data provide sufficient evidence to indicate that the dissolved oxygen content is less than 5 ppm? Test using a .05. 10.8 Lobsters In a study of the infestation of the

Thenus orientalis lobster by two types of barnacles, Octolasmis tridens and O. lowei, the carapace lengths (in millimeters) of 10 randomly selected lobsters caught in the seas near Singapore are measured:2 78

Find a 95% conﬁdence interval for the mean carapace length of the T. orientalis lobsters. 10.9 Smoking and Lung Capacity It is recognized that cigarette smoking has a deleterious effect on lung function. In a study of the effect of cigarette smoking on the carbon monoxide diffusing capacity (DL) of the lung, researchers found that current smokers had DL readings signiﬁcantly lower than those of either exsmokers or nonsmokers. The carbon monoxide diffusing capacities for a random sample of n 20 current smokers are listed here:

EX1009

103.768 92.295 100.615 102.754

88.602 61.675 88.017 108.579

73.003 90.677 71.210 73.154

123.086 84.023 82.115 106.755

91.052 76.014 89.222 90.479

a. Do these data indicate that the mean DL reading for current smokers is significantly lower than 100 DL, the average for nonsmokers? Use a .01. b. Find a 99% upper one-sided confidence bound for the mean DL reading for current smokers. Does this bound confirm your conclusions in part a? 10.10 Brett Favre In Exercise 2.36 (EX0236), the number of passes completed by Brett Favre, quarterback for the Green Bay Packers, was recorded for each of the 16 regular season games in the fall of 2006 (ESPN.com):3

EX1010

15 17 22

31 28 20

25 24 26

22 5 21

22 22

19 24

a. A stem and leaf plot of the n 16 observations is shown below: Stem-and-Leaf Display: Favre Stem-and-leaf of Favre Leaf Unit = 1.0 LO 5 2 1 5 3 1 7 4 1 9 6 2 01 (4) 2 2222 6 2 445 3 2 6 2 2 8 1 3 1

= 16

Based on this plot, is it reasonable to assume that the underlying population is approximately normal, as required for the one-sample t-test? Explain. b. Calculate the mean and standard deviation for Brett Favre’s per game pass completions. c. Construct a 95% conﬁdence interval to estimate the per game pass completions per game for Brett Favre. 10.11 Purifying Organic Compound Organic

chemists often purify organic compounds by a method known as fractional crystallization. An experimenter wanted to prepare and purify 4.85 grams (g) of aniline. Ten 4.85-g quantities of aniline were individually prepared and puriﬁed to acetanilide. The following dry yields were recorded: 3.85 3.36

3.80 3.62

3.88 4.01

3.85 3.72

3.90 3.82

Estimate the mean grams of acetanilide that can be recovered from an initial amount of 4.85 g of aniline. Use a 95% conﬁdence interval. 10.12 Organic Compounds, continued Refer to

Exercise 10.11. Approximately how many 4.85-g specimens of aniline are required if you wish to estimate the mean number of grams of acetanilide correct to within .06 g with probability equal to .95? 10.13 Bulimia Although there are many treatments

for bulimia nervosa, some subjects fail to beneﬁt from treatment. In a study to determine which factors predict who will beneﬁt from treatment, an article in the British Journal of Clinical Psychology indicates that self-esteem was one of these important predictors.4 The table gives the mean and standard deviation of self-esteem scores prior to treatment, at posttreatment, and during a follow-up: Pretreatment Posttreatment Follow-up Sample Mean x苶 Standard Deviation s Sample Size n

20.3 5.0 21

26.6 7.4 21

27.7 8.2 20

a. Use a test of hypothesis to determine whether there is sufficient evidence to conclude that the true pretreatment mean is less than 25. b. Construct a 95% conﬁdence interval for the true posttreatment mean. c. In Section 10.4, we will introduce small-sample techniques for making inferences about the difference between two population means. Without the formality of a statistical test, what are you willing to conclude about the differences among the three sampled population means represented by the results in the table?

10.4 SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS

10.14 RBC Counts Here are the red blood 6

cell counts (in 10 cells per microliter) of a healthy person measured on each of 15 days:

EX1014

5.4 5.3 5.3

5.2 5.4 4.9

5.0 5.2 5.4

5.2 5.1 5.2

5.5 5.3 5.2

10.15 Hamburger Meat These data are the weights (in pounds) of 27 packages of ground beef in a supermarket meat display:

EX1015

.99 1.14 .89 .98

.97 1.38 .96 1.14

1.18 .75 1.12 .92

1.41 .96 1.12 1.18

1.28 1.08 .93 1.17

.83 .87 1.24

a. Interpret the accompanying MINITAB printouts for the one-sample test and estimation procedures. MINITAB output for Exercise 10.15

One-Sample T: Weight Test of mu = 1

not = 1

Variable Weight

N 27

Variable Weight

95% CI (0.9867, 1.1178)

10.4

399

b. Verify the calculated values of t and the upper and lower conﬁdence limits. 10.16 Cholesterol The serum cholesterol

levels of 50 subjects randomly selected from the L.A. Heart Data, data from an epidemiological heart disease study on Los Angeles County employees,5 follow.

EX1016

Find a 95% conﬁdence interval estimate of m, the true mean red blood cell count for this person during the period of testing.

1.08 1.06 .89 .89

❍

Mean 1.0522

StDev 0.1657 T 1.64

SE Mean 0.0319 P 0.113

148 303 262 278 305

304 315 284 227 225

300 174 275 220 306

240 209 229 260 184

368 253 261 221 242

139 169 239 247 282

203 170 254 178 311

249 254 222 204 271

265 212 273 250 276

229 255 299 256 248

a. Construct a histogram for the data. Are the data approximately mound-shaped? b. Use a t-distribution to construct a 95% conﬁdence interval for the average serum cholesterol levels for L.A. County employees. 10.17 Cholesterol, continued Refer to Exercise 10.16. Since n 30, use the methods of Chapter 8 to create a large-sample 95% conﬁdence interval for the average serum cholesterol level for L.A. County employees. Compare the two intervals. (HINT: The two intervals should be quite similar. This is the reason we choose to approximate the sample distribution of x苶 m with a z-distribution when n 30.) s/兹n苶

SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS: INDEPENDENT RANDOM SAMPLES The physical setting for the problem considered in this section is the same as the one in Section 8.6, except that the sample sizes are no longer large. Independent random samples of n1 and n2 measurements are drawn from two populations, with means and variances m1, s 12, m 2, and s 22, and your objective is to make inferences about (m1 m 2 ), the difference between the two population means. When the sample sizes are small, you can no longer rely on the Central Limit Theorem to ensure that the sample means will be normal. If the original populations are normal, however, then the sampling distribution of the difference in the sample means, ( 苶x1 x苶2), will be normal (even for small samples) with mean (m1 m 2 ) and standard error

冪莦莦

s2 s2 1 2 n1 n2

400 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

Assumptions for the two-sample (independent) t-test: • • •

Random independent samples Normal distributions s1 s2

In Chapters 7 and 8, you used the sample variances, s12 and s 22, to calculate an estimate of the standard error, which was then used to form a large-sample conﬁdence interval or a test of hypothesis based on the large-sample z statistic: ( 苶x1 x苶2) (m1 m2) z ⬇ s2 s2 1 2 n1 n2

冪莦莦

Unfortunately, when the sample sizes are small, this statistic does not have an approximately normal distribution—nor does it have a Student’s t distribution. In order to form a statistic with a sampling distribution that can be derived theoretically, you must make one more assumption. Suppose that the variability of the measurements in the two normal populations is the same and can be measured by a common variance s 2. That is, both populations have exactly the same shape, and s 12 s 22 s 2. Then the standard error of the difference in the two sample means is 冣冢n莦冪莦莦冪s莦 n s2 s2 1 2 n1 n2

It can be proven mathematically that, if you use the appropriate sample estimate s2 for the population variance s 2, then the resulting test statistic, ( 苶x1 x苶2) (m1 m2 ) t 1 1 s2 n1 n2

冢莦冣冪莦

has a Student’s t distribution. The only remaining problem is to find the sample estimate s2 and the appropriate number of degrees of freedom for the t statistic. Remember that the population variance s 2 describes the shape of the normal distributions from which your samples come, so that either s12 or s 22 would give you an estimate of s 2. But why use just one when information is provided by both? A better procedure is to combine the information in both sample variances using a weighted average, in which the weights are determined by the relative amount of information (the number of measurements) in each sample. For example, if the first sample contained twice as many measurements as the second, you might consider giving the first sample variance twice as much weight. To achieve this result, use this formula: (n1 1)s12 (n2 1)s 22 s 2 n1 n 2 2 Remember from Section 10.3 that the degrees of freedom for the one-sample t statistic are (n 1), the denominator of the sample estimate s 2. Since s 21 has (n1 1) df and s 22 has (n2 1) df, the total number of degrees of freedom is the sum (n1 1) (n2 1) n1 n2 2 shown in the denominator of the formula for s 2.

10.4 SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS

❍

401

CALCULATION OF s 2 •

If you have a scientiﬁc calculator, calculate each of the two sample standard deviations s1 and s2 separately, using the data entry procedure for your particular calculator. These values are squared and used in this formula: (n1 1)s12 (n2 1)s 22 s 2 n1 n 2 2

It can be shown that s 2 is an unbiased estimator of the common population variance s 2. If s 2 is used to estimate s 2 and if the samples have been randomly and independently drawn from normal populations with a common variance, then the statistic For the two-sample (independent) t-test, df n1 n2 2

( 苶x1 x苶2) (m1 m2 ) t 1 1 s2 n1 n2

冢莦冣冪莦

has a Student’s t distribution with (n1 n2 2) degrees of freedom. The small-sample estimation and test procedures for the difference between two means are given next. TEST OF HYPOTHESIS CONCERNING THE DIFFERENCE BETWEEN TWO MEANS: INDEPENDENT RANDOM SAMPLES 1. Null hypothesis: H0 : (m1 m2) D0, where D0 is some speciﬁed difference that you wish to test. For many tests, you will hypothesize that there is no difference between m1 and m2; that is, D0 0. 2. Alternative hypothesis: One-Tailed Test

Two-Tailed Test

Ha : (m1 m2) D0 [or Ha : (m1 m2) D0]

Ha : (m1 m2 ) D0

( x苶1 x苶2) D0 3. Test statistic: t 1 1 s2 n1 n2

冢莦冣冪莦

where

(n1 1)s12 (n2 1)s22 s 2 n1 n2 2 4. Rejection region: Reject H0 when One-Tailed Test

Two-Tailed Test

t ta [or t ta when the alternative hypothesis is Ha : (m1 m2) D0]

t ta/2

or when p-value a

t ta/2

(continued)

402 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

TEST OF HYPOTHESIS CONCERNING THE DIFFERENCE BETWEEN TWO MEANS: INDEPENDENT RANDOM SAMPLES (continued) The critical values of t, ta, and ta/2 are based on (n1 n2 2) df. The tabulated values can be found using Table 4 of Appendix I or the Student’s t Probabilities applet. Assumptions: The samples are randomly and independently selected from normally distributed populations. The variances of the populations s 12 and s 22 are equal.

SMALL-SAMPLE (1 a)100% CONFIDENCE INTERVAL FOR (m1 m2) BASED ON INDEPENDENT RANDOM SAMPLES (x苶1 苶x2 ) ta/2

冢n 莦冪s莦 n 冣 2

where s 2 is the pooled estimate of s 2. EXAMPLE

TABLE 10.2

10.5

A course can be taken for credit either by attending lecture sessions at fixed times and days, or by doing online sessions that can be done at the student’s own pace and at those times the student chooses. The course coordinator wants to determine if these two ways of taking the course resulted in a significant difference in achievement as measured by the final exam for the course. The following data gives the scores on an examination with 45 possible points for one group of n1 9 students who took the course online, and a second group of n2 9 students who took the course with conventional lectures. Do these data present sufficient evidence to indicate that the grades for students who took the course online are significantly higher than those who attended a conventional class? ●

Test Scores for Online and Classroom Presentations Online

Classroom

32 37 35 28 41 44 35 31 34

35 31 29 25 34 40 27 32 31

Let m1 and m2 be the mean scores for the online group and the classroom group, respectively. Then, since you seek evidence to support the theory that m1 m2, you can test the null hypothesis

Solution

H0 : m1 m2

[or H0 : (m1 m2) 0]

10.4 SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS

❍

403

versus the alternative hypothesis Ha : m1 m2

[or Ha : (m1 m2) 0]

To conduct the t-test for these two independent samples, you must assume that the sampled populations are both normal and have the same variance s 2. Is this reasonable? Stem and leaf plots of the data in Figure 10.10 show at least a “mounding” pattern, so that the assumption of normality is not unreasonable. FIGU R E 1 0 . 1 0

Stem and leaf plots for Example 10.5

● Online 2 3 3 4

8 124 557 14

Classroom 2 3 3 4

579 1124 5 0

Furthermore, the standard deviations of the two samples, calculated as s1 4.9441 Stem and leaf plots can help you decide if the normality assumption is reasonable.

and

s2 4.4752

are not different enough for us to doubt that the two distributions may have the same shape. If you make these two assumptions and calculate (using full accuracy) the pooled estimate of the common variance as (n1 1)s12 (n2 1)s 22 8(4.9441)2 8(4.4752)2 s 2 22.2361 n1 n2 2 992 you can then calculate the test statistic, x苶1 x苶2 35.22 31.56 t 1.65 1 1 1 1 s 2 22.2361 n1 n2 9 9

冢莦冣冪莦莦冪莦冢莦冣

If you are using a calculator, don’t round off until the ﬁnal step!

FIGU R E 1 0 . 1 1

Rejection region for Example 10.5

The alternative hypothesis Ha : m1 m2 or, equivalently, Ha : (m1 m2) 0 implies that you should use a one-tailed test in the upper tail of the t distribution with (n1 n2 2) 16 degrees of freedom. You can ﬁnd the appropriate critical value for a rejection region with a .05 in Table 4 of Appendix I, and H0 will be rejected if t 1.746. Comparing the observed value of the test statistic t 1.65 with the critical value t.05 1.746, you cannot reject the null hypothesis (see Figure 10.11). There is insufficient evidence to indicate that the online course grades are higher than the conventional course grades at the 5% level of signiﬁcance. ● f(t)

α = .05

1.746 Reject H0

404 ❍

EXAMPLE

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

10.6

Find the p-value that would be reported for the statistical test in Example 10.5. Solution

The observed value of t for this one-tailed test is t 1.65. Therefore,

p-value P(t 1.65) for a t statistic with 16 degrees of freedom. Remember that you cannot obtain this probability directly from Table 4 in Appendix I; you can only bound the p-value using the critical values in the table. Since the observed value, t 1.65, lies between t.100 1.337 and t.050 1.746, the tail area to the right of 1.65 is between .05 and .10. The p-value for this test would be reported as .05 p-value .10 Because the p-value is greater than .05, most researchers would report the results as not signiﬁcant.

You can use the Two-Sample t-Test: Independent Samples applet, shown in Figure 10.12, to visualize the p-values for either one- or two-tailed tests of the difference between two population means. The procedure follows the same pattern as with previous applets. You need to enter summary statistics—the values of x苶1, x苶2, n1, n2, s1, and s2 and press “Enter” after each entry; the applet will calculate t (assuming equal variances) and give you the option of choosing one- or two-tailed p-values, (Area to Left, Area to Right, or Two Tails), as well as a Middle area that you will not need. F IGU R E 1 0 . 1 2

Two-Sample t-Test: Independent Samples applet

●

For the data of Example 10.5, the p-value is the one-tailed area to the right of t 1.65. Does the p-value conﬁrm the conclusions for the test in Example 10.5?

10.4 SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS

EXAMPLE

10.7

❍

405

Use a lower 95% confidence bound to estimate the difference (m1 m2) in Example 10.5. Does the lower confidence bound indicate that the online average is significantly higher than the classroom average? The lower confidence bound takes a familiar form—the point estimator ( 苶x1 x苶2) minus an amount equal to ta times the standard error of the estimator. Substituting into the formula, you can calculate the 95% lower confidence bound:

Solution

冢莦冣冪莦 1 1 (35.22 31.56) 1.746 22.2361 冢9 莦冪莦莦 9冣 (x苶1 苶x2) ta

1 1 s 2 n1 n2

3.66 3.88 or (m1 m2) .22. Since the value (m1 m2) 0 is included in the conﬁdence interval, it is possible that the two means are equal. There is insufficient evidence to indicate that the online average is higher than the classroom average. The two-sample procedure that uses a pooled estimate of the common variance s 2 relies on four important assumptions: • Larger s 2/Smaller s 2 3 ⇔ variance assumption is reasonable

•

The samples must be randomly selected. Samples not randomly selected may introduce bias into the experiment and thus alter the signiﬁcance levels you are reporting. The samples must be independent. If not, this is not the appropriate statistical procedure. We discuss another procedure for dependent samples in Section 10.5. The populations from which you sample must be normal. However, moderate departures from normality do not seriously affect the distribution of the test statistic, especially if the sample sizes are nearly the same. The population variances should be equal or nearly equal to ensure that the procedures are valid.

If the population variances are far from equal, there is an alternative procedure for estimation and testing that has an approximate t distribution in repeated sampling. As a rule of thumb, you should use this procedure if the ratio of the two sample variances, Larger s 2 2 3 Smaller s Since the population variances are not equal, the pooled estimator s2 is no longer appropriate, and each population variance must be estimated by its corresponding sample variance. The resulting test statistic is ( 苶x1 x苶2 ) D0 s2 s2 1 2 n1 n2

冪莦莦

406

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

When the sample sizes are small, critical values for this statistic are found using degrees of freedom approximated by the formula s2 s2 2 1 2 n1 n2 df ⬇ 2 (s1/n1)2 (s22/n2)2 (n1 1) (n2 1)

冢

冣

The degrees of freedom are taken to be the integer part of this result. Computer packages such as MINITAB can be used to implement this procedure, sometimes called Satterthwaite’s approximation, as well as the pooled method described earlier. In fact, some experimenters choose to analyze their data using both methods. As long as both analyses lead to the same conclusions, you need not concern yourself with the equality or inequality of variances. The MINITAB output resulting from the pooled method of analysis for the data of Example 10.5 is shown in Figure 10.13. Notice that the ratio of the two sample variances, (4.94/4.48)2 1.22, is less than 3, which makes the pooled method appropriate. The calculated value of t 1.65 and the exact p-value .059 with 16 degrees of freedom are shown in the last line of the output. The exact p-value makes it quite easy for you to determine the significance or nonsignificance of the sample results. You will find instructions for generating this MINITAB output in the section “My MINITAB” at the end of this chapter. FI GU R E 1 0 . 1 3

MINITAB output for Example 10.5

● Two-Sample T-Test and CI: Online, Classroom Two-sample T for Online vs Classroom N Mean StDev SE Mean Online 9 35.22 4.94 1.6 Classroom 9 31.56 4.48 1.5 Difference = mu (Online) - mu (Classroom) Estimate for difference: 3.67 95% lower bound for difference: -0.21 T-Test of difference = 0 (vs >): T-Value = 1.65 P-Value = 0.059 DF = 16 Both use Pooled StDev = 4.7155

If there is reason to believe that the normality assumptions have been violated, you can test for a shift in location of two population distributions using the nonparametric Wilcoxon rank sum test of Chapter 15. This test procedure, which requires fewer assumptions concerning the nature of the population probability distributions, is almost as sensitive in detecting a difference in population means when the conditions necessary for the t-test are satisﬁed. It may be more sensitive when the normality assumption is not satisﬁed.

10.4

EXERCISES

BASIC TECHNIQUES 10.18 Give the number of degrees of freedom for s2,

10.19 Calculate s 2, the pooled estimator for s 2, in

the pooled estimator of s , in these cases:

these cases:

a. n1 16, n2 8 b. n1 10, n2 12 c. n1 15, n2 3

a. n1 10, n2 4, s12 3.4, s22 4.9 b. n1 12, n2 21, s12 18, s22 23

10.4 SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS

10.20 Two independent random samples of sizes n1 4 and n2 5 are selected from each of two normal populations: Population 1

Population 2

a. Calculate s , the pooled estimator of s 2. b. Find a 90% conﬁdence interval for (m1 m2), the difference between the two population means. c. Test H0 : (m1 m2) 0 against Ha : (m1 m2) 0 for a .05. State your conclusions. 2

❍

407

b. What is the observed value of the test statistic? What is the p-value associated with this test? c. What is the pooled estimate s 2 of the population variance? d. Use the answers to part b to draw conclusions about the difference in the two population means. e. Find the 95% conﬁdence interval for the difference in the population means. Does this interval conﬁrm your conclusions in part d?

10.21 Independent random samples of n1 16 and

n2 13 observations were selected from two normal populations with equal variances: Population

Sample Size Sample Mean Sample Variance

16 34.6 4.8

13 32.2 5.9

a. Suppose you wish to detect a difference between the population means. State the null and alternative hypotheses for the test. b. Find the rejection region for the test in part a for a .01. c. Find the value of the test statistic. d. Find the approximate p-value for the test. e. Conduct the test and state your conclusions. 10.22 Refer to Exercise 10.21. Find a 99% conﬁdence interval for (m1 m2). 10.23 The MINITAB printout shows a test for the dif-

ference in two population means. MINITAB output for Exercise 10.23

Two-Sample T-Test and CI: Sample 1, Sample 2 Two-sample T for Sample 1 vs Sample 2 N Mean StDev SE Mean Sample 1 6 29.00 4.00 1.6 Sample 2 7 28.86 4.67 1.8 Difference = mu (Sample 1) - mu (Sample 2) Estimate for difference: 0.14 95% CI for difference: (-5.2, 5.5) T-Test of difference = 0 (vs not =): T-Value = 0.06 P-Value = 0.95 DF = 11 Both use Pooled StDev = 4.38

a. Do the two sample standard deviations indicate that the assumption of a common population variance is reasonable?

APPLICATIONS 10.24 Healthy Teeth Jan Lindhe conducted a study

on the effect of an oral antiplaque rinse on plaque buildup on teeth.6 Fourteen people whose teeth were thoroughly cleaned and polished were randomly assigned to two groups of seven subjects each. Both groups were assigned to use oral rinses (no brushing) for a 2-week period. Group 1 used a rinse that contained an antiplaque agent. Group 2, the control group, received a similar rinse except that, unknown to the subjects, the rinse contained no antiplaque agent. A plaque index x, a measure of plaque buildup, was recorded at 4, 7, and 14 days. The mean and standard deviation for the 14-day plaque measurements are shown in the table for the two groups.

Sample Size Mean Standard Deviation

Control Group

Antiplaque Group

7 1.26 .32

7 .78 .32

a. State the null and alternative hypotheses that should be used to test the effectiveness of the antiplaque oral rinse. b. Do the data provide sufficient evidence to indicate that the oral antiplaque rinse is effective? Test using a .05. c. Find the approximate p-value for the test.

10.25 Tuna, again In Exercise 10.6 we presented data on the estimated average price for a 6-ounce can or a 7.06-ounce pouch of tuna, based on prices paid nationally in supermarkets. A portion of the data is reproduced in the table below. Use the MINITAB printout to answer the questions.

EX1025

408 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

Light Tuna in Water

Light Tuna in Oil

.99 1.92 1.23 .85 .65 .69 .60

2.56 1.92 1.30 1.79 1.23

.53 1.41 1.12 .63 .67 .60 .66

b. Construct a 95% conﬁdence interval estimate of the difference in means for runners and cyclists under the condition of exercising at 80% of maximal oxygen consumption. c. To test for a signiﬁcant difference in compartment pressure at maximal oxygen consumption, should you use the pooled or unpooled t-test? Explain.

.62 .66 .62 .65 .60 .67

10.27 Disinfectants An experiment published in MINITAB output for Exercise 10.25

Two-Sample T-Test and CI: Water, Oil Two-sample T for Water vs Oil N Mean StDev Water 14 0.896 0.400 Oil 11 1.147 0.679

SE Mean 0.11 0.20

Difference = mu (Water) - mu (Oil) Estimate for difference: -0.251 95% CI for difference: (-0.700, 0.198) T-Test of difference = 0 (vs not =): T-Value = -1.16 P-Value = 0.260 DF = 23 Both use Pooled StDev = 0.5389

a. Do the data in the table present sufficient evidence to indicate a difference in the average prices of light tuna in water versus oil? Test using a .05. b. What is the p-value for the test? c. The MINITAB analysis uses the pooled estimate of s 2. Is the assumption of equal variances reasonable? Why or why not? 10.26 Runners and Cyclists Chronic anterior com-

partment syndrome is a condition characterized by exercise-induced pain in the lower leg. Swelling and impaired nerve and muscle function also accompany this pain, which is relieved by rest. Susan Beckham and colleagues conducted an experiment involving ten healthy runners and ten healthy cyclists to determine whether there are signiﬁcant differences in pressure measurements within the anterior muscle compartment for runners and cyclists.7 The data summary—compartment pressure in millimeters of mercury (Hg)—is as follows: Runners Condition Rest 80% maximal O2 consumption Maximal O2 consumption

Mean

Standard Deviation

Cyclists Mean

Standard Deviation

14.5

3.92

11.1

3.98

12.2 19.1

3.49 16.9

11.5 12.2

4.95 4.47

a. Test for a signiﬁcant difference in compartment pressure between runners and cyclists under the resting condition. Use a .05.

The American Biology Teacher studied the efficacy of using 95% ethanol or 20% bleach as a disinfectant in removing bacterial and fungal contamination when culturing plant tissues. The experiment was repeated 15 times with each disinfectant, using eggplant as the plant tissue being cultured.8 Five cuttings per plant were placed on a petri dish for each disinfectant and stored at 25°C for 4 weeks. The observation reported was the number of uncontaminated eggplant cuttings after the 4-week storage. Disinfectant Mean Variance n

95% Ethanol

20% Bleach

3.73 2.78095 15 Pooled variance 1.47619

4.80 .17143 15

a. Are you willing to assume that the underlying variances are equal? b. Using the information from part a, are you willing to conclude that there is a signiﬁcant difference in the mean numbers of uncontaminated eggplants for the two disinfectants tested? 10.28 Titanium A geologist collected 20 different ore samples, all of the same weight, and randomly divided them into two groups. The titanium contents of the samples, found using two different methods, are listed in the table:

EX1028

Method 1 .011 .013

.013 .010

.013 .013

Method 2 .015 .011

.014 .012

.011 .012

.016 .017

.013 .013

.012 .014

.015 .015

a. Use an appropriate method to test for a signiﬁcant difference in the average titanium contents using the two different methods. b. Determine a 95% conﬁdence interval estimate for (m1 m2). Does your interval estimate substantiate your conclusion in part a? Explain. 10.29 Raisins The numbers of raisins in

each of 14 miniboxes (1/2-ounce size) were counted for a generic brand and for Sunmaid® brand raisins:

EX1029

10.4 SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO POPULATION MEANS

Generic Brand 25 26 26 26

26 28 27 26

25 28 24

Sunmaid 28 27 25

25 28 25 28

29 24 28 24

24 28 30

24 22 27

a. Although counts cannot have a normal distribution, do these data have approximately normal distributions? (HINT: Use a histogram or stem and leaf plot.) b. Are you willing to assume that the underlying population variances are equal? Why? c. Use the p-value approach to determine whether there is a significant difference in the mean numbers of raisins per minibox. What are the implications of your conclusion? 10.30 Dissolved O2 Content, continued Refer to Exercise 10.7, in which we measured the dissolved oxygen content in river water to determine whether a stream had sufficient oxygen to support aquatic life. A pollution control inspector suspected that a river community was releasing amounts of semitreated sewage into a river. To check his theory, he drew five randomly selected specimens of river water at a location above the town, and another five below. The dissolved oxygen readings (in parts per million) are as follows: Above Town

4.8

5.2

5.0

4.9

5.1

Below Town

5.0

4.7

4.9

4.8

4.9

a. Do the data provide sufficient evidence to indicate that the mean oxygen content below the town is less than the mean oxygen content above? Test using a .05. b. Suppose you prefer estimation as a method of inference. Estimate the difference in the mean dissolved oxygen contents for locations above and below the town. Use a 95% conﬁdence interval. 10.31 Freestyle Swimmers In an effort to

compare the average swimming times for two swimmers, each swimmer was asked to swim freestyle for a distance of 100 yards at randomly selected times. The swimmers were thoroughly rested between laps and did not race against each other, so that each sample of times was an independent random sample. The times for each of 10 trials are shown for the two swimmers.

EX1031

Swimmer 1

Swimmer 2

59.62 59.48 59.65 59.50 60.01

59.81 59.32 59.76 59.64 59.86

59.74 59.43 59.72 59.63 59.68

59.41 59.63 59.50 59.83 59.51

❍

409

Suppose that swimmer 2 was last year’s winner when the two swimmers raced. Does it appear that the average time for swimmer 2 is still faster than the average time for swimmer 1 in the 100-yard freestyle? Find the approximate p-value for the test and interpret the results. 10.32 Freestyle Swimmers, continued Refer to Exercise 10.31. Construct a lower 95% one-sided conﬁdence bound for the difference in the average times for the two swimmers. Does this interval conﬁrm your conclusions in Exercise 10.31? 10.33 Comparing NFL Quarterbacks

How does Brett Favre, quarterback for the Green Bay Packers, compare to Peyton Manning, quarterback for the Indianapolis Colts? The table below shows the number of completed passes for each athlete during the 2006 NFL football season:3

EX1033

Brett Favre 15 31 25 22 22 19

17 28 24 5 22 24

Peyton Manning 22 20 26 21

25 26 14 21 20 25

32 30 27 20 14 21

25 29 21 22

a. Does the data indicate that there is a difference in the average number of completed passes for the two quarterbacks? Test using a .05. b. Construct a 95% confidence interval for the difference in the average number of completed passes for the two quarterbacks. Does the confidence interval confirm your conclusion in part a? Explain. 10.34 An Archeological Find An article in Archaeometry involved an analysis of 26 samEX1034 ples of Romano-British pottery, found at four different kiln sites in the United Kingdom.9 The samples were analyzed to determine their chemical composition and the percentage of aluminum oxide in each of 10 samples at two sites is shown below. Island Thorns 18.3 15.8 18.0 18.0 20.8

Ashley Rails 17.7 18.3 16.7 14.8 19.1

Does the data provide sufficient information to indicate that there is a difference in the average percentage of aluminum oxide at the two sites? Test at the 5% level of signiﬁcance.

410

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO MEANS: A PAIRED-DIFFERENCE TEST

10.5

To compare the wearing qualities of two types of automobile tires, A and B, a tire of type A and one of type B are randomly assigned and mounted on the rear wheels of each of ﬁve automobiles. The automobiles are then operated for a speciﬁed number of miles, and the amount of wear is recorded for each tire. These measurements appear in Table 10.3. Do the data present sufficient evidence to indicate a difference in the average wear for the two tire types? TABLE 10.3

●

Average Wear for Two Types of Tires Automobile

Tire A

Tire B

1 2 3 4 5

10.6 9.8 12.3 9.7 8.8

10.2 9.4 11.8 9.1 8.3

x苶1 10.24 s1 1.316

x苶2 9.76 s2 1.328

Table 10.3 shows a difference of (x苶1 x苶2 ) (10.24 9.76) .48 between the two sample means, while the standard deviations of both samples are approximately 1.3. Given the variability of the data and the small number of measurements, this is a rather small difference, and you would probably not suspect a difference in the average wear for the two types of tires. Let’s check your suspicions using the methods of Section 10.4. Look at the MINITAB analysis in Figure 10.14. The two-sample pooled t-test is used for testing the difference in the means based on two independent random samples. The calculated value of t used to test the null hypothesis H0 : m1 m2 is t .57 with p-value .582, a value that is not nearly small enough to indicate a significant difference in the two population means. The corresponding 95% confidence interval, given as 1.448 (m1 m2) 2.408 is quite wide and also does not indicate a significant difference in the population means. FI GU R E 1 0 . 1 4

MINITAB output using t-test for independent samples for the tire data

● Two-Sample T-Test and CI: Tire A, Tire B Two-sample T for Tire A vs Tire B N Mean StDev SE Mean Tire A 5 10.24 1.32 0.59 Tire B 5 9.76 1.33 0.59 Difference = mu (Tire A) - mu (Tire B) Estimate for difference: 0.480 95% CI for difference: (-1.448, 2.408) T-Test of difference = 0 (vs not =): T-Value = 0.57 Both use Pooled StDev = 1.3221

P-Value = 0.582 DF = 8

10.5 SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO MEANS: A PAIRED-DIFFERENCE TEST

❍

411

Take a second look at the data and you will notice that the wear measurement for type A is greater than the corresponding value for type B for each of the five automobiles. Wouldn’t this be unlikely, if there’s really no difference between the two tire types? Consider a simple intuitive test, based on the binomial distribution of Chapter 5. If there is no difference in the mean tire wear for the two types of tires, then it is just as likely as not that tire A shows more wear than tire B. The five automobiles then correspond to five binomial trials with p P(tire A shows more wear than tire B) .5. Is the observed value of x 5 positive differences shown in Table 10.4 unusual? The probability of observing x 5 or the equally unlikely value x 0 can be found in Table 1 in Appendix I to be 2(.031) .062, which is quite small compared to the likelihood of the more powerful t-test, which had a p-value of .58. Isn’t it peculiar that the t-test, which uses more information (the actual sample measurements) than the binomial test, fails to supply sufficient information for rejecting the null hypothesis? TABLE 10.4

●

Differences in Tire Wear, Using the Data of Table 10.3 Automobile

dAB

1 2 3 4 5

10.6 9.8 12.3 9.7 8.8

10.2 9.4 11.8 9.1 8.3

.4 .4 .5 .6 .5 d苵 .48

There is an explanation for this inconsistency. The t-test described in Section 10.4 is not the proper statistical test to be used for our example. The statistical test procedure of Section 10.4 requires that the two samples be independent and random. Certainly, the independence requirement is violated by the manner in which the experiment was conducted. The (pair of) measurements, an A and a B tire, for a particular automobile are definitely related. A glance at the data shows that the readings have approximately the same magnitude for a particular automobile but vary markedly from one automobile to another. This, of course, is exactly what you might expect. Tire wear is largely determined by driver habits, the balance of the wheels, and the road surface. Since each automobile has a different driver, you would expect a large amount of variability in the data from one automobile to another. In designing the tire wear experiment, the experimenter realized that the measurements would vary greatly from automobile to automobile. If the tires (five of type A and five of type B) were randomly assigned to the ten wheels, resulting in independent random samples, this variability would result in a large standard error and make it difficult to detect a difference in the means. Instead, he chose to “pair” the measurements, comparing the wear for type A and type B tires on each of the five automobiles. This experimental design, sometimes called a paired-difference or matched pairs design, allows us to eliminate the car-to-car variability by looking at only the five difference measurements shown in Table 10.4. These five differences form a single random sample of size n 5. Notice that in Table 10.4 the sample mean of the differences, d A B, is calculated as Sd d苶 i .48 n

412

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

and is exactly the same as the difference of the sample means: (x苶1 苶x2) (10.24 9.76) .48. It should not surprise you that this can be proven to be true in general, and also that the same relationship holds for the population means. That is, the average of the population differences is md (m1 m2) Because of this fact, you can use the sample differences to test for a signiﬁcant difference in the two population means, (m1 m2) md. The test is a single-sample t-test of the difference measurements to test the null hypothesis H0 : md 0

[or H0 : (m1 m2) 0]

versus the alternative hypothesis Ha : md 0

[or Ha : (m1 m2) 0]

The test procedures take the same form as the procedures used in Section 10.3 and are described next.

PAIRED-DIFFERENCE TEST OF HYPOTHESIS FOR (m1 m2) md: DEPENDENT SAMPLES 1. Null hypothesis: H0 : md 0 2. Alternative hypothesis: One-Tailed Test

Two-Tailed Test

Ha : md 0 (or Ha : md 0)

Ha : md 0

苶d 0 苶d 3. Test statistic: t 苶 sd /兹苶n sd/兹n where n Number of paired differences d苶 Mean of the sample differences sd Standard deviation of the sample differences 4. Rejection region: Reject H0 when One-Tailed Test

Two-Tailed Test

t ta t ta/2 (or t ta when the alternative hypothesis is Ha : md 0)

t ta/2

or when p-value a The critical values of t, ta, and ta/2 are based on (n 1) df. These tabulated values can be found using Table 4 or the Student’s t Probabilities applet.

10.5 SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO MEANS: A PAIRED-DIFFERENCE TEST

❍

413

(1 a)100% SMALL-SAMPLE CONFIDENCE INTERVAL FOR (m1 m2) md, BASED ON A PAIRED-DIFFERENCE EXPERIMENT sd d苶 ta/2 兹n苶

冢冣

Assumptions: The experiment is designed as a paired-difference test so that the n differences represent a random sample from a normal population. EXAMPLE

10.8

Do the data in Table 10.3 provide sufficient evidence to indicate a difference in the mean wear for tire types A and B? Test using a .05. You can verify using your calculator that the average and standard deviation of the ﬁve difference measurements are

Solution

苶d .48

and

sd .0837

Then H0 : md 0

Ha : md 0

and

and 苶d 0 .48 t 12.8 苶 sd /兹苶n .0837/兹5 The critical value of t for a two-tailed statistical test, a .05 and 4 df, is 2.776. Certainly, the observed value of t 12.8 is extremely large and highly signiﬁcant. Hence, you can conclude that there is a difference in the mean wear for tire types A and B.

EXAMPLE

10.9

Find a 95% conﬁdence interval for (m1 m2) md using the data in Table 10.3. Solution

A 95% conﬁdence interval for the difference between the mean levels of

wear is sd 苶d ta/2 兹n苶

冢冣

.0837 .48 2.776 5 兹苶

冢

冣

.48 .10 Conﬁdence intervals are always interpreted in the same way! In repeated sampling, intervals constructed in this way enclose the true value of the parameter 100(1 a)% of the time.

or .38 (m1 m2) .58. How does the width of this interval compare with the width of an interval you might have constructed if you had designed the experiment in an unpaired manner? It probably would have been of the same magnitude as the interval calculated in Figure 10.14, where the observed data were incorrectly analyzed using the unpaired analysis. This interval, 1.45 (m1 m2) 2.41, is much wider than the paired interval, which indicates that the paired difference design increased the accuracy of our estimate, and we have gained valuable information by using this design. The paired-difference test or matched pairs design used in the tire wear experiment is a simple example of an experimental design called a randomized block design.

414

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

Paired difference test: df n 1

FI GU R E 1 0 . 1 5

MINITAB output for paired-difference analysis of tire wear data

When there is a great deal of variability among the experimental units, even before any experimental procedures are implemented, the effect of this variability can be minimized by blocking—that is, comparing the different procedures within groups of relatively similar experimental units called blocks. In this way, the “noise” caused by the large variability does not mask the true differences between the procedures. We will discuss randomized block designs in more detail in Chapter 11. It is important for you to remember that the pairing or blocking occurs when the experiment is planned, and not after the data are collected. An experimenter may choose to use pairs of identical twins to compare two learning methods. A physician may record a patient’s blood pressure before and after a particular medication is given. Once you have used a paired design for an experiment, you no longer have the option of using the unpaired analysis of Section 10.4. The independence assumption has been purposely violated, and your only choice is to use the paired analysis described here! Although pairing was very beneficial in the tire wear experiment, this may not always be the case. In the paired analysis, the degrees of freedom for the t-test are cut in half—from (n n 2) 2(n 1) to (n 1). This reduction increases the critical value of t for rejecting H0 and also increases the width of the confidence interval for the difference in the two means. If pairing is not effective, this increase is not offset by a decrease in the variability, and you may in fact lose rather than gain information by pairing. This, of course, did not happen in the tire experiment—the large reduction in the standard error more than compensated for the loss in degrees of freedom. Except for notation, the paired-difference analysis is the same as the singlesample analysis presented in Section 10.3. However, MINITAB provides a single procedure called Paired t to analyze the differences, as shown in Figure 10.15. The p-value for the paired analysis, .000, indicates a highly significant difference in the means. You will find instructions for generating this MINITAB output in the “My MINITAB ” section at the end of this chapter. ● Paired T-Test and CI: Tire A, Tire B Paired T for Tire A - Tire B N Mean Tire A 5 10.240 Tire B 5 9.760 Difference 5 0.4800

StDev 1.316 1.328 0.0837

SE Mean 0.589 0.594 0.0374

95% CI for mean difference: (0.3761, 0.5839) T-Test of mean difference = 0 (vs not = 0): T-Value = 12.83

10.5

P-Value = 0.000

EXERCISES

BASIC TECHNIQUES 10.35 A paired-difference experiment was conducted using n 10 pairs of observations.

a. Test the null hypothesis H0 : (m1 m2) 0 against Ha : (m1 m2) 0 for a .05, 苶d .3, and sd2 .16. Give the approximate p-value for the test. b. Find a 95% conﬁdence interval for (m1 m2).

c. How many pairs of observations do you need if you want to estimate (m1 m2) correct to within .1 with probability equal to .95? 10.36 A paired-difference experiment consists of n 18 pairs, d苶 5.7, and sd2 256. Suppose you wish to detect md 0.

a. Give the null and alternative hypotheses for the test. b. Conduct the test and state your conclusions.

10.5 SMALL-SAMPLE INFERENCES FOR THE DIFFERENCE BETWEEN TWO MEANS: A PAIRED-DIFFERENCE TEST

10.37 A paired-difference experiment was conducted to compare the means of two populations: Pairs Population

1 2

1.3 1.2

1.6 1.5

1.1 1.1

1.4 1.2

1.7 1.8

a. Do the data provide sufficient evidence to indicate that m1 differs from m2? Test using a .05. b. Find the approximate p-value for the test and interpret its value. c. Find a 95% conﬁdence interval for (m1 m2). Compare your interpretation of the conﬁdence interval with your test results in part a. d. What assumptions must you make for your inferences to be valid? APPLICATIONS 10.38 Auto Insurance The cost of automo-

bile insurance has become a sore subject in California because the rates are dependent on so many variables, such as the city in which you live, the number of cars you insure, and the company with which you are insured. Here are the annual 2006–2007 premiums for a single male, licensed for 6–8 years, who drives a Honda Accord 12,600 to 15,000 miles per year and has no violations or accidents.10

EX1038

City

Allstate

Long Beach Pomona San Bernardino Moreno Valley

$2617 2305 2286 2247

21st Century $2228 2098 2064 1890

Source: www.insurance.ca.gov

a. Why would you expect these pairs of observations to be dependent? b. Do the data provide sufficient evidence to indicate that there is a difference in the average annual premiums between Allstate and 21st Century insurance? Test using a .01. c. Find the approximate p-value for the test and interpret its value. d. Find a 99% conﬁdence interval for the difference in the average annual premiums for Allstate and 21st Century insurance. e. Can we use the information in the table to make valid comparisons between Allstate and 21st Century insurance throughout the United States? Why or why not?

❍

415

10.39 Runners and Cyclists II Refer to Exercise 10.26. In addition to the compartment pressures, the level of creatine phosphokinase (CPK) in blood samples, a measure of muscle damage, was determined for each of 10 runners and 10 cyclists before and after exercise.7 The data summary—CPK values in units/liter—is as follows: Runners

Cyclists

Condition

Mean

Standard Deviation

Mean

Standard Deviation

Before exercise After exercise Difference

255.63 284.75 29.13

115.48 132.64 21.01

173.8 177.1 3.3

60.69 64.53 6.85

a. Test for a significant difference in mean CPK values for runners and cyclists before exercise under the assumption that s 12 s 22; use a .05. Find a 95% confidence interval estimate for the corresponding difference in means. b. Test for a significant difference in mean CPK values for runners and cyclists after exercise under the assumption that s 12 s 22; use a .05. Find a 95% confidence interval estimate for the corresponding difference in means. c. Test for a significant difference in mean CPK values for runners before and after exercise. d. Find a 95% confidence interval estimate for the difference in mean CPK values for cyclists before and after exercise. Does your estimate indicate that there is no significant difference in mean CPK levels for cyclists before and after exercise? 10.40 America’s Market Basket An advertisement for Albertsons, a supermarket chain in the western United States, claims that Albertsons has had consistently lower prices than four other fullservice supermarkets. As part of a survey conducted by an “independent market basket price-checking company,” the average weekly total, based on the prices of approximately 95 items, is given for two different supermarket chains recorded during 4 consecutive weeks in a particular month.

EX1040

Week

Albertsons

Ralphs

1 2 3 4

254.26 240.62 231.90 234.13

256.03 255.65 255.12 261.18

a. Is there a signiﬁcant difference in the average prices for these two different supermarket chains? b. What is the approximate p-value for the test conducted in part a?

416 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

c. Construct a 99% conﬁdence interval for the difference in the average prices for the two supermarket chains. Interpret this interval. 10.41 No Left Turn An experiment was conducted to compare the mean reaction times to two types of traffic signs: prohibitive (No Left Turn) and permissive (Left Turn Only). Ten drivers were included in the experiment. Each driver was presented with 40 traffic signs, 20 prohibitive and 20 permissive, in random order. The mean time to reaction and the number of correct actions were recorded for each driver. The mean reaction times (in milliseconds) to the 20 prohibitive and 20 permissive traffic signs are shown here for each of the 10 drivers:

EX1041

Driver

Prohibitive

Permissive

1 2 3 4 5 6 7 8 9 10

824 866 841 770 829 764 857 831 846 759

702 725 744 663 792 708 747 685 742 610

a. Explain why this is a paired-difference experiment and give reasons why the pairing should be useful in increasing information on the difference between the mean reaction times to prohibitive and permissive traffic signs. b. Do the data present sufficient evidence to indicate a difference in mean reaction times to prohibitive and permissive traffic signs? Use the p-value approach. c. Find a 95% conﬁdence interval for the difference in mean reaction times to prohibitive and permissive traffic signs. 10.42 Healthy Teeth II Exercise 10.24 describes a

dental experiment conducted to investigate the effectiveness of an oral rinse used to inhibit the growth of plaque on teeth. Subjects were divided into two groups: One group used a rinse with an antiplaque ingredient, and the control group used a rinse containing inactive ingredients. Suppose that the plaque growth on each person’s teeth was measured after using the rinse after 4 hours and then again after 8 hours. If you wish to estimate the difference in plaque growth from 4 to 8 hours, should you use a conﬁdence interval based on a paired or an unpaired analysis? Explain.

10.43 Ground or Air? The earth’s temperature (which affects seed germination, crop survival in bad weather, and many other aspects of agricultural production) can be measured using either ground-based sensors or infrared-sensing devices mounted in aircraft or space satellites. Ground-based sensoring is tedious, requiring many replications to obtain an accurate estimate of ground temperature. On the other hand, airplane or satellite sensoring of infrared waves appears to introduce a bias in the temperature readings. To determine the bias, readings were obtained at ﬁve different locations using both ground- and air-based temperature sensors. The readings (in degrees Celsius) are listed here: Location

Ground

Air

1 2 3 4 5

46.9 45.4 36.3 31.0 24.7

47.3 48.1 37.9 32.7 26.2

a. Do the data present sufficient evidence to indicate a bias in the air-based temperature readings? Explain. b. Estimate the difference in mean temperatures between ground- and air-based sensors using a 95% conﬁdence interval. c. How many paired observations are required to estimate the difference between mean temperatures for ground- versus air-based sensors correct to within .2°C, with probability approximately equal to .95? 10.44 Red Dye To test the comparative

brightness of two red dyes, nine samples of cloth were taken from a production line and each sample was divided into two pieces. One of the two pieces in each sample was randomly chosen and red dye 1 applied; red dye 2 was applied to the remaining piece. The following data represent a “brightness score” for each piece. Is there sufficient evidence to indicate a difference in mean brightness scores for the two dyes? Use a .05.

EX1044

Sample

Dye 1 Dye 2

10 8

12 11

9 10

8 6

15 12

12 13

9 9

10 8

15 13

10.45 Tax Assessors In response to a com-

plaint that a particular tax assessor (A) was biased, an experiment was conducted to compare the assessor named in the complaint with another tax assessor (B) from the same office. Eight properties were selected, and each was assessed by both assessors. The assessments (in thousands of dollars) are shown in the table.

EX1045

10.6 INFERENCES CONCERNING A POPULATION VARIANCE

Property

Assessor A

Assessor B

1 2 3 4 5 6 7 8

76.3 88.4 80.2 94.7 68.7 82.8 76.1 79.0

75.1 86.8 77.3 90.6 69.1 81.0 75.3 79.1

417

10.46 Memory Experiments A psychol-

ogy class performed an experiment to compare whether a recall score in which instructions to form images of 25 words were given is better than an initial recall score for which no imagery instructions were given. Twenty students participated in the experiment with the following results:

EX1046

Use the MINITAB printout to answer the questions. MINITAB output for Exercise 10.45

Paired T-Test and CI: Assessor A, Assessor B Paired T for Assessor A - Assessor B N Mean StDev SE Mean Assessor A 8 80.77 7.99 2.83 Assessor B 8 79.29 6.85 2.42 Difference 8 1.488 1.491 0.527 95% lower bound for mean difference: 0.489 T-Test of mean difference = 0 (vs > 0): T-Value = 2.82 P-value = 0.013

a. Do the data provide sufficient evidence to indicate that assessor A tends to give higher assessments than assessor B? b. Estimate the difference in mean assessments for the two assessors. c. What assumptions must you make in order for the inferences in parts a and b to be valid? d. Suppose that assessor A had been compared with a more stable standard—say, the average 苶x of the assessments given by four assessors selected from the tax office. Thus, each property would be assessed by A and also by each of the four other assessors and (xA 苶x ) would be calculated. If the test in part a is valid, can you use the paired-difference t-test to test the hypothesis that the bias, the mean difference between A’s assessments and the mean of the assessments of the four assessors, is equal to 0? Explain.

10.6

❍

Student

With Imagery

Without Imagery

Student

With Imagery

Without Imagery

1 2 3 4 5 6 7 8 9 10

20 24 20 18 22 19 20 19 17 21

5 9 5 9 6 11 8 11 7 9

11 12 13 14 15 16 17 18 19 20

17 20 20 16 24 22 25 21 19 23

8 16 10 12 7 9 21 14 12 13

Does it appear that the average recall score is higher when imagery is used? 10.47 Music in the Workplace Before con-

tracting to have stereo music piped into each of his suites of offices, an executive had his office manager randomly select seven offices in which to have the system installed. The average time (in minutes) spent outside these offices per excursion among the employees involved was recorded before and after the music system was installed with the following results.

EX1047

Office Number

No Music Music

8 5

9 6

5 7

6 5

5 6

10 7

7 8

Would you suggest that the executive proceed with the installation? Conduct an appropriate test of hypothesis. Find the approximate p-value and interpret your results.

INFERENCES CONCERNING A POPULATION VARIANCE You have seen in the preceding sections that an estimate of the population variance s 2 is usually needed before you can make inferences about population means. Sometimes, however, the population variance s 2 is the primary objective in an experimental investigation. It may be more important to the experimenter than the population mean! Consider these examples: •

Scientiﬁc measuring instruments must provide unbiased readings with a very small error of measurement. An aircraft altimeter that measures the correct

418

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

altitude on the average is fairly useless if the measurements are in error by as much as 1000 feet above or below the correct altitude. Machined parts in a manufacturing process must be produced with minimum variability in order to reduce out-of-size and hence defective parts. Aptitude tests must be designed so that scores will exhibit a reasonable amount of variability. For example, an 800-point test is not very discriminatory if all students score between 601 and 605.

• •

In previous chapters, you have used S(xi x苶)2 s2 n1 as an unbiased estimator of the population variance s 2. This means that, in repeated sampling, the average of all your sample estimates will equal the target parameter, s 2. But how close or far from the target is your estimator s2 likely to be? To answer this question, we use the sampling distribution of s2, which describes its behavior in repeated sampling. Consider the distribution of s2 based on repeated random sampling from a normal distribution with a speciﬁed mean and variance. We can show theoretically that the distribution begins at s2 0 (since the variance cannot be negative) with a mean equal to s 2. Its shape is nonsymmetric and changes with each different sample size and each different value of s 2. Finding critical values for the sampling distribution of s2 would be quite difficult and would require separate tables for each population variance. Fortunately, we can simplify the problem by standardizing, as we did with the z distribution. Definition

The standardized statistic

(n 1)s2 x 2 s2 is called a chi-square variable and has a sampling distribution called the chi-square probability distribution, with n 1 degrees of freedom. The equation of the density function for this statistic is quite complicated to look at, but it traces the curve shown in Figure 10.16. FI GU R E 1 0 . 1 6

A chi-square distribution

●

f(χ2)

a 0

χ2a

χ2

Certain critical values of the chi-square statistic, which are used for making inferences about the population variance, have been tabulated by statisticians and appear in Table 5 of Appendix I. Since the shape of the distribution varies with the sample

10.6 INFERENCES CONCERNING A POPULATION VARIANCE

❍

419

size n or, more precisely, the degrees of freedom, n 1, associated with s2, Table 5, partially reproduced in Table 10.5, is constructed in exactly the same way as the t table, with the degrees of freedom in the ﬁrst and last columns. The symbol x 2a indicates that the tabulated x 2-value has an area a to its right (see Figure 10.16). TABLE 10.5

Testing one variance: df n 1

●

Format of the Chi-Square Table from Table 5 in Appendix I df

x 2.995

1 2 3 4 5 6 . . . 15 16 17 18 19 . . .

.0000393 .0100251 .0717212 .206990 .411740 .0675727 . . . 4.60094 5.14224 5.69724 6.26481 6.84398 . . .

x 2.950

x 2.900

x 2.100

x 2.050

.0039321 .102587 .351846 .710721 1.145476 1.63539 . . . 7.26094 7.96164 8.67176 9.39046 10.1170 . . .

.0157908 .210720 .584375 1.063623 1.610310 2.204130 . . . 8.54675 9.31223 10.0852 10.8649 11.6509 . . .

2.70554 4.60517 6.25139 7.77944 9.23635 10.6446 . . . 22.3072 23.5418 24.7690 25.9894 27.2036 . . .

3.84146 5.99147 7.81473 9.48773 11.0705 12.5916 . . . 24.9958 26.2962 27.5871 28.8693 30.1435 . . .

x 2.005

7.87944 10.5966 12.8381 14.8602 16.7496 18.5476 . . . 32.8013 34.2672 35.7185 37.1564 38.5822 . . .

1 2 3 4 5 6 . . . 15 16 17 18 19 . . .

You can see in Table 10.5 that, because the distribution is nonsymmetric and starts at 0, both upper and lower tail areas must be tabulated for the chi-square statistic. For example, the value x 2.95 is the value that has 95% of the area under the curve to its right and 5% of the area to its left. This value cuts off an area equal to .05 in the lower tail of the chi-square distribution. EXAMPLE

10.10

Check your ability to use Table 5 in Appendix I by verifying the following statements: 1. The probability that x 2, based on n 16 measurements (df 15), exceeds 24.9958 is .05. 2. For a sample of n 6 measurements, 95% of the area under the x 2 distribution lies to the right of 1.145476. These values are shaded in Table 10.5.

You can use the Chi-Square Probabilities applet to ﬁnd the x 2-value described in Example 10.10. Since the applet provides x 2-values and their one-tailed probabilities for the degrees of freedom that you select using the slider on the right side of the applet, you should choose df 5 and type .95 in the box marked “prob:” at the bottom of the applet. The applet will provide the value of x 2 that puts .95 in the right tail of the x 2 distribution and hence .05 in the left tail. The applet in Figure 10.17 shows x 2 1.14, which differs only slightly from the value in Example 10.10. We will use this applet for the MyApplet Exercises at the end of the chapter.

420

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

F IGU R E 1 0 . 1 7

Chi-Square Probabilities applet

●

The statistical test of a null hypothesis concerning a population variance H0 : s 2 s 20 uses the test statistic (n 1)s2 x 2 s 20 Notice that when H0 is true, s2/s 20 should be near 1, so x 2 should be close to (n 1), the degrees of freedom. If s 2 is really greater than the hypothesized value s 20, the test statistic will tend to be larger than (n 1) and will probably fall toward the upper tail of the distribution. If s 2 s 20, the test statistic will tend to be smaller than (n 1) and will probably fall toward the lower tail of the chi-square distribution. As in other testing situations, you may use either a one- or a two-tailed statistical test, depending on the alternative hypothesis. This test of hypothesis and the (1 a)100% conﬁdence interval for s 2 are both based on the chi-square distribution and are described next.

TEST OF HYPOTHESIS CONCERNING A POPULATION VARIANCE 1. Null hypothesis: H0 : s 2 s 20 2. Alternative hypothesis: One-Tailed Test

Two-Tailed Test

Ha : s 2 s 20 (or Ha : s 2 s 20)

Ha : s 2 s 20

(n 1)s2 3. Test statistic: x 2 s 20

10.6 INFERENCES CONCERNING A POPULATION VARIANCE

❍

421

4. Rejection region: Reject H0 when One-Tailed Test

Two-Tailed Test

x 2 x 2a (or x 2 x 2(1a) when the alternative hypothesis is Ha : s 2 s 20), where x 2a and x 2(1a) are, respectively, the upper- and lower-tail values of x 2 that place a in the tail areas

2 x 2 x a/2 or x 2 x 2(1a/2), 2 where x a/2 and x 2(1a/2) are, respectively, the upper- and lower-tail values of x 2 that place a/2 in the tail areas

or when p-value a The critical values of x 2 are based on (n 1) df. These tabulated values can be found using Table 5 of Appendix I or the Chi-Square Probabilities applet.

α/2

χ2α

χ2α/2

χ2(1 – α/2)

(1 a)100% CONFIDENCE INTERVAL FOR s 2 (n 1)s2 (n 1)s2 2 s 2 x a/2 x 2(1a/2) 2 where x a/2 and x 2(1a/2) are the upper and lower x 2-values, which locate one-half of a in each tail of the chi-square distribution. Assumption: The sample is randomly selected from a normal population.

EXAMPLE

10.11

A cement manufacturer claims that concrete prepared from his product has a relatively stable compressive strength and that the strength measured in kilograms per square centimeter (kg/cm2) lies within a range of 40 kg/cm2. A sample of n 10 measurements produced a mean and variance equal to, respectively, 2 苶x 312 and s 195 Do these data present sufficient evidence to reject the manufacturer’s claim? In Section 2.5, you learned that the range of a set of measurements should be approximately four standard deviations. The manufacturer’s claim that the range of the strength measurements is within 40 kg/cm2 must mean that the standard deviation of the measurements is roughly 10 kg/cm2 or less. To test his claim, the appropriate hypotheses are H0 : s 2 102 100 versus Ha : s 2 100 Solution

If the sample variance is much larger than the hypothesized value of 100, then the test statistic (n 1)s2 1755 17.55 x 2 s 20 100 will be unusually large, favoring rejection of H0 and acceptance of Ha. There are two ways to use the test statistic to make a decision for this test.

422

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

•

FI GU R E 1 0 . 1 8

Rejection region and p-value (shaded) for Example 10.11

●

The critical value approach: The appropriate test requires a one-tailed rejection region in the right tail of the x 2 distribution. The critical value for a .05 and (n 1) 9 df is x 2.05 16.9190 from Table 5 in Appendix I. Figure 10.18 shows the rejection region; you can reject H0 if the test statistic exceeds 16.9190. Since the observed value of the test statistic is x 2 17.55, you can conclude that the null hypothesis is false and that the range of concrete strength measurements exceeds the manufacturer’s claim.

f(χ2)

.050 .025 0

16.9190 19.0228

χ2

Reject H0

•

EXAMPLE

10.12

The p-value approach: The p-value for a statistical test is the smallest value of a for which H0 can be rejected. It is calculated, as in other one-tailed tests, as the area in the tail of the x 2 distribution to the right of the observed value, x 2 17.55. Although computer packages allow you to calculate this area exactly, Table 5 in Appendix I allows you only to bound the p-value. Since the value 17.55 lies between x 2.050 16.9190 and x 2.025 19.0228, the p-value lies between .025 and .05. Most researchers would reject H0 and report these results as signiﬁcant at the 5% level, or P .05. Again, you can reject H0 and conclude that the range of measurements exceeds the manufacturer’s claim.

An experimenter is convinced that her measuring instrument had a variability measured by standard deviation s 2. During an experiment, she recorded the measurements 4.1, 5.2, and 10.2. Do these data confirm or disprove her assertion? Test the appropriate hypothesis, and construct a 90% confidence interval to estimate the true value of the population variance. Since there is no preset level of significance, you should choose to use the p-value approach in testing these hypotheses:

Solution

H0 : s 2 4

versus Ha : s 2 4

Use your scientiﬁc calculator to verify that the sample variance is s2 10.57 and the test statistic is (n 1)s2 2(10.57) 5.285 x 2 s 20 4 Since this is a two-tailed test, the rejection region is divided into two parts, half in each tail of the x 2 distribution. If you approximate the area to the right of the observed test statistic, x 2 5.285, you will have only half of the p-value for the test. Since an equally unlikely value of x 2 might occur in the lower tail of the distribution,

10.6 INFERENCES CONCERNING A POPULATION VARIANCE

❍

423

with equal probability, you must double the upper area to obtain the p-value. With 2 df, the observed value, 5.29, falls between x 2.10 and x 2.05 so that 1 .05 ( p-value) .10 2

.10 p-value .20

Since the p-value is greater than .10, the results are not statistically signiﬁcant. There is insufficient evidence to reject the null hypothesis H0 : s 2 4. The corresponding 90% conﬁdence interval is (n 1)s2 (n 1) s2 2 s 2 x a/2 x 2(1a/2) The values of x 2(1a/2) and x 2a/2 are x 2(1 a/2) x 2.95 .102587 x 2a/2 x 2.05 5.99147 Substituting these values into the formula for the interval estimate, you get 2(10.57) 2(10.57) s 2 or 5.99147 .102587

3.53 s 2 206.07

Thus, you can estimate the population variance to fall into the interval 3.53 to 206.07. This very wide conﬁdence interval indicates how little information on the population variance is obtained from a sample of only three measurements. Consequently, it is not surprising that there is insufficient evidence to reject the null hypothesis s 2 4. To obtain more information on s 2, the experimenter needs to increase the sample size. The MINITAB command Stat 씮 Basic Statistics 씮 1 Variance allows you to enter raw data or a summary statistic to perform the F-test for a single variance, and calculate a conﬁdence interval. The MINITAB printout corresponding to Example 10.12 is shown in Figure 10.19. F I GU R E 1 0 . 1 9

MINITAB output for Example 10.12

10.6

● Chi-Square Method (Normal Distribution) Variable Measurements

N 3

Variance 10.6

90% CI (3.5, 206.1)

Chi-Square 5.28

P 0.142

EXERCISES

BASIC TECHNIQUES 10.48 A random sample of n 25 observations from a normal population produced a sample variance equal to 21.4. Do these data provide sufficient evidence to indicate that s 2 15? Test using a .05. 10.49 A random sample of n 15 observations was selected from a normal population. The sample mean and variance were x苶 3.91 and s 2 .3214. Find a 90% conﬁdence interval for the population variance s 2. 10.50 A random sample of size n 7 from a normal

population produced these measurements: 1.4, 3.6, 1.7, 2.0, 3.3, 2.8, 2.9.

a. Calculate the sample variance, s 2. b. Construct a 95% conﬁdence interval for the population variance, s 2. c. Test H0 : s 2 .8 versus Ha : s 2 .8 using a .05. State your conclusions. d. What is the approximate p-value for the test in part c? APPLICATIONS 10.51 Instrument Precision A precision instru-

ment is guaranteed to read accurately to within 2 units. A sample of four instrument readings on the same object yielded the measurements 353, 351, 351, and

424 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

355. Test the null hypothesis that s .7 against the alternative s .7. Use a .05.

estimate the variance of the manufacturer’s potency measurements.

10.52 Instrument Precision, continued Find a

10.55 Hard Hats A manufacturer of hard safety hats for construction workers is concerned about the mean and the variation of the forces helmets transmit to wearers when subjected to a standard external force. The manufacturer desires the mean force transmitted by helmets to be 800 pounds (or less), well under the legal 1000-pound limit, and s to be less than 40. A random sample of n 40 helmets was tested, and the sample mean and variance were found to be equal to 825 pounds and 2350 pounds 2, respectively. a. If m 800 and s 40, is it likely that any helmet, subjected to the standard external force, will transmit a force to a wearer in excess of 1000 pounds? Explain. b. Do the data provide sufficient evidence to indicate that when the helmets are subjected to the standard external force, the mean force transmitted by the helmets exceeds 800 pounds?

90% confidence interval for the population variance in Exercise 10.51. 10.53 Drug Potency To properly treat patients, drugs prescribed by physicians must have a potency that is accurately defined. Consequently, not only must the distribution of potency values for shipments of a drug have a mean value as specified on the drug’s container, but also the variation in potency must be small. Otherwise, pharmacists would be distributing drug prescriptions that could be harmfully potent or have a low potency and be ineffective. A drug manufacturer claims that his drug is marketed with a potency of 5 .1 milligram per cubic centimeter (mg/cc). A random sample of four containers gave potency readings equal to 4.94, 5.09, 5.03, and 4.90 mg/cc. a. Do the data present sufficient evidence to indicate that the mean potency differs from 5 mg/cc? b. Do the data present sufficient evidence to indicate that the variation in potency differs from the error limits specified by the manufacturer? [HINT: It is sometimes difficult to determine exactly what is meant by limits on potency as specified by a manufacturer. Since he implies that the potency values will fall into the interval 5 .1 mg/cc with very high probability—the implication is always—let us assume that the range .2; or (4.9 to 5.1), represents 6s, as suggested by the Empirical Rule. Note that letting the range equal 6s rather than 4s places a stringent interpretation on the manufacturer’s claim. We want the potency to fall into the interval 5 .1 with very high probability.] 10.54 Drug Potency, continued Refer to Exercise

10.53. Testing of 60 additional randomly selected containers of the drug gave a sample mean and variance equal to 5.04 and .0063 (for the total of n 64 containers). Using a 95% conﬁdence interval,

10.7

10.56 Hard Hats, continued Refer to Exercise

10.55. Do the data provide sufficient evidence to indicate that s exceeds 40? 10.57 Light Bulbs A manufacturer of

industrial light bulbs likes its bulbs to have a mean life that is acceptable to its customers and a variation in life that is relatively small. If some bulbs fail too early in their life, customers become annoyed and shift to competitive products. Large variations above the mean reduce replacement sales, and variation in general disrupts customers’ replacement schedules. A sample of 20 bulbs tested produced the following lengths of life (in hours):

EX1057

2100 2302 1951 2067 2415 1883 2101 2146 2278 2019 1924 2183 2077 2392 2286 2501 1946 2161 2253 1827

The manufacturer wishes to control the variability in length of life so that s is less than 150 hours. Do the data provide sufficient evidence to indicate that the manufacturer is achieving this goal? Test using a .01.

COMPARING TWO POPULATION VARIANCES Just as a single population variance is sometimes important to an experimenter, you might also need to compare two population variances. You might need to compare the precision of one measuring device with that of another, the stability of one manufacturing process with that of another, or even the variability in the grading procedure of one college professor with that of another.

10.7 COMPARING TWO POPULATION VARIANCES

❍

425

One way to compare two population variances, s 21 and s 22, is to use the ratio of the sample variances, s12/s 22. If s12/s 22 is nearly equal to 1, you will ﬁnd little evidence to indicate that s 12 and s 22 are unequal. On the other hand, a very large or very small value for s12/s22 provides evidence of a difference in the population variances. How large or small must s 21/s 22 be for sufficient evidence to exist to reject the following null hypothesis? H0 : s 12 s 22 The answer to this question may be found by studying the distribution of s12/s 22 in repeated sampling. When independent random samples are drawn from two normal populations with equal variances—that is, s 12 s 22—then s 21/s 22 has a probability distribution in repeated sampling that is known to statisticians as an F distribution, shown in Figure 10.20. FIGU R E 1 0 . 2 0

An F distribution with df1 10 and df2 10

● f(F)

10 F

2 ASSUMPTIONS FOR s 2 1/s 2 TO HAVE AN F DISTRIBUTION

• •

Testing two variances: df1 n1 1 and df2 n2 1

Random and independent samples are drawn from each of two normal populations. The variability of the measurements in the two populations is the same and can be measured by a common variance, s 2; that is, s 21 s 22 s 2.

It is not important for you to know the complex equation of the density function for F. For your purposes, you need only to use the well-tabulated critical values of F given in Table 6 in Appendix I. Critical values of F and p-values for signiﬁcance tests can also be found using the F Probabilities applet shown in Figure 10.21. Like the x 2 distribution, the shape of the F distribution is nonsymmetric and depends on the number of degrees of freedom associated with s 21 and s 22, represented as df1 (n1 1) and df2 (n2 1), respectively. This complicates the tabulation of critical values of the F distribution because a table is needed for each different combination of df1, df2, and a. In Table 6 in Appendix I, critical values of F for right-tailed areas corresponding to a .100, .050, .025, .010, and .005 are tabulated for various combinations of df1 numerator degrees of freedom and df2 denominator degrees of freedom. A portion of Table 6 is reproduced in Table 10.6. The numerator degrees of freedom df1 are listed across the top margin, and the denominator degrees of freedom df2 are listed along the side margin. The values of a are listed in the second column. For a ﬁxed combination of df1 and df2, the appropriate critical values of F are found in the line indexed by the value of a required.

426

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

FI GU R E 1 0 . 2 1

F Probabilities applet

EXAMPLE

●

10.13

Check your ability to use Table 6 in Appendix I by verifying the following statements: 1. The value of F with area .05 to its right for df1 6 and df2 9 is 3.37. 2. The value of F with area .05 to its right for df1 5 and df2 10 is 3.33. 3. The value of F with area .01 to its right for df1 6 and df2 9 is 5.80. These values are shaded in Table 10.6.

TABLE 10.6

●

Format of the F Table from Table 6 in Appendix I df1 df2

.100 .050 .025 .010 .005 .100 .050 .025 .010 .005

39.86 161.4 647.8 4052 16211 8.53 18.51 38.51 98.50 198.5

49.50 199.5 799.5 4999.5 20000 9.00 19.00 39.00 99.00 199.0

53.59 215.7 864.2 5403 21615 9.16 19.16 39.17 99.17 199.2

55.83 224.6 899.6 5625 22500 9.24 19.25 39.25 99.25 199.2

57.24 230.2 921.8 5764 23056 9.29 19.30 39.30 99.30 199.3

58.20 234.0 937.1 5859 23437 9.33 19.33 39.33 99.33 199.3

.100 .050 .025 .010 .005 . . . .100 .050 .025 .010 .005

5.54 10.13 17.44 34.12 55.55

5.46 9.55 16.04 30.82 49.80

5.34 9.12 15.10 28.71 46.19

5.31 9.01 14.88 28.24 45.39

3.36 5.12 7.21 10.56 13.61

3.01 4.26 5.71 8.02 10.11

5.39 9.28 15.44 29.46 47.47 . . . 2.81 3.86 5.08 6.99 8.72

2.69 3.63 4.72 6.42 7.96

2.61 3.48 4.48 6.06 7.47

5.28 8.94 14.73 27.91 44.84 . . . 2.55 3.37 4.32 5.80 7.13

.100 .050 .025 .010 .005

3.29 4.96 6.94 10.04 12.83

2.92 4.10 5.46 7.56 9.43

2.73 3.71 4.83 6.55 8.08

2.61 3.48 4.47 5.99 7.34

2.52 3.33 4.24 5.64 6.87

2.46 3.22 4.07 5.39 6.54

. . . 9

10.7 COMPARING TWO POPULATION VARIANCES

❍

427

The statistical test of the null hypothesis H0 : s 12 s 22 uses the test statistic s2 F 21 s2 When the alternative hypothesis implies a one-tailed test—that is, Ha : s 12 s 22 you can ﬁnd the right-tailed critical value for rejecting H0 directly from Table 6 in Appendix I. However, when the alternative hypothesis requires a two-tailed test—that is, H0 : s 21 s 22 the rejection region is divided between the upper and lower tails of the F distribution. These left-tailed critical values are not given in Table 6 for the following reason: You are free to decide which of the two populations you want to call “Population 1.” If you always choose to call the population with the larger sample variance “Population 1,” then the observed value of your test statistic will always be in the right tail of the F distribution. Even though half of the rejection region, the area a/2 to its left, will be in the lower tail of the distribution, you will never need to use it! Remember these points, though, for a two-tailed test: • •

The area in the right tail of the rejection region is only a/2. The area to the right of the observed test statistic is only ( p-value)/2.

The formal procedures for a test of hypothesis and a (1 a)100% conﬁdence interval for two population variances are shown next. TEST OF HYPOTHESIS CONCERNING THE EQUALITY OF TWO POPULATION VARIANCES 1. Null hypothesis: H0 : s 21 s 22 2. Alternative hypothesis: One-Tailed Test

Two-Tailed Test

Ha : s 21 s 22 (or Ha : s 21 s 22)

Ha : s 21 s 22

3. Test statistic: One-Tailed Test

s2 12 s2

Two-Tailed Test

s2 F 12 s2

where s12 is the larger sample variance 4. Rejection region: Reject H0 when One-Tailed Test

Two-Tailed Test

F Fa

F Fa/2

or when p-value a

(continued)

428

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

TEST OF HYPOTHESIS CONCERNING THE EQUALITY OF TWO POPULATION VARIANCES (continued)

The critical values of Fa and Fa/2 are based on df1 (n1 1) and df2 (n2 1). These tabulated values, for a .100, .050, .025, .010, and .005, can be found using Table 6 in Appendix I, or the F Probabilities applet.

α/2 α

Fα

Fα/2

Assumptions: The samples are randomly and independently selected from normally distributed populations.

2 CONFIDENCE INTERVAL FOR s 2 1/s 2

s2 1 s2 s2 21 12 12 Fdf2,df1 s 2 Fdf1,df2 s2 s2

冢冣

where df1 (n1 1) and df2 (n2 1). Fdf1,df2 is the tabulated critical value of F corresponding to df1 and df2 degrees of freedom in the numerator and denominator of F, respectively, with area a/2 to its right. Assumptions: The samples are randomly and independently selected from normally distributed populations. EXAMPLE

10.14

An experimenter is concerned that the variability of responses using two different experimental procedures may not be the same. Before conducting his research, he conducts a prestudy with random samples of 10 and 8 responses and gets s12 7.14 and s22 3.21, respectively. Do the sample variances present sufficient evidence to indicate that the population variances are unequal? Assume that the populations have probability distributions that are reasonably mound-shaped and hence satisfy, for all practical purposes, the assumption that the populations are normal. You wish to test these hypotheses:

Solution

H0 : s 12 s 22

versus Ha : s 12 s 22

Using Table 6 in Appendix I for a/2 .025, you can reject H0 when F 4.82 with a .05. The calculated value of the test statistic is 7.14 s2 F 12 2.22 3.21 s2 Because the test statistic does not fall into the rejection region, you cannot reject H0 : s 21 s 22. Thus, there is insufficient evidence to indicate a difference in the population variances.

10.7 COMPARING TWO POPULATION VARIANCES

EXAMPLE

10.15

❍

429

Refer to Example 10.14 and ﬁnd a 90% conﬁdence interval for s 12/s 22. Solution

The 90% conﬁdence interval for s 12/s 22 is

1 s2 s2 s2 12 12 12 Fdf2,df1 s 2 Fdf1,df2 s2 s2

冢冣

where s 21 7.14

s 22 3.21

df1 (n1 1) 9

df2 (n2 1) 7

F9,7 3.68

F7,9 3.29

Substituting these values into the formula for the conﬁdence interval, you get 7.14 1 s2 7.14 21 3.29 3.21 3.68 s2 3.21

冢冣

s2 .60 21 7.32 s2

The calculated interval estimate .60 to 7.32 includes 1.0, the value hypothesized in H0. This indicates that it is quite possible that s 21 s 22 and therefore agrees with the test conclusions. Do not reject H0 : s 21 s 22. The MINITAB command Stat 씮 Basic Statistics 씮 2 Variances allows you to enter either raw data or summary statistics to perform the F-test for the equality of variances and calculates conﬁdence intervals for the two individual standard deviations (which we have not discussed). The relevant printout, containing the F statistic and its p-value, is shaded in Figure 10.22. FIGU R E 1 0 . 2 2

MINITAB output for Example 10.14

● Test for Equal Variances 95% Bonferroni conﬁdence intervals for standard deviations Sample N Lower StDev Upper 1 10 1.74787 2.67208 5.38064 2 8 1.12088 1.79165 4.10374 F-Test (Normal Distribution) Test statistic = 2.22, p-value = 0.304

EXAMPLE

10.16

The variability in the amount of impurities present in a batch of chemical used for a particular process depends on the length of time the process is in operation. A manufacturer using two production lines, 1 and 2, has made a slight adjustment to line 2, hoping to reduce the variability as well as the average amount of impurities in the chemical. Samples of n1 25 and n2 25 measurements from the two batches yield these means and variances: 苶x1 3.2 苶x2 3.0

s12 1.04 s 22 .51

Do the data present sufficient evidence to indicate that the process variability is less for line 2? The experimenter believes that the average levels of impurities are the same for the two production lines but that her adjustment may have decreased the variability of the levels for line 2, as illustrated in Figure 10.23. This adjustment would be good for the company because it would decrease the probability of producing shipments of the chemical with unacceptably high levels of impurities.

Solution

430

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

FI GU R E 1 0 . 2 3

Distributions of impurity measurements for two production lines

●

f(x)

Distribution for production line 2

Distribution for production line 1

Level of impurities

To test for a decrease in variability, the test of hypothesis is H0 : s 12 s 22

versus Ha : s 12 s 22

and the observed value of the test statistic is s2 1.04 F 21 2.04 s2 .51 Using the p-value approach, you can bound the one-tailed p-value using Table 6 in Appendix I with df1 df2 (25 1) 24. The observed value of F falls between F.050 1.98 and F.025 2.27, so that .025 p-value .05. The results are judged signiﬁcant at the 5% level, and H0 is rejected. You can conclude that the variability of line 2 is less than that of line 1. The F-test for the difference in two population variances completes the battery of tests you have learned in this chapter for making inferences about population parameters under these conditions: • •

The sample sizes are small. The sample or samples are drawn from normal populations.

You will ﬁnd that the F and x 2 distributions, as well as the Student’s t distribution, are very important in other applications in the chapters that follow. They will be used for different estimators designed to answer different types of inferential questions, but the basic techniques for making inferences remain the same. In the next section, we review the assumptions required for all of these inference tools, and discuss options that are available when the assumptions do not seem to be reasonably correct.

10.7

EXERCISES

BASIC TECHNIQUES 10.58 Independent random samples from two normal populations produced the variances listed here: Sample Size

Sample Variance

16 20

55.7 31.4

a. Do the data provide sufficient evidence to indicate that s 21 differs from s 22? Test using a .05. b. Find the approximate p-value for the test and interpret its value. 10.59 Refer to Exercise 10.58 and ﬁnd a 95% conﬁdence interval for s 21/s 22.

10.7 COMPARING TWO POPULATION VARIANCES

❍

431

10.60 Independent random samples from two normal populations produced the given variances:

b. Find the approximate p-value for the test and interpret its value.

Sample Size

10.63 Construct a 90% conﬁdence interval for the variance ratio in Exercise 10.62.

13 13

Sample Variance 18.3 7.9

a. Do the data provide sufficient evidence to indicate that s 12 s 22? Test using a .05. b. Find the approximate p-value for the test and interpret its value. APPLICATIONS 10.61 SAT Scores The SAT subject tests in chem-

istry and physics11 for two groups of 15 students each electing to take these tests are given below. Chemistry

Physics

x苶 629 s 110 n 15

x苶 643 s 107 n 15

To use the two-sample t-test with a pooled estimate of 2, you must assume that the two population variances are equal. Test this assumption using the F-test for equality of variances. What is the approximate p-value for the test?

10.64 Tuna III In Exercise 10.25 and dataset EX1025, you conducted a test to detect a difference in the average prices of light tuna in water versus light tuna in oil.

a. What assumption had to be made concerning the population variances so that the test would be valid? b. Do the data present sufficient evidence to indicate that the variances violate the assumption in part a? Test using a .05. 10.65 Runners and Cyclists III Refer to Exer-

cise 10.26. Susan Beckham and colleagues conducted an experiment involving 10 healthy runners and 10 healthy cyclists to determine if there are signiﬁcant differences in pressure measurements within the anterior muscle compartment for runners and cyclists.7 The data—compartment pressure, in millimeters of mercury (Hg)—are reproduced here: Runners Condition

10.62 Product Quality The stability of measurements on a manufactured product is important in maintaining product quality. In fact, it is sometimes better to have small variation in the measured value of some important characteristic of a product and have the process mean be slightly off target than to suffer wide variation with a mean value that perfectly ﬁts requirements. The latter situation may produce a higher percentage of defective products than the former. A manufacturer of light bulbs suspected that one of her production lines was producing bulbs with a wide variation in length of life. To test this theory, she compared the lengths of life for n 50 bulbs randomly sampled from the suspect line and n 50 from a line that seemed to be “in control.” The sample means and variances for the two samples were as follows: “Suspect Line”

Line “in Control”

x苶1 1520 s 12 92,000

x苶2 1476 s 22 37,000

a. Do the data provide sufficient evidence to indicate that bulbs produced by the “suspect line” have a larger variance in length of life than those produced by the line that is assumed to be in control? Test using a .05.

Rest 80% maximal O2 consumption Maximal O2 consumption

Cyclists

Mean

Standard Deviation

Mean

Standard Deviation

14.5

3.92

11.1

3.98

12.2 19.1

3.49 16.9

11.5 12.2

4.95 4.47

For each of the three variables measured in this experiment, test to see whether there is a signiﬁcant difference in the variances for runners versus cyclists. Find the approximate p-values for each of these tests. Will a two-sample t-test with a pooled estimate of s 2 be appropriate for all three of these variables? Explain. 10.66 Impurities A pharmaceutical manufacturer purchases a particular material from two different suppliers. The mean level of impurities in the raw material is approximately the same for both suppliers, but the manufacturer is concerned about the variability of the impurities from shipment to shipment. If the level of impurities tends to vary excessively for one source of supply, it could affect the quality of the pharmaceutical product. To compare the variation in percentage impurities for the two suppliers, the manufacturer selects 10 shipments from each of the two suppliers and measures the percentage of impurities in the raw material for each shipment. The sample means and variances are shown in the table.

432 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

Supplier A

Supplier B

x苶1 1.89 s 12 .273 n1 10

x苶2 1.85 s 22 .094 n2 10

a. Do the data provide sufficient evidence to indicate a difference in the variability of the shipment

impurity levels for the two suppliers? Test using a .01. Based on the results of your test, what recommendation would you make to the pharmaceutical manufacturer? b. Find a 99% conﬁdence interval for s 22 and interpret your results.

How Do I Decide Which Test to Use? Are you interested in testing means? If the design involves: a. One random sample, use the one-sample t statistic. b. Two independent random samples, are the population variances equal? i. If equal, use the two-sample t statistic with pooled s 2. ii. If unequal, use the unpooled t with estimated df. c. Two paired samples with random pairs, use a one-sample t for analyzing differences. Are you interested in testing variances? If the design involves: a. One random sample, use the x 2 test for a single variance. b. Two independent random samples, use the F-test to compare two variances.

10.8

REVISITING THE SMALL-SAMPLE ASSUMPTIONS All of the tests and estimation procedures discussed in this chapter require that the data satisfy certain conditions in order that the error probabilities (for the tests) and the confidence coefficients (for the confidence intervals) be equal to the values you have specified. For example, if you construct what you believe to be a 95% confidence interval, you want to be certain that, in repeated sampling, 95% (and not 85% or 75% or less) of all such intervals will contain the parameter of interest. These conditions are summarized in these assumptions: ASSUMPTIONS 1. For all tests and confidence intervals described in this chapter, it is assumed that samples are randomly selected from normally distributed populations. 2. When two samples are selected, it is assumed that they are selected in an independent manner except in the case of the paired-difference experiment. 3. For tests or confidence intervals concerning the difference between two population means m1 and m2 based on independent random samples, it is assumed that s 12 s 22.

CHAPTER REVIEW

❍

433

In reality, you will never know everything about the sampled population. If you did, there would be no need for sampling or statistics. It is also highly unlikely that a population will exactly satisfy the assumptions given in the box. Fortunately, the procedures presented in this chapter give good inferences even when the data exhibit moderate departures from the necessary conditions. A statistical procedure that is not sensitive to departures from the conditions on which it is based is said to be robust. The Student’s t-tests are quite robust for moderate departures from normality. Also, as long as the sample sizes are nearly equal, there is not much difference between the pooled and unpooled t statistics for the difference in two population means. However, if the sample sizes are not clearly equal, and if the population variances are unequal, the pooled t statistic provides inaccurate conclusions. If you are concerned that your data do not satisfy the assumptions, other options are available: •

•

If you can select relatively large samples, you can use one of the largesample procedures of Chapters 8 and 9, which do not rely on the normality or equal variance assumptions. You may be able to use a nonparametric test to answer your inferential questions. These tests have been developed speciﬁcally so that few or no distributional assumptions are required for their use. Tests that can be used to compare the locations or variability of two populations are presented in Chapter 15.

CHAPTER REVIEW Key Concepts and Formulas I.

Experimental Designs for Small Samples

III. Small-Sample Test Statistics

1. Single random sample: The sampled population must be normal. 2. Two independent random samples: Both sampled populations must be normal. a. Populations have a common variance s 2. s 21

b. Populations have different variances: and s 22. 3. Paired-difference or matched pairs design: The samples are not independent. II. Statistical Tests of Significance

1. Based on the t, F, and x distributions 2

2. Use the same procedure as in Chapter 9 3. Rejection region—critical values and signiﬁcance levels: based on the t, F, or x 2 distributions with the appropriate degrees of freedom 4. Tests of population parameters: a single mean, the difference between two means, a single variance, and the ratio of two variances

To test one of the population parameters when the sample sizes are small, use the following test statistics: Parameter m

Test Statistic x苶 m0 t s/兹苶n

(x苶1 x苶2) (m1 m2) t m1 m2 1 1 (equal variances) s 2 n1 n2

冢莦冣冪莦

m1 m2 (unequal variances)

(x苶1 x苶2) (m1 m2) t ⬇ s2 s2 1 2 n1 n2

冪莦莦

d苶 md m1 m2 t sd /兹n苶 (paired samples) s2

(n 1)s 2 x 2 s 02

s 12/s 22

F s 12 /s 22

Degrees of Freedom n1 n1 n2 2

Satterthwaite’s approximation

n1 n1 n1 1 and n 2 1

434

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

Small-Sample Testing and Estimation The tests and confidence intervals for population means based on the Student’s t distribution are found in a MINITAB submenu by choosing Stat 씮 Basic Statistics. You will see choices for 1-Sample t, 2-Sample t, and Paired t, which will generate Dialog boxes for the procedures in Sections 10.3, 10.4, and 10.5, respectively. You must choose the columns in which the data are stored and the null and alternative hypotheses to be tested (or the confidence coefficient for a confidence interval). In the case of the two-sample t-test, you must indicate whether the population variances are assumed equal or unequal, so that MINITAB can perform the correct test. We will display some of the Dialog boxes and Session window outputs for the examples in this chapter, beginning with the one-sample t-test of Example 10.3. First, enter the six recorded weights—.46, .61, .52, .48, .57, .54—in column C1 and name them “Weights.” Use Stat 씮 Basic Statistics 씮 1-Sample t to generate the Dialog box in Figure 10.24. To test H0 : m .5 versus Ha : m .5, use the list on the left to select “Weights” for the box marked “Samples in Columns.” Check the box marked “Perform hypothesis test.” Then, place your cursor in the box marked “Hypothesized mean:” and enter .5 as the test value. Finally, use Options and the drop-down menu marked “Alternative” to select “greater than.” Click OK twice to obtain the output in Figure 10.25. Notice that MINITAB produces a one- or a two-sided confidence interval for the single population mean, consistent with the alternative hypothesis you have chosen. You can change the confidence coefficient from the default of .95 in the Options box. Also, the Graphs option will produce a histogram, a box plot, or an individual value plot of the data in column C1. Data for a two-sample t-test with independent samples can be entered into the worksheet in one of two ways: FI GU R E 1 0 . 2 4

●

MY MINITAB

FIGU R E 1 0 . 2 5

❍

435

●

•

Enter measurements from both samples into a single column and enter numbers (1 or 2) in a second column to identify the sample from which the measurement comes. • Enter the samples in two separate columns. If you do not have the raw data, but rather have summary statistics—the sample mean, standard deviation, and sample size—MINITAB 15 will allow you to use these values by selecting the radio button marked “Summarized data” and entering the appropriate values in the boxes. Use the second method and enter the data from Example 10.5 into columns C2 and C3. Then use Stat 씮 Basic Statistics 씮 2-Sample t to generate the Dialog box in Figure 10.26. Check “Samples in different columns,” selecting C2 and C3 from the box on the left. Check the “Assume equal variances” box and select the proper alternative hypothesis in the Options box. (Otherwise, MINITAB will perform Satterthwaite’s approximation for unequal variances.) The two-sample output when you click OK twice automatically contains a 95% one- or two-sided conﬁdence interval as well as the test statistic and p-value (you can change the conﬁdence coefficient if you like). The output for Example 10.5 is shown in Figure 10.13. For a paired-difference test, the two samples are entered into separate columns, which we did with the tire wear data in Table 10.3. Use Stat 씮 Basic Statistics 씮 Paired t FIGU R E 1 0 . 2 6

●

436 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

to generate the Dialog box in Figure 10.27. If you have only summary statistics—the sample mean and standard deviation of the differences and sample size—MINITAB will allow you to use these values by selecting the radio button marked “Summarized data” and entering the appropriate values in the boxes. Select C4 and C5 from the box on the left, and use Options to pick the proper alternative hypothesis. You may change the conﬁdence coefficient or the test value (the default value is zero). When you click OK twice, you will obtain the output shown in Figure 10.15. The MINITAB command Stat 씮 Basic Statistics 씮 2 Variances allows you to enter either raw data or summary statistics to perform the F-test for the equality of variances, as shown in Figure 10.28. The MINITAB command Stat 씮 Basic Statistics 씮 1 Variance will allow you to perform the x 2 test and construct a conﬁdence interval for a single population variance, s 2. FI GU R E 1 0 . 2 7

●

FI GU R E 1 0 . 2 8

●

SUPPLEMENTARY EXERCISES

Supplementary Exercises 10.67 What assumptions are made when Student’s t-test is used to test a hypothesis concerning a population mean?

❍

437

of the three titrations are as follows: 82.10, 75.75, and 75.44 milliliters. Use a 99% conﬁdence interval to estimate the mean number of milliliters required to neutralize 1 gram of the acid. 10.73 Sodium Chloride Measurements of water

10.68 What assumptions are made about the popula-

tions from which random samples are obtained when the t distribution is used in making small-sample inferences concerning the difference in population means? 10.69 Why use paired observations to estimate the difference between two population means rather than estimation based on independent random samples selected from the two populations? Is a paired experiment always preferable? Explain. 10.70 Impurities II A manufacturer can tolerate a

small amount (.05 milligrams per liter (mg/l)) of impurities in a raw material needed for manufacturing its product. Because the laboratory test for the impurities is subject to experimental error, the manufacturer tests each batch 10 times. Assume that the mean value of the experimental error is 0 and hence that the mean value of the ten test readings is an unbiased estimate of the true amount of the impurities in the batch. For a particular batch of the raw material, the mean of the ten test readings is .058 mg/l, with a standard deviation of .012 mg/l. Do the data provide sufficient evidence to indicate that the amount of impurities in the batch exceeds .05 mg/l? Find the p-value for the test and interpret its value. 10.71 Red Pine The main stem growth measured for a sample of seventeen 4-year-old red pine trees produced a mean and standard deviation equal to 11.3 and 3.4 inches, respectively. Find a 90% conﬁdence interval for the mean growth of a population of 4-year-old red pine trees subjected to similar environmental conditions. 10.72 Sodium Hydroxide The object of a general

chemistry experiment is to determine the amount (in milliliters) of sodium hydroxide (NaOH) solution needed to neutralize 1 gram of a speciﬁed acid. This will be an exact amount, but when the experiment is run in the laboratory, variation will occur as the result of experimental error. Three titrations are made using phenolphthalein as an indicator of the neutrality of the solution (pH equals 7 for a neutral solution). The three volumes of NaOH required to attain a pH of 7 in each

intake, obtained from a sample of 17 rats that had been injected with a sodium chloride solution, produced a mean and standard deviation of 31.0 and 6.2 cubic centimeters (cm3), respectively. Given that the average water intake for noninjected rats observed over a comparable period of time is 22.0 cm3, do the data indicate that injected rats drink more water than noninjected rats? Test at the 5% level of signiﬁcance. Find a 90% conﬁdence interval for the mean water intake for injected rats. 10.74 Sea Urchins An experimenter was interested in determining the mean thickness of the cortex of the sea urchin egg. The thickness was measured for n 10 sea urchin eggs. These measurements were obtained: 4.5 5.2

6.1 2.6

3.2 3.7

3.9 4.6

4.7 4.1

Estimate the mean thickness of the cortex using a 95% conﬁdence interval. 10.75 Fabricating Systems A production plant has two extremely complex fabricating systems; one system is twice as old as the other. Both systems are checked, lubricated, and maintained once every 2 weeks. The number of ﬁnished products fabricated daily by each of the systems is recorded for 30 working days. The results are given in the table. Do these data present sufficient evidence to conclude that the variability in daily production warrants increased maintenance of the older fabricating system? Use the p-value approach. New System

Old System

x苶1 246 s1 15.6

x苶2 240 s2 28.2

10.76 Fossils The data in the table are the diameters and heights of ten fossil specimens of a species of small shellﬁsh, Rotularia (Annelida) fallax, that were unearthed in a mapping expedition near the Antarctic Peninsula.12 The table gives an identiﬁcation symbol for the fossil specimen, the fossil’s diameter and height in millimeters, and the ratio of diameter to height.

EX1076

438

❍

Specimen

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

Height

D/H

10.80 Drug Absorption An experiment was con-

OSU 36651 OSU 36652 OSU 36653 OSU 36654 OSU 36655 OSU 36656 OSU 36657 OSU 36658 OSU 36659 OSU 36660

Diameter 185 194 173 200 179 213 134 191 177 199

78 65 77 76 72 76 75 77 69 65

2.37 2.98 2.25 2.63 2.49 2.80 1.79 2.48 2.57 3.06

ducted to compare the mean lengths of time required for the bodily absorption of two drugs A and B. Ten people were randomly selected and assigned to receive one of the drugs. The length of time (in minutes) for the drug to reach a speciﬁed level in the blood was recorded, and the data summary is given in the table:

x苶: s:

184.5 21.5

73 5

2.54 .37

a. Find a 95% confidence interval for the mean diameter of the species. b. Find a 95% confidence interval for the mean height of the species. c. Find a 95% confidence interval for the mean ratio of diameter to height. d. Compare the three intervals constructed in parts a, b, and c. Is the average of the ratios the same as the ratio of the average diameter to average height? 10.77 Fossils, continued Refer to Exercise 10.76

and data set EX1076. Suppose you want to estimate the mean diameter of the fossil specimens correct to within 5 millimeters with probability equal to .95. How many fossils do you have to include in your sample? 10.78 Alcohol and Reaction Times To test the effect of alcohol in increasing the reacEX1078 tion time to respond to a given stimulus, the reaction times of seven people were measured. After consuming 3 ounces of 40% alcohol, the reaction time for each of the seven people was measured again. Do the following data indicate that the mean reaction time after consuming alcohol was greater than the mean reaction time before consuming alcohol? Use a .05. Person

Before After

4 7

5 8

5 3

4 5

3 4

6 5

2 5

10.79 Cheese, Please Here are the prices per ounce of n 13 different brands of individually wrapped cheese slices:

EX1079

29.0 28.7 21.6

24.1 28.0 25.9

23.7 23.8 27.4

19.6 18.9

27.5 23.9

Construct a 95% conﬁdence interval estimate of the underlying average price per ounce of individually wrapped cheese slices.

Drug A

Drug B

x苶1 27.2 s 12 16.36

x苶2 33.5 s 22 18.92

a. Do the data provide sufficient evidence to indicate a difference in mean times to absorption for the two drugs? Test using a .05. b. Find the approximate p-value for the test. Does this value confirm your conclusions? c. Find a 95% confidence interval for the difference in mean times to absorption. Does the interval confirm your conclusions in part a? 10.81 Drug Absorption, continued Refer to Exer-

cise 10.80. Suppose you wish to estimate the difference in mean times to absorption correct to within 1 minute with probability approximately equal to .95. a. Approximately how large a sample is required for each drug (assume that the sample sizes are equal)? b. If conducting the experiment using the sample sizes of part a will require a large amount of time and money, can anything be done to reduce the sample sizes and still achieve the 1-minute margin of error for estimation? 10.82 Ring-Necked Pheasants The weights in grams of 10 males and 10 female juvenile ring-necked pheasants are given below.

EX1082

Males 1384 1286 1503 1627 1450

1672 1370 1659 1725 1394

Females 1073 1053 1038 1018 1146

1058 1123 1089 1034 1281

a. Use a statistical test to determine if the population variance of the weights of the male birds differs from that of the females. b. Test whether the average weight of juvenile male ring-necked pheasants exceeds that of the females by more than 300 grams. (HINT: The procedure that you use should take into account the results of the analysis in part a.)

SUPPLEMENTARY EXERCISES

Full Sun

10.83 Bees Insects hovering in ﬂight expend

enormous amounts of energy for their size and weight. The data shown here were taken from a much larger body of data collected by T.M. Casey and colleagues.13 They show the wing stroke frequencies (in hertz) for two different species of bees, n1 4 Euglossa mandibularis Friese and n2 6 Euglossa imperialis Cockerell.

EX1083

E. mandibularis Friese

E. imperialis Cockerell

235 225 190 188

180 169 180 185 178 182

a. Based on the observed ranges, do you think that a difference exists between the two population variances? b. Use an appropriate test to determine whether a difference exists. c. Explain why a Student’s t-test with a pooled estimator s 2 is unsuitable for comparing the mean wing stroke frequencies for the two species of bees. 10.84 Calcium The calcium (Ca) content of

a powdered mineral substance was analyzed 10 times with the following percent compositions recorded:

EX1084

.0271 .0271

.0282 .0281

.0279 .0269

.0281 .0275

.0268 .0276

a. Find a 99% confidence interval for the true calcium content of this substance. b. What does the phrase “99% confident” mean? c. What assumptions must you make about the sampling procedure so that this confidence interval will be valid? What does this mean to the chemist who is performing the analysis? 10.85 Sun or Shade? Karl Niklas and T.G. Owens examined the differences in a particular plant, Plantago Major L., when grown in full sunlight versus shade conditions.14 In this study, shaded plants received direct sunlight for less than 2 hours each day, whereas full-sun plants were never shaded. A partial summary of the data based on n1 16 full-sun plants and n2 15 shade plants is shown here:

Leaf Area (cm ) Overlap Area (cm2) Leaf Number Thickness (mm) Length (cm) Width (cm)

❍

439

Shade

x苶

苶x

128.00 46.80 9.75 .90 8.70 5.24

43.00 2.21 2.27 .03 1.64 .98

78.70 8.10 6.93 .50 8.91 3.41

41.70 1.26 1.49 .02 1.23 .61

a. What assumptions are required in order to use the small-sample procedures given in this chapter to compare full-sun versus shade plants? From the summary presented, do you think that any of these assumptions have been violated? b. Do the data present sufficient evidence to indicate a difference in mean leaf area for full-sun versus shade plants? c. Do the data present sufficient evidence to indicate a difference in mean overlap area for full-sun versus shade plants? 10.86 Orange Juice A comparison of the precisions of two machines developed for extracting juice from oranges is to be made using the following data: Machine A

Machine B

s 3.1 ounces n 25 2

s 2 1.4 ounces2 n 25

a. Is there sufficient evidence to indicate that there is a difference in the precision of the two machines at the 5% level of significance? b. Find a 95% confidence interval for the ratio of the two population variances. Does this interval confirm your conclusion from part a? Explain. 10.87 At Home or at School? Four sets of identical twins (pairs A, B, C, and D) were selected at random from a computer database of identical twins. One child was selected at random from each pair to form an “experimental group.” These four children were sent to school. The other four children were kept at home as a control group. At the end of the school year, the following IQ scores were obtained: Pair

Experimental Group

Control Group

A B C D

110 125 139 142

111 120 128 135

Does this evidence justify the conclusion that lack of school experience has a depressing effect on IQ scores? Use the p-value approach.

440 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

10.88 Dieting Eight obese persons were

placed on a diet for 1 month, and their weights, at the beginning and at the end of the month, were recorded:

EX1088

Weights Subjects

Initial

Final

1 2 3 4 5 6 7 8

310 295 287 305 270 323 277 299

263 251 249 259 233 267 242 265

10.89 Repair Costs Car manufacturers try

to design the bumpers of their automobiles to prevent costly damage in parking-lot type accidents. To compare repair costs of front versus back bumpers for several brands of cars, the cars were subject to a front and rear impacts at 5 mph, and the repair costs recorded.15

EX1089

Vehicle

Front

Rear

VW Jetta Daewoo Nubira Acura 3.4 RL Dodge Neon Nissan Sentra

$396 451 1123 687 583

$602 404 968 748 571

Do the data provide sufficient evidence to indicate that there is a signiﬁcant difference in average repair costs for front versus rear bumper repairs costs? Test using a .05. 10.90 Breathing Patterns Research psychologists measured the baseline breathing patterns—the total ventilation (in liters of air per minute) adjusted for body size—for each of n 30 patients, so that they could estimate the average total ventilation for patients before any experimentation was done. The data, along with some MINITAB output, are presented here:

EX1090

5.72 4.79 6.04 5.38 5.17

5.77 5.16 5.83 5.48 6.34

Stem-and-Leaf Display: Ltrs/min Stem-and-leaf of Ltrs/min Leaf Unit = 0.10 1 2 5 8 12 (4) 14 11 7 4 2 1

4 4 4 4 5 5 5 5 5 6 6 6

N = 30

3 5 677 899 1111 2333 455 6777 889 01 3 5

Descriptive Statistics: Ltrs/min

Estimate the mean weight loss for obese persons when placed on the diet for a 1-month period. Use a 95% conﬁdence interval and interpret your results. What assumptions must you make so that your inference is valid?

5.23 5.54 5.92 4.72 4.67

MINITAB output for Exercise 10.90

4.99 5.84 5.32 5.37 6.58

5.12 4.51 6.19 4.96 4.35

4.82 5.14 5.70 5.58 5.63

Variable Ltrs/min Minimum 4.3500

N 30

N* 0

Q1 4.9825

Mean 5.3953

SE Mean 0.0997

StDev 0.5462

Median 5.3750

Q3 5.7850

Maximum 6.5800

a. What information does the stem and leaf plot give you about the data? Why is this important? b. Use the MINITAB output to construct a 99% conﬁdence interval for the average total ventilation for patients. 10.91 Reaction Times A comparison of reaction times (in seconds) for two different stimuli in a psychological word-association experiment produced the following results when applied to a random sample of 16 people: Stimulus 1

Stimulus 2

Do the data present sufficient evidence to indicate a difference in mean reaction times for the two stimuli? Test using a .05. 10.92 Reaction Times II Refer to Exercise 10.91. Suppose that the word-association experiment is conducted using eight people as blocks and making a comparison of reaction times within each person; that is, each person is subjected to both stimuli in a random order. The reaction times (in seconds) for the experiment are as follows: Person

Stimulus 1

Stimulus 2

1 2 3 4 5 6 7 8

3 1 1 2 1 2 3 2

4 2 3 1 2 3 3 3

SUPPLEMENTARY EXERCISES

Do the data present sufficient evidence to indicate a difference in mean reaction times for the two stimuli? Test using a .05. 10.93 Refer to Exercises 10.91 and 10.92. Calculate a 95% conﬁdence interval for the difference in the two population means for each of these experimental designs. Does it appear that blocking increased the amount of information available in the experiment? 10.94 Impact Strength The following data are readings (in foot-pounds) of the impact EX1094 strengths of two kinds of packaging material: A

1.25 1.16 1.33 1.15 1.23 1.20 1.32 1.28 1.21

.89 1.01 .97 .95 .94 1.02 .98 1.06 .98

❍

441

a. Do the data present sufficient evidence to indicate a difference between the average densities of cakes prepared using the two types of batter? b. Construct a 95% conﬁdence interval for the difference between the average densities for the two mixes. 10.96 Under what assumptions can the F distribution

be used in making inferences about the ratio of population variances? 10.97 Got Milk? A dairy is in the market for a new

container-filling machine and is considering two models, manufactured by company A and company B. Ruggedness, cost, and convenience are comparable in the two models, so the deciding factor is the variability of fills. The model that produces fills with the smaller variance is preferred. If you obtain samples of fills for each of the two models, an F-test can be used to test for the equality of population variances. Which type of rejection region would be most favored by each of these individuals? a. The manager of the dairy—Why? b. A sales representative for company A—Why? c. A sales representative for company B—Why?

MINITAB output for Exercise 10.94

Two-Sample T-Test and CI: A, B Two-sample T for A vs B N Mean StDev SE Mean A 9 1.2367 0.0644 0.021 B 9 0.9778 0.0494 0.016 Difference = mu (A) - mu (B) Estimate for difference: 0.2589 95% CI for difference: (0.2015, 0.3163) T-Test of difference = 0 (vs not =): T-Value = 9.56 P-Value = 0.000 DF = 16 Both use Pooled StDev = 0.0574

a. Use the MINITAB printout to determine whether there is evidence of a difference in the mean strengths for the two kinds of material. b. Are there practical implications to your results? 10.95 Cake Mixes An experiment was conducted to compare the densities (in ounces per cubic inch) of cakes prepared from two different cake mixes. Six cake pans were filled with batter A, and six were filled with batter B. Expecting a variation in oven temperature, the experimenter placed a pan filled with batter A and another with batter B side by side at six different locations in the oven. The six paired observations of densities are as follows: Batter A

.135

.102

.098

.141

.131

.144

Batter B

.129

.120

.112

.152

.135

.163

10.98 Got Milk II Refer to Exercise 10.97. Wishing to demonstrate that the variability of fills is less for her model than for her competitor’s, a sales representative for company A acquired a sample of 30 fills from her company’s model and a sample of 10 fills from her competitor’s model. The sample variances were sA2 .027 and sB2 .065, respectively. Does this result provide statistical support at the .05 level of significance for the sales representative’s claim? 10.99 Chemical Purity A chemical manufacturer

claims that the purity of his product never varies by more than 2%. Five batches were tested and given purity readings of 98.2, 97.1, 98.9, 97.7, and 97.9%. a. Do the data provide sufficient evidence to contradict the manufacturer’s claim? (HINT: To be generous, let a range of 2% equal 4s.) b. Find a 90% conﬁdence interval for s 2. 10.100 16-Ounce Cans? A cannery prints “weight

16 ounces” on its label. The quality control supervisor selects nine cans at random and weighs them. She ﬁnds 苶x 15.7 and s .5. Do the data present sufficient evidence to indicate that the mean weight is less than that claimed on the label?

442

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

10.101 Reaction Time III A psychologist wishes to

Sea Level

12,000 Feet

verify that a certain drug increases the reaction time to a given stimulus. The following reaction times (in tenths of a second) were recorded before and after injection of the drug for each of four subjects:

.07 .10 .09 .12 .09 .13

.13 .17 .15 .14 .10 .14

Reaction Time Subject

Before

After

1 2 3 4

7 2 12 12

13 3 18 13

Test at the 5% level of significance to determine whether the drug significantly increases reaction time. 10.102 Food Production At a time when energy conservation is so important, some scientists think closer scrutiny should be given to the cost (in energy) of producing various forms of food. Suppose you wish to compare the mean amount of oil required to produce 1 acre of corn versus 1 acre of cauliflower. The readings (in barrels of oil per acre), based on 20-acre plots, seven for each crop, are shown in the table. Use these data to find a 90% confidence interval for the difference between the mean amounts of oil required to produce these two crops.

EX10102

Corn

Cauliﬂower

5.6 7.1 4.5 6.0 7.9 4.8 5.7

15.9 13.4 17.6 16.8 15.8 16.3 17.1

10.103 Alcohol and Altitude The effect of alcohol

consumption on the body appears to be much greater at high altitudes than at sea level. To test this theory, a scientist randomly selects 12 subjects and randomly divides them into two groups of six each. One group is put into a chamber that simulates conditions at an altitude of 12,000 feet, and each subject ingests a drink containing 100 cubic centimeters (cc) of alcohol. The second group receives the same drink in a chamber that simulates conditions at sea level. After 2 hours, the amount of alcohol in the blood (grams per 100 cc) for each subject is measured. The data are shown in the table. Do the data provide sufficient evidence to support the theory that retention of alcohol in the blood is greater at high altitudes?

10.104 Stock Risks The closing prices of two common stocks were recorded for a period of 15 days. The means and variances are x苶1 40.33 s 12 1.54

x苶2 42.54 s 22 2.96

a. Do these data present sufficient evidence to indicate a difference between the variabilities of the closing prices of the two stocks for the populations associated with the two samples? Give the p-value for the test and interpret its value. b. Construct a 99% conﬁdence interval for the ratio of the two population variances. 10.105 Auto Design An experiment is conducted to compare two new automobile designs. Twenty people are randomly selected, and each person is asked to rate each design on a scale of 1 (poor) to 10 (excellent). The resulting ratings will be used to test the null hypothesis that the mean level of approval is the same for both designs against the alternative hypothesis that one of the automobile designs is preferred. Do these data satisfy the assumptions required for the Student’s t-test of Section 10.4? Explain. 10.106 Safety Programs The data shown here were collected on lost-time accidents (the ﬁgures given are mean work-hours lost per month over a period of 1 year) before and after an industrial safety program was put into effect. Data were recorded for six industrial plants. Do the data provide sufficient evidence to indicate whether the safety program was effective in reducing lost-time accidents? Test using a .01. Plant Number

Before Program After Program

38 31

64 58

42 43

70 65

58 52

30 29

10.107 Two Different Entrees To compare the demand for two different entrees, the manager of a cafeteria recorded the number of purchases of each entree on seven consecutive days. The data are shown in the table. Do the data provide

EX10107

SUPPLEMENTARY EXERCISES

sufficient evidence to indicate a greater mean demand for one of the entrees? Use the MINITAB printout. Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

420 374 434 395 637 594 679

391 343 469 412 538 521 625

❍

443

a. Is there sufficient evidence to reject his claim at the a .05 level of signiﬁcance? b. Find a 95% conﬁdence interval for the variance of the rod diameters. 10.111 Sleep and the College Student How much sleep do you get on a typical school night? A group of 10 college students were asked to report the number of hours that they slept on the previous night with the following results:

MINITAB output for Exercise 10.107

Paired T-Test and CI: A, B

a. Find a 99% conﬁdence interval for the average number of hours that college students sleep. b. What assumptions are required in order for this conﬁdence interval to be valid?

Paired T for A - B N Mean A 7 504.7 B 7 471.3 Difference 7 33.4

StDev 127.2 97.4 47.5

SE Mean 48.1 36.8 18.0

95% CI for mean difference: (-10.5, 77.4) T-Test of mean difference = 0 (vs not = 0): T-Value = 1.86 P-Value = 0.112

10.108 Pollution Control The EPA limit on the

allowable discharge of suspended solids into rivers and streams is 60 milligrams per liter (mg/l) per day. A study of water samples selected from the discharge at a phosphate mine shows that over a long period, the mean daily discharge of suspended solids is 48 mg/l, but day-to-day discharge readings are variable. State inspectors measured the discharge rates of suspended solids for n 20 days and found s 2 39 (mg/l)2. Find a 90% conﬁdence interval for s 2. Interpret your results. 10.109 Enzymes Two methods were used to mea-

sure the speciﬁc activity (in units of enzyme activity per milligram of protein) of an enzyme. One unit of enzyme activity is the amount that catalyzes the formation of 1 micromole of product per minute under speciﬁed conditions. Use an appropriate test or estimation procedure to compare the two methods of measurement. Comment on the validity of any assumptions you need to make. Method 1

125

137

130

151

142

Method 2

137

143

151

156

149

10.110 Connector Rods A producer of machine

parts claimed that the diameters of the connector rods produced by his plant had a variance of at most .03 inch2. A random sample of 15 connector rods from his plant produced a sample mean and variance of .55 inch and .053 inch2, respectively.

7.25,

8.5,

6.75,

10.112 Arranging Objects The following EX10112

5.2 4.2 3.1 3.6 4.7

data are the response times in seconds for n 25 ﬁrst graders to arrange three objects by size. 3.8 4.1 2.5 3.9 3.3

5.7 4.3 3.0 4.8 4.2

3.9 4.7 4.4 5.3 3.8

3.7 4.3 4.8 4.2 5.4

Find a 95% conﬁdence interval for the average response time for ﬁrst graders to arrange three objects by size. Interpret this interval. 10.113 Finger-Lickin’ Good! Maybe too good,

according to tests performed by the consumer testing division of Good Housekeeping. Nutritional information provided by Kentucky Fried Chicken claims that each small bag of Potato Wedges contains 4.8 ounces of food, for a total of 280 calories. A sample of 10 orders from KFC restaurants in New York and New Jersey averaged 358 calories.16 If the standard deviation of this sample was s 54, is there sufficient evidence to indicate that the average number of calories in small bags of KFC Potato Wedges is greater than advertised? Test at the 1% level of signiﬁcance. 10.114 Mall Rats An article in American Demo-

graphics investigated consumer habits at the mall. We tend to spend the most money shopping on the weekends, and, in particular, on Sundays from 4 to 6 P.M. Wednesday morning shoppers spend the least!17 Suppose that a random sample of 20 weekend shoppers and a random sample of 20 weekday shoppers were selected, and the amount spent per trip to the mall was recorded.

444

❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

Sample Size Sample Mean Sample Standard Deviation

Weekends

Weekdays

20 $78 $22

20 $67 $20

a. Is it reasonable to assume that the two population variances are equal? Use the F-test to test this hypothesis with a .05. b. Based on the results of part a, use the appropriate test to determine whether there is a difference in the average amount spent per trip on weekends versus weekdays. Use a .05. 10.115 Border Wars As the costs of pre-

scription drugs escalate, more and more senior citizens are ordering prescriptions from Canada, or actually crossing the border to buy prescription drugs. The price of a typical prescription for nine best-selling

EX10115

drugs was recorded at randomly selected stores in both the United States and in Canada.18 Drug

U.S.

Canada

Lipitor® Zocor® Prilosec® Norvasc® Zyprexa® Paxil® Prevacid® Celebrex® Zoloft®

$290 412 117 139 571 276 484 161 235

$179 211 72 125 396 171 196 67 156

a. Is there sufficient evidence to indicate that the average cost of prescription drugs in the United States is different from the average cost in Canada? Use a .01. b. What is the approximate the p-value for this test? Does this conﬁrm your conclusions in part a?

Exercises 10.116 Use the Student’s t Probabilities applet to find the following probabilities: a. P(t 1.2) with 5 df b. P(t 2) P(t 2) with 10 df c. P(t 3.3) with 8 df d. P(t .6) with 12 df 10.117 Use the Student’s t Probabilities applet to find the following critical values: a. an upper one-tailed rejection region with a .05 and 11 df b. a two-tailed rejection region with a .05 and 7 df c. a lower one-tailed rejection region with a .01 and 15 df 10.118 Refer to the Interpreting Confidence Intervals applet. a. Suppose that you have a random sample of size n 10 from a population with unknown mean m. What formula would you use to construct a 95% confidence interval for the unknown population mean? b. Use the button in the first applet to create a single 95% confidence interval for m. Use the formula in part a and the information given in the

applet to verify the confidence limits provided. (The applet rounds to the nearest integer.) Did this confidence interval enclose the true value, m 100? 10.119 Refer to the Interpreting Confidence Intervals applet. a. Use the button in the first applet to create ten 95% confidence intervals for m. b. Are the widths of these intervals all the same? Explain why or why not. c. How many of the intervals work properly and enclose the true value of m? d. Try this simulation again by clicking the button a few more times and counting the number of intervals that work correctly. Is it close to our 95% confidence level? e. Use the button in the second applet to create ten 99% confidence intervals for m. How many of these intervals work properly? 10.120 Refer to the Interpreting Confidence Intervals applet. a. Use the button to create one hundred 95% confidence intervals for m. How many of the

CASE STUDY

intervals work properly and enclose the true value of m? b. Repeat the instructions of part a to construct 99% confidence intervals. How many of the intervals work properly and enclose the true value of m? c. Try this simulation again by clicking the button a few more times and counting the number of intervals that work correctly. Use both the 95% and 99% confidence intervals. Do the percentage of intervals that work come close to our 95% and 99% confidence levels? 10.121 A random sample of n 12 observations

from a normal population produced x苶 47.1 and s 2 4.7. Test the hypothesis H0: m 48 against Ha: m 48. Use the Small-Sample Test of a Population Mean applet and a 5% signiﬁcance level. 10.122 SAT Scores In Exercise 9.73, we reported that the national average SAT scores for the class of 2005 were 508 on the verbal portion and 520 on the math portion. Suppose that we have a small random sample of 15 California students in the class of 2005; their SAT scores are recorded in the following table. Sample Average Sample Standard Deviation

Verbal

Math

499 98

516 96

a. Use the Small-Sample Test of a Population Mean applet. Do the data provide sufficient evidence to indicate that the average verbal score for all California students in the class of 2005 is different from the national average? Test using a .05.

CASE STUDY Flextime

❍

445

b. Use the Small-Sample Test of a Population Mean applet. Do the data provide sufficient evidence to indicate that the average math score for all California students in the class of 2005 is different from the national average? Test using a .05. 10.123 Surgery Recovery Times The length of time to recovery was recorded for patients randomly assigned and subjected to two different surgical procedures. The data (recorded in days) are as follows:

Sample Average Sample Variance Sample Size

Procedure I

Procedure II

7.3 1.23 11

8.9 1.49 13

Do the data present sufficient evidence to indicate a difference between the mean recovery times for the two surgical procedures? Perform the test of hypothesis, calculating the test statistic and the approximate p-value by hand. Then check your results using the Two-Sample T-Test: Independent Samples applet. 10.124 Stock Prices Refer to Exercise 10.104 in which we reported the closing prices of two common stocks, recorded over a period of 15 days. x苶1 40.33 s 12 1.54

x苶2 42.54 s 22 2.96

Use the Two-Sample T-Test: Independent Samples applet. Do the data provide sufficient evidence to indicate that the average prices of the two common stocks are different? Use the p-value to access the signiﬁcance of the test.

How Would You Like a Four-Day Workweek? Will a flexible workweek schedule result in positive benefits for both employer and employee? Is a more rested employee, who spends less time commuting to and from work, likely to be more efficient and take less time off for sick leave and personal leave? A report on the benefits of flexible work schedules that appeared in Environmental Health looked at the records of n 11 employees who worked in a satellite office in a county health department in Illinois under a 4-day workweek schedule.19 Employees worked a conventional workweek in year 1 and a 4-day workweek in year 2. Some statistics for these employees are shown in the following table:

446 ❍

CHAPTER 10 INFERENCE FROM SMALL SAMPLES

Personal Leave

Sick Leave

Employee

Year 2

Year 1

Year 2

Year 1

1 2 3 4 5 6 7 8 9 10 11

26 18 24 19 17 34 19 18 9 36 26

33 37 20 26 1 2 13 22 22 13 18

30 61 59 2 79 63 71 83 35 81 79

37 45 56 9 92 65 21 62 26 73 21

1. A 4-day workweek ensures that employees will have one more day that need not be spent at work. One possible result is a reduction in the average number of personal-leave days taken by employees on a 4-day work schedule. Do the data indicate that this is the case? Use the p-value approach to testing to reach your conclusion. 2. A 4-day workweek schedule might also have an effect on the average number of sick-leave days an employee takes. Should a directional alternative be used in this case? Why or why not? 3. Construct a 95% conﬁdence interval to estimate the average difference in days taken for sick leave between these 2 years. What do you conclude about the difference between the average number of sick-leave days for these two work schedules? 4. Based on the analysis of these two variables, what can you conclude about the advantages of a 4-day workweek schedule?

Case Study from “Four-Day Work Week Improves Environment,” by C.S. Catlin, Environmental Health, Vol. 59, No. 7, March 1997. Copyright 1997 National Environmental Health Association. Reprinted by permission.

The Analysis of Variance

GENERAL OBJECTIVE The quantity of information contained in a sample is affected by various factors that the experimenter may or may not be able to control. This chapter introduces three different experimental designs, two of which are direct extensions of the unpaired and paired designs of Chapter 10. A new technique called the analysis of variance is used to determine how the different experimental factors affect the average response.

CHAPTER INDEX ● The analysis of variance (11.2) ● The completely randomized design (11.4, 11.5) ● Factorial experiments (11.9, 11.10) ● The randomized block design (11.7, 11.8) ● Tukey’s method of paired comparisons (11.6)

How Do I Know Whether My Calculations Are Accurate?

“A Fine Mess” Do you risk a fine by parking your car in red zones or next to fire hydrants? Do you fail to put enough money in a parking meter? If so, you are among the thousands of drivers who receive parking tickets every day in almost every city in the United States. Depending on the city in which you receive a ticket, your fine can be as little as $8 for overtime parking in San Luis Obispo, California, or as high as $340 for illegal parking in a handicapped space in San Diego, California. The case study at the end of this chapter statistically analyzes the variation in parking fines in southern California cities.

447

448 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

11.1

THE DESIGN OF AN EXPERIMENT The way that a sample is selected is called the sampling plan or experimental design and determines the amount of information in the sample. Some research involves an observational study, in which the researcher does not actually produce the data but only observes the characteristics of data that already exist. Most sample surveys, in which information is gathered with a questionnaire, fall into this category. The researcher forms a plan for collecting the data—called the sampling plan—and then uses the appropriate statistical procedures to draw conclusions about the population or populations from which the sample comes. Other research involves experimentation. The researcher may deliberately impose one or more experimental conditions on the experimental units in order to determine their effect on the response. Here are some new terms we will use to discuss the design of a statistical experiment. Definition An experimental unit is the object on which a measurement (or measurements) is taken. A factor is an independent variable whose values are controlled and varied by the experimenter. A level is the intensity setting of a factor. A treatment is a speciﬁc combination of factor levels. The response is the variable being measured by the experimenter.

EXAMPLE

11.1

A group of people is randomly divided into an experimental and a control group. The control group is given an aptitude test after having eaten a full breakfast. The experimental group is given the same test without having eaten any breakfast. What are the factors, levels, and treatments in this experiment? Solution The experimental units are the people on which the response (test score) is measured. The factor of interest could be described as “meal” and has two levels: “breakfast” and “no breakfast.” Since this is the only factor controlled by the experimenter, the two levels—“breakfast” and “no breakfast”—also represent the treatments of interest in the experiment.

EXAMPLE

11.2

Suppose that the experimenter in Example 11.1 began by randomly selecting 20 men and 20 women for the experiment. These two groups were then randomly divided into 10 each for the experimental and control groups. What are the factors, levels, and treatments in this experiment? Now there are two factors of interest to the experimenter, and each factor has two levels:

Solution

• •

“Gender” at two levels: men and women “Meal” at two levels: breakfast and no breakfast

In this more complex experiment, there are four treatments, one for each speciﬁc combination of factor levels: men without breakfast, men with breakfast, women without breakfast, and women with breakfast.

11.3 THE ASSUMPTIONS FOR AN ANALYSIS OF VARIANCE

❍

449

In this chapter, we will concentrate on experiments that have been designed in three different ways, and we will use a technique called the analysis of variance to judge the effects of various factors on the experimental response. Two of these experimental designs are extensions of the unpaired and paired designs from Chapter 10.

WHAT IS AN ANALYSIS OF VARIANCE?

11.2

The responses that are generated in an experimental situation always exhibit a certain amount of variability. In an analysis of variance, you divide the total variation in the response measurements into portions that may be attributed to various factors of interest to the experimenter. If the experiment has been properly designed, these portions can then be used to answer questions about the effects of the various factors on the response of interest. You can better understand the logic underlying an analysis of variance by looking at a simple experiment. Consider two sets of samples randomly selected from populations 1 (䉬) and 2 (䊊), each with identical pairs of means, 苶x1 and x苶2. The two sets are shown in Figure 11.1. Is it easier to detect the difference in the two means when you look at set A or set B? You will probably agree that set A shows the difference much more clearly. In set A, the variability of the measurements within the groups (䉬s and 䊊s) is much smaller than the variability between the two groups. In set B, there is more variability within the groups (䉬s and 䊊s), causing the two groups to “mix” together and making it more difficult to see the identical difference in the means. FIGU R E 1 1 . 1

Two sets of samples with the same means

●

Set A

Set B

The comparison you have just done intuitively is formalized by the analysis of variance. Moreover, the analysis of variance can be used not only to compare two means but also to make comparisons of more than two population means and to determine the effects of various factors in more complex experimental designs. The analysis of variance relies on statistics with sampling distributions that are modeled by the F distribution of Section 10.7.

11.3

THE ASSUMPTIONS FOR AN ANALYSIS OF VARIANCE The assumptions required for an analysis of variance are similar to those required for the Student’s t and F statistics of Chapter 10. Regardless of the experimental design used to generate the data, you must assume that the observations within each treatment group are normally distributed with a common variance s 2. As in Chapter 10, the analysis of variance procedures are fairly robust when the sample sizes are equal and when the data are fairly mound-shaped. Violating the assumption of a common variance is more serious, especially when the sample sizes are not nearly equal.

450 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

ASSUMPTIONS FOR ANALYSIS OF VARIANCE TEST AND ESTIMATION PROCEDURES • •

The observations within each population are normally distributed with a common variance s 2. Assumptions regarding the sampling procedure are speciﬁed for each design in the sections that follow.

This chapter describes the analysis of variance for three different experimental designs. The ﬁrst design is based on independent random sampling from several populations and is an extension of the unpaired t-test of Chapter 10. The second is an extension of the paired-difference or matched pairs design and involves a random assignment of treatments within matched sets of observations. The third is a design that allows you to judge the effect of two experimental factors on the response. The sampling procedures necessary for each design are restated in their respective sections.

11.4

THE COMPLETELY RANDOMIZED DESIGN: A ONE-WAY CLASSIFICATION One of the simplest experimental designs is the completely randomized design, in which random samples are selected independently from each of k populations. This design involves only one factor, the population from which the measurement comes— hence the designation as a one-way classiﬁcation. There are k different levels corresponding to the k populations, which are also the treatments for this one-way classiﬁcation. Are the k population means all the same, or is at least one mean different from the others? Why do you need a new procedure, the analysis of variance, to compare the population means when you already have the Student’s t-test available? In comparing k 3 means, you could test each of three pairs of hypotheses: H0 : m1 m2

H0 : m1 m3

H0 : m2 m3

to ﬁnd out where the differences lie. However, you must remember that each test you perform is subject to the possibility of error. To compare k 4 means, you would need six tests, and you would need 10 tests to compare k 5 means. The more tests you perform on a set of measurements, the more likely it is that at least one of your conclusions will be incorrect. The analysis of variance procedure provides one overall test to judge the equality of the k population means. Once you have determined whether there is actually a difference in the means, you can use another procedure to ﬁnd out where the differences lie. How can you select these k random samples? Sometimes the populations actually exist in fact, and you can use a computerized random number generator or a random number table to randomly select the samples. For example, in a study to compare the average sizes of health insurance claims in four different states, you could use a computer database provided by the health insurance companies to select random samples from the four states. In other situations, the populations may be hypothetical, and responses can be generated only after the experimental treatments have been applied. EXAMPLE

11.3

A researcher is interested in the effects of five types of insecticides for use in controlling the boll weevil in cotton fields. Explain how to implement a completely randomized design to investigate the effects of the five insecticides on crop yield.

11.5 THE ANALYSIS OF VARIANCE FOR A COMPLETELY RANDOMIZED DESIGN

❍

451

The only way to generate the equivalent of five random samples from the hypothetical populations corresponding to the five insecticides is to use a method called a randomized assignment. A fixed number of cotton plants are chosen for treatment, and each is assigned a random number. Suppose that each sample is to have an equal number of measurements. Using a randomization device, you can assign the first n plants chosen to receive insecticide 1, the second n plants to receive insecticide 2, and so on, until all five treatments have been assigned. Solution

Whether by random selection or random assignment, both of these examples result in a completely randomized design, or one-way classiﬁcation, for which the analysis of variance is used.

THE ANALYSIS OF VARIANCE FOR A COMPLETELY RANDOMIZED DESIGN

11.5

FIGU R E 1 1 . 2

Normal populations with a common variance but different means

Suppose you want to compare k population means, m1, m2, . . . , mk, based on independent random samples of size n1, n2, . . . , nk from normal populations with a common variance s 2. That is, each of the normal populations has the same shape, but their locations might be different, as shown in Figure 11.2.

●

... µ1

µ2

µk

Partitioning the Total Variation in an Experiment Let xij be the jth measurement ( j 1, 2, . . . , ni) in the ith sample. The analysis of variance procedure begins by considering the total variation in the experiment, which is measured by a quantity called the total sum of squares (TSS): (Sxij )2 Total SS S(xij 苶x )2 Sx 2ij n This is the familiar numerator in the formula for the sample variance for the entire set of n n1 n2 nk measurements. The second part of the calculational formula is sometimes called the correction for the mean (CM). If we let G represent the grand total of all n observations, then (Sxij )2 G2 CM n n This Total SS is partitioned into two components. The ﬁrst component, called the sum of squares for treatments (SST), measures the variation among the k sample means: T2 x )2 Si CM SST Sni ( 苶xi 苶 ni

452 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

where Ti is the total of the observations for treatment i. The second component, called the sum of squares for error (SSE), is used to measure the pooled variation within the k samples: SSE (n1 1)s12 (n2 1)s 22 (nk 1)s 2k This formula is a direct extension of the numerator in the formula for the pooled estimate of s 2 from Chapter 10. We can show algebraically that, in the analysis of variance, Total SS SST SSE Therefore, you need to calculate only two of the three sums of squares—Total SS, SST, and SSE—and the third can be found by subtraction. Each of the sources of variation, when divided by its appropriate degrees of freedom, provides an estimate of the variation in the experiment. Since Total SS involves n squared observations, its degrees of freedom are df (n 1). Similarly, the sum of squares for treatments involves k squared observations, and its degrees of freedom are df (k 1). Finally, the sum of squares for error, a direct extension of the pooled estimate in Chapter 10, has df (n1 1) (n2 1) (nk 1) n k Notice that the degrees of freedom for treatments and error are additive—that is, df (total) df (treatments) df(error) These two sources of variation and their respective degrees of freedom are combined to form the mean squares as MS SS/df. The total variation in the experiment is then displayed in an analysis of variance (or ANOVA) table. ANOVA TABLE FOR k INDEPENDENT RANDOM SAMPLES: COMPLETELY RANDOMIZED DESIGN The column labeled “SS” satisﬁes: Total SS SST SSE.

Source

Treatments Error

k1 nk

SST SSE

MST SST/(k 1) MSE SSE/(n k)

MST/MSE

Total

Total SS

where Total SS Sx 2ij CM (Sum of squares of all x-values) CM with (Sxij )2 G2 CM n n The column labeled “df” always adds up to n 1.

T2 SST Si CM ni

SST MST k1

SSE Total SS SST

SSE MSE nk

and

G Grand total of all n observations Ti Total of all observations in sample i ni Number of observations in sample i n n1 n2 nk

11.5 THE ANALYSIS OF VARIANCE FOR A COMPLETELY RANDOMIZED DESIGN

EXAMPLE

TABLE 11.1

11.4

❍

453

In an experiment to determine the effect of nutrition on the attention spans of elementary school students, a group of 15 students were randomly assigned to each of three meal plans: no breakfast, light breakfast, and full breakfast. Their attention spans (in minutes) were recorded during a morning reading period and are shown in Table 11.1. Construct the analysis of variance table for this experiment. ●

Attention Spans of Students After Three Meal Plans No Breakfast

Light Breakfast

Full Breakfast

8 7 9 13 10

14 16 12 17 11

10 12 16 15 12

T1 47

T2 70

T3 65

To use the calculational formulas, you need the k 3 treatment totals together with n1 n2 n3 5, n 15, and S xij 182. Then

Solution

(182)2 CM 2208.2667 15 Total SS (82 72 122 ) CM 2338 2208.2667 129.7333 with (n 1) (15 1) 14 degrees of freedom, 472 702 652 SST CM 2266.8 2208.2667 58.5333 5 with (k 1) (3 1) 2 degrees of freedom, and by subtraction, SSE Total SS SST 129.7333 58.5333 71.2 with (n k) (15 3) 12 degrees of freedom. These three sources of variation, their degrees of freedom, sums of squares, and mean squares are shown in the shaded area of the ANOVA table generated by MINITAB and given in Figure 11.3. You will ﬁnd instructions for generating this output in the “My MINITAB ” section at the end of this chapter. F IGU R E 1 1 . 3

MINITAB output for Example 11.4

● One-way ANOVA: Span versus Meal Source Meal Error Total

DF 2 12 14

S = 2.436

Level 1 2 3

N 5 5 5

SS 58.53 71.20 129.73

MS 29.27 5.93

R-Sq = 45.12%

Mean 9.400 14.000 13.000

StDev 2.302 2.550 2.449

Pooled StDev = 2.436

F 4.93

P 0.027

R-Sq(adj) = 35.97% Individual 95% CIs For Mean Based on Pooled StDev --+---------+---------+---------+------(---------*--------) (--------*--------) (--------*--------) --+---------+---------+---------+------7.5 10.0 12.5 15.0

454 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

The MINITAB output gives some additional information about the variation in the experiment. The second section shows the means and standard deviations for the three meal plans. More important, you can see in the ﬁrst section of the printout two columns marked “F” and “P.” We can use these values to test a hypothesis concerning the equality of the three treatment means.

Testing the Equality of the Treatment Means The mean squares in the analysis of variance table can be used to test the null hypothesis H0 : m1 m2 mk MS SS/df

versus the alternative hypothesis Ha : At least one of the means is different from the others using the following theoretical argument: •

Remember that s 2 is the common variance for all k populations. The quantity SSE MSE nk

•

F I GU R E 1 1 . 4

Sample means drawn from identical versus different populations

●

is a pooled estimate of s 2, a weighted average of all k sample variances, whether or not H0 is true. If H0 is true, then the variation in the sample means, measured by MST [SST/(k 1)], also provides an unbiased estimate of s 2. However, if H0 is false and the population means are different, then MST—which measures the variation in the sample means—will be unusually large, as shown in Figure 11.4.

H0 true

x1 x2

H0 false

µ1

µ2 x2

µ3

µ1 = µ2= µ3

•

The test statistic MST F MSE

F-tests for ANOVA tables are always upper (right) tailed.

tends to be larger than usual if H0 is false. Hence, you can reject H0 for large values of F, using a right-tailed statistical test. When H0 is true, this test statistic has an F distribution with df1 (k 1) and df2 (n k) degrees of freedom, and right-tailed critical values of the F distribution (from Table 6 in Appendix I) or computer-generated p-values can be used to draw statistical conclusions about the equality of the population means.

11.5 THE ANALYSIS OF VARIANCE FOR A COMPLETELY RANDOMIZED DESIGN

❍

455

F TEST FOR COMPARING k POPULATION MEANS 1. Null hypothesis: H0 : m1 m2 mk 2. Alternative hypothesis: Ha : One or more pairs of population means differ 3. Test statistic: F MST/MSE, where F is based on df1 (k 1) and df2 (n k) 4. Rejection region: Reject H0 if F Fa, where Fa lies in the upper tail of the F distribution (with df1 k 1 and df2 n k) or if the p-value a. f(F)

α Fα

Assumptions • •

EXAMPLE

11.5

The samples are randomly and independently selected from their respective populations. The populations are normally distributed with means m1, m2, . . . , mk and equal variances, s 12 s 22 s 2k s 2.

Do the data in Example 11.4 provide sufficient evidence to indicate a difference in the average attention spans depending on the type of breakfast eaten by the student? To test H0 : m1 m2 m3 versus the alternative hypothesis that the average attention span is different for at least one of the three treatments, you use the analysis of variance F statistic, calculated as

Solution

MST 29.2667 F 4.93 MSE 5.9333 and shown in the column marked “F” in Figure 11.3. It will not surprise you to know that the value in the column marked “P” in Figure 11.3 is the exact p-value for this statistical test. The test statistic MST/MSE calculated above has an F distribution with df1 2 and df2 12 degrees of freedom. Using the critical value approach with a .05, you can reject H0 if F F.05 3.89 from Table 6 in Appendix I (see Figure 11.5). Since the observed value, F 4.93, exceeds the critical value, you reject H0. There is sufﬁcient evidence to indicate that at least one of the three average attention spans is different from at least one of the others.

456 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

FI GU R E 1 1 . 5

Rejection region for Example 11.5

●

f(F ) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

= .05

F 0

5 3.89

10 Rejection region

You could have reached this same conclusion using the exact p-value, P .027, given in Figure 11.3. Since the p-value is less than a .05, the results are statistically signiﬁcant at the 5% level. You still conclude that at least one of the three average attention spans is different from at least one of the others.

Computer printouts give the exact p-value—use the p-value to make your decision.

You can use the F Probabilities applet to find critical values of F or p-values for the analysis of variance F-test. Look at the two applets in Figure 11.6. Use the sliders on the left and right of the applets to select the appropriate degrees of freedom (df1 and df2). To find the critical value for rejection of H0, enter the significance level a in the box marked “Prob” and press Enter. To find the p-value, enter the observed value of the test statistic in the box marked “F” and press Enter. Can you identify the critical value for rejection and the p-value for Example 11.5? FI GU R E 1 1 . 6

F Probabilities applet

● F Distribution

F Distribution

Estimating Differences in the Treatment Means The next obvious question you might ask involves the nature of the differences in the population means. Which means are different from the others? How can you estimate the difference, or possibly the individual means for each of the three treatments?

11.5 THE ANALYSIS OF VARIANCE FOR A COMPLETELY RANDOMIZED DESIGN

❍

457

In Section 11.6, we will present a procedure that you can use to compare all possible pairs of treatment means simultaneously. However, if you have a special interest in a particular mean or pair of means, you can construct conﬁdence intervals using the small-sample procedures of Chapter 10, based on the Student’s t distribution. For a single population mean, mi, the conﬁdence interval is s 苶xi ta/2 兹n苶i

冢冣

where x苶i is the sample mean for the i th treatment. Similarly, for a comparison of two population means—say, mi and mj—the conﬁdence interval is (x苶i 苶xj ) ta/2

冢莦冣冪莦 1 1 s2 ni nj

Before you can use these conﬁdence intervals, however, two questions remain: • •

How do you calculate s or s 2, the best estimate of the common variance s 2? How many degrees of freedom are used for the critical value of t?

To answer these questions, remember that in an analysis of variance, the mean square for error, MSE, always provides an unbiased estimator of s 2 and uses information from the entire set of measurements. Hence, it is the best available estimator of s 2, regardless of what test or estimation procedure you are using. You should always use s 2 MSE

with df (n k)

苶, to estimate s ! You can ﬁnd the positive square root of this estimator, s 兹MSE on the last line of Figure 11.3 labeled “Pooled StDev.” 2

Degrees of freedom for conﬁdence intervals are the df for error.

COMPLETELY RANDOMIZED DESIGN: (1 a)100% CONFIDENCE INTERVALS FOR A SINGLE TREATMENT MEAN AND THE DIFFERENCE BETWEEN TWO TREATMENT MEANS Single treatment mean: s 苶xi ta/2 兹n苶i

冢冣

Difference between two treatment means: ( 苶xi 苶xj ) ta/2

冣冢n 莦冪s莦 n 2

with 苶 s 兹s苶2 兹MSE

冪莦 nk SSE

where n n1 n2 nk and ta/2 is based on (n k) df. EXAMPLE

11.6

The researcher in Example 11.4 believes that students who have no breakfast will have significantly shorter attention spans but that there may be no difference between those who eat a light or a full breakfast. Find a 95% confidence interval for the average attention span for students who eat no breakfast, as well as a 95% confidence interval for the difference in the average attention spans for light versus full breakfast eaters.

458 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

苶 2.436 with df For s 2 MSE 5.9333 so that s 兹5.9333 (n k) 12, you can calculate the two conﬁdence intervals:

Solution

•

For no breakfast: s 苶x1 ta/2 n1 兹苶

冢冣

2.436 9.4 2.179 兹5苶

冢

冣

9.4 2.37 or between 7.03 and 11.77 minutes. For light versus full breakfast:

•

s 冢冣冪莦 n 莦 n 1 1 (14 13) 2.179 5.9333 冢5 莦冪莦莦 5冣 (x苶2 苶x3) ta/2

1 3.36 a difference of between 2.36 and 4.36 minutes. You can see that the second conﬁdence interval does not indicate a difference in the average attention spans for students who ate light versus full breakfasts, as the researcher suspected. If the researcher, because of prior beliefs, wishes to test the other two possible pairs of means—none versus light breakfast, and none versus full breakfast—the methods given in Section 11.6 should be used for testing all three pairs. Some computer programs have graphics options that provide a powerful visual description of data and the k treatment means. One such option in the MINITAB program is shown in Figure 11.7. The treatment means are indicated by the symbol ⊕ and are connected with straight lines. Notice that the “no breakfast” mean appears to be somewhat different from the other two means, as the researcher suspected, although there is a bit of overlap in the box plots. In the next section, we present a formal procedure for testing the signiﬁcance of the differences between all pairs of treatment means. Box plots for Example 11.6

● Boxplot of Span by Meal 18 16 14 Span

FI GU R E 1 1 . 7

12 10 8 6 1

2 Meal

11.5 THE ANALYSIS OF VARIANCE FOR A COMPLETELY RANDOMIZED DESIGN

❍

459

How Do I Know Whether My Calculations Are Accurate? The following suggestions apply to all the analyses of variance in this chapter: 1. When calculating sums of squares, be certain to carry at least six signiﬁcant ﬁgures before performing subtractions. 2. Remember, sums of squares can never be negative. If you obtain a negative sum of squares, you have made a mistake in arithmetic. 3. Always check your analysis of variance table to make certain that the degrees of freedom sum to the total degrees of freedom (n 1) and that the sums of squares sum to Total SS.

11.5

EXERCISES

BASIC TECHNIQUES 11.1 Suppose you wish to compare the means of six

populations based on independent random samples, each of which contains 10 observations. Insert, in an ANOVA table, the sources of variation and their respective degrees of freedom. 11.2 The values of Total SS and SSE for the experi-

ment in Exercise 11.1 are Total SS 21.4 and SSE 16.2. a. Complete the ANOVA table for Exercise 11.1. b. How many degrees of freedom are associated with the F statistic for testing H0 : m1 m2 m6? c. Give the rejection region for the test in part b for a .05. d. Do the data provide sufficient evidence to indicate differences among the population means? e. Estimate the p-value for the test. Does this value conﬁrm your conclusions in part d? 11.3 The sample means corresponding to popu-

lations 1 and 2 in Exercise 11.1 are 苶x1 3.07 and x苶2 2.52. a. Find a 95% conﬁdence interval for m1. b. Find a 95% conﬁdence interval for the difference (m1 m2). 11.4 Suppose you wish to compare the means of four

populations based on independent random samples, each of which contains six observations. Insert, in an ANOVA table, the sources of variation and their respective degrees of freedom.

11.5 The values of Total SS and SST for the experi-

ment in Exercise 11.4 are Total SS 473.2 and SST 339.8. a. Complete the ANOVA table for Exercise 11.4. b. How many degrees of freedom are associated with the F statistic for testing H0 : m1 m2 m3 m4? c. Give the rejection region for the test in part b for a .05. d. Do the data provide sufficient evidence to indicate differences among the population means? e. Approximate the p-value for the test. Does this conﬁrm your conclusions in part d? 11.6 The sample means corresponding to populations

1 and 2 in Exercise 11.4 are 苶x1 88.0 and 苶x2 83.9. a. Find a 90% conﬁdence interval for m1. b. Find a 90% conﬁdence interval for the difference (m1 m2). 11.7 These data are observations collected EX1107

using a completely randomized design:

Sample 1

Sample 2

Sample 3

3 2 4 3 2

4 3 5 2 5

2 0 2 1

a. b. c. d.

Calculate CM and Total SS. Calculate SST and MST. Calculate SSE and MSE. Construct an ANOVA table for the data.

460 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

e. State the null and alternative hypotheses for an analysis of variance F-test. f. Use the p-value approach to determine whether there is a difference in the three population means. 11.8 Refer to Exercise 11.7 and data set EX1107. Do

the data provide sufficient evidence to indicate a difference between m2 and m3? Test using the t-test of Section 10.4 with a .05. 11.9 Refer to Exercise 11.7 and data set EX1107.

a. Find a 90% conﬁdence interval for m1. b. Find a 90% conﬁdence interval for the difference (m1 m3). APPLICATIONS 11.10 Reducing Hostility A clinical psychologist wished to compare three methods for reducing hostility levels in university students using a certain psychological test (HLT). High scores on this test were taken to indicate great hostility. Eleven students who got high and nearly equal scores were used in the experiment. Five were selected at random from among the 11 problem cases and treated by method A, three were taken at random from the remaining six students and treated by method B, and the other three students were treated by method C. All treatments continued throughout a semester, when the HLT test was given again. The results are shown in the table. Method

Scores on the HLT Test

A B C

73 54 79

83 74 95

76 71 87

a. Perform an analysis of variance for this experiment. b. Do the data provide sufficient evidence to indicate a difference in mean student response to the three methods after treatment? 11.11 Hostility, continued Refer to Exercise 11.10. Let mA and mB, respectively, denote the mean scores at the end of the semester for the populations of extremely hostile students who were treated throughout that semester by method A and method B. a. Find a 95% confidence interval for mA. b. Find a 95% confidence interval for mB. c. Find a 95% confidence interval for (mA mB). d. Is it correct to claim that the confidence intervals found in parts a, b, and c are jointly valid? 11.12 Assembling Electronic Equipment EX1112

An experiment was conducted to compare the

effectiveness of three training programs, A, B, and C, in training assemblers of a piece of electronic equipment. Fifteen employees were randomly assigned, ﬁve each, to the three programs. After completion of the courses, each person was required to assemble four pieces of the equipment, and the average length of time required to complete the assembly was recorded. Several of the employees resigned during the course of the program; the remainder were evaluated, producing the data shown in the accompanying table. Use the MINITAB printout to answer the questions. Training Program

Average Assembly Time (min)

A B C

59 52 58

64 58 65

57 54 71

62 63

a. Do the data provide sufficient evidence to indicate a difference in mean assembly times for people trained by the three programs? Give the p-value for the test and interpret its value. b. Find a 99% conﬁdence interval for the difference in mean assembly times between persons trained by programs A and B. c. Find a 99% conﬁdence interval for the mean assembly times for persons trained in program A. d. Do you think the data will satisfy (approximately) the assumption that they have been selected from normal populations? Why? MINITAB output for Exercise 11.12 One-way ANOVA: Time versus Program Source Program Error Total S = 3.865

Level 1 2 3

DF SS 2 170.5 9 134.5 11 304.9 R-Sq = 55.90%

N 4 3 5

Pooled StDev =

Mean 60.500 54.667 64.200 3.865

MS 85.2 14.9

F 5.70

P 0.025

R-Sq(adj) = 46.10% Individual 95% CIs For Mean Based on Pooled StDev StDev -+---------+---------+---------+----3.109 (--------*--------) 3.055 (---------*---------) 4.658 (------*-------) -+---------+---------+---------+----50.0 55.0 60.0 65.0

11.13 Swampy Sites An ecological study

was conducted to compare the rates of growth of vegetation at four swampy undeveloped sites and to determine the cause of any differences that might be observed. Part of the study involved measuring the leaf lengths of a particular plant species on a preselected date in May. Six plants were randomly selected at each of the four sites to be used in the comparison. The data in the table are the mean leaf length per plant (in centimeters) for a random sample of ten leaves per plant. The MINITAB analysis of variance computer printout for these data is also provided.

EX1113

11.5 THE ANALYSIS OF VARIANCE FOR A COMPLETELY RANDOMIZED DESIGN

Location

Mean Leaf Length (cm)

1 2 3 4

5.7 6.2 5.4 3.7

6.3 5.3 5.0 3.2

6.1 5.7 6.0 3.9

6.0 6.0 5.6 4.0

5.8 5.2 4.9 3.5

6.2 5.5 5.2 3.6

MINITAB output for Exercise 11.13

❍

461

were randomly selected at each location, but one specimen, corresponding to location 4, was lost in the laboratory. The data and a MINITAB analysis of variance computer printout are provided here (the greater the pollution, the lower the dissolved oxygen readings). Location

Mean Dissolved Oxygen Content

1 2 3 4

5.9 6.3 4.8 6.0

One-way ANOVA: Length versus Location Source DF SS Location 3 19.740 Error 20 2.293 Total 23 22.033 S = 0.3386 R-Sq = 89.59%

Level 1 2 3 4

N 6 6 6 6

Pooled StDev =

Mean 6.0167 5.6500 5.3500 3.6500 0.3386

MS 6.580 0.115

F 57.38

P 0.000

R-Sq(adj) = 88.03% Individual 95% CIs For Mean Based on Pooled StDev StDev --------+---------+---------+---------+0.2317 (--*---) 0.3937 (---*--) 0.4087 (---*--) 0.2881 (---*--) --------+---------+---------+---------+4.00 4.80 5.60 6.40

a. You will recall that the test and estimation procedures for an analysis of variance require that the observations be selected from normally distributed (at least, roughly so) populations. Why might you feel reasonably conﬁdent that your data satisfy this assumption? b. Do the data provide sufficient evidence to indicate a difference in mean leaf length among the four locations? What is the p-value for the test? c. Suppose, prior to seeing the data, you decided to compare the mean leaf lengths of locations 1 and 4. Test the null hypothesis m1 m4 against the alternative m1 m4. d. Refer to part c. Construct a 99% conﬁdence interval for (m1 m4). e. Rather than use an analysis of variance F-test, it would seem simpler to examine one’s data, select the two locations that have the smallest and largest sample mean lengths, and then compare these two means using a Student’s t-test. If there is evidence to indicate a difference in these means, there is clearly evidence of a difference among the four. (If you were to use this logic, there would be no need for the analysis of variance F-test.) Explain why this procedure is invalid. 11.14 Dissolved O2 Content Water samples were taken at four different locations in a river to determine whether the quantity of dissolved oxygen, a measure of water pollution, varied from one location to another. Locations 1 and 2 were selected above an industrial plant, one near the shore and the other in midstream; location 3 was adjacent to the industrial water discharge for the plant; and location 4 was slightly downriver in midstream. Five water specimens

6.1 6.6 4.3 6.2

6.3 6.4 5.0 6.1

6.1 6.4 4.7 5.8

6.0 6.5 5.1

MINITAB output for Exercise 11.14 One-way ANOVA: Oxygen versus Location Source DF SS MS F P Location 3 7.8361 2.6120 63.66 0.000 Error 15 0.6155 0.0410 Total 18 8.4516 S = 0.2026 R-Sq = 92.72% R-Sq(adj) = 91.26% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ----+---------+---------+---------+-1 5 6.0800 0.1483 (--*---) 2 5 6.4400 0.1140 (--*---) 3 5 4.7800 0.3114 (---*--) 4 4 6.0250 0.1708 (--*---) ----+---------+---------+---------+-Pooled StDev = 0.2026 4.80 5.40 6.00 6.60

a. Do the data provide sufficient evidence to indicate a difference in the mean dissolved oxygen contents for the four locations? b. Compare the mean dissolved oxygen content in midstream above the plant with the mean content adjacent to the plant (location 2 versus location 3). Use a 95% conﬁdence interval. 11.15 Calcium The calcium content of a

powdered mineral substance was analyzed ﬁve times by each of three methods, with similar standard deviations:

EX1115

Method

Percent Calcium

1 2 3

.0279 .0268 .0280

.0276 .0274 .0279

.0270 .0267 .0282

.0275 .0263 .0278

.0281 .0267 .0283

Use an appropriate test to compare the three methods of measurement. Comment on the validity of any assumptions you need to make.

EX1114

11.16 Tuna Fish In Exercise 10.6, we

reported the estimated average prices for a 6-ounce can or a 7.06-ounce pouch of tuna ﬁsh, based on prices paid nationally for a variety of different brands of tuna.1

EX1116

462

❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

Light Tuna in Water

White Tuna in Oil

White Tuna in Water

Light Tuna in Oil

.99 .53 1.92 1.41 1.23 1.12 .85 .63 .65 .67 .69 .60 .60 .66

1.27 1.22 1.19 1.22

1.49 1.29 1.27 1.35 1.29 1.00 1.27 1.28

2.56 1.92 1.30 1.79 1.23

.62 .66 .62 .65 .60 .67

Source: From “Pricing of Tuna” Copyright 2001 by Consumers Union of U.S., Inc., Yonkers, NY 10703-1057, a nonproﬁt organization. Reprinted with permission from the June 2001 issue of Consumer Reports® for educational purposes only. No commercial use or reproduction permitted. www.ConsumerReports.org®.

a. Use an analysis of variance for a completely randomized design to determine if there are significant differences in the prices of tuna packaged in these four different ways. Can you reject the hypothesis of no difference in average price for these packages at the a .05 level of significance? At the a .01 level of significance? b. Find a 95% confidence interval estimate of the difference in price between light tuna in water and light tuna in oil. Does there appear to be a significant difference in the price of these two kinds of packaged tuna? c. Find a 95% confidence interval estimate of the difference in price between white tuna in water and white tuna in oil. Does there appear to be a significant difference in the price of these two kinds of packaged tuna? d. What other confidence intervals might be of interest to the researcher who conducted this experiment? 11.17 The Cost of Lumber A national

home builder wants to compare the prices per 1,000 board feet of standard or better grade Douglas ﬁr framing lumber. He randomly selects ﬁve suppliers in each of the four states where the builder is planning to begin construction. The prices are given in the table.

EX1117

11.6

State 1

$241 235 238 247 250

$216 220 205 213 220

$230 225 235 228 240

$245 250 238 255 255

a. What type of experimental design has been used? b. Construct the analysis of variance table for this data. c. Do the data provide sufficient evidence to indicate that the average price per 1000 board feet of Douglas ﬁr differs among the four states? Test using a .05. 11.18 Good at Math? Twenty third graders were randomly separated into four equal groups, and each group was taught a mathematical concept using a different teaching method. At the end of the teaching period, progress was measured by a unit test. The scores are shown below (one child in group 3 was absent on the day that the test was administered).

EX1118

Group 1

112 92 124 89 97

111 129 102 136 99

140 121 130 106

101 116 105 126 119

a. What type of design has been used in this experiment? b. Construct an ANOVA table for the experiment. c. Do the data present sufficient evidence to indicate a difference in the average scores for the four teaching methods? Test using a .05.

RANKING POPULATION MEANS Many experiments are exploratory in nature. You have no preconceived notions about the results and have not decided (before conducting the experiment) to make speciﬁc treatment comparisons. Rather, you want to rank the treatment means, determine which means differ, and identify sets of means for which no evidence of difference exists.

11.6 RANKING POPULATION MEANS

❍

463

One option might be to order the sample means from the smallest to the largest and then to conduct t-tests for adjacent means in the ordering. If two means differ by more than ta/2

s 冢n 莦冪莦 n 冣 2

you conclude that the pair of population means differ. The problem with this procedure is that the probability of making a Type I error—that is, concluding that two means differ when, in fact, they are equal—is a for each test. If you compare a large number of pairs of means, the probability of detecting at least one difference in means, when in fact none exists, is quite large. A simple way to avoid the high risk of declaring differences when they do not exist is to use the studentized range, the difference between the smallest and the largest in a set of k sample means, as the yardstick for determining whether there is a difference in a pair of population means. This method, often called Tukey’s method for paired comparisons, makes the probability of declaring that a difference exists between at least one pair in a set of k treatment means, when no difference exists, equal to a. Tukey’s method for making paired comparisons is based on the usual analysis of variance assumptions. In addition, it assumes that the sample means are independent and based on samples of equal size. The yardstick that determines whether a difference exists between a pair of treatment means is the quantity v (Greek lowercase omega), which is presented next.

YARDSTICK FOR MAKING PAIRED COMPARISONS s v qa(k, df ) 兹n苶t where

冢冣

k Number of treatments s 2 MSE Estimator of the common variance s 2 and s 兹s苶2 df Number of degrees of freedom for s 2 nt Common sample size—that is, the number of observations in each of the k treatment means qa(k, df ) Tabulated value from Tables 11(a) and 11(b) in Appendix I, for a .05 and .01, respectively, and for various combinations of k and df Rule: Two population means are judged to differ if the corresponding sample means differ by v or more. Tables 11(a) and 11(b) in Appendix I list the values of qa(k, df ) for a .05 and .01, respectively. To illustrate the use of the tables, refer to the portion of Table 11(a) reproduced in Table 11.2. Suppose you want to make pairwise comparisons of k 5 means with a .05 for an analysis of variance, where s2 possesses 9 df. The tabulated value for k 5, df 9, and a .05, shaded in Table 11.2, is q.05(5, 9) 4.76.

464 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

TABLE 11.2

EXAMPLE

●

11.7

A Partial Reproduction of Table 11(a) in Appendix I; Upper 5% Points df

1 2 3 4 5 6 7 8 9 10 11 12

17.97 6.08 4.50 3.93 3.64 3.46 3.34 3.26 3.20 3.15 3.11 3.08

26.98 8.33 5.91 5.04 4.60 4.34 4.16 4.04 3.95 3.88 3.82 3.77

32.82 9.80 6.82 5.76 5.22 4.90 4.68 4.53 4.41 4.33 4.26 4.20

37.08 10.88 7.50 6.29 5.67 5.30 5.06 4.89 4.76 4.65 4.57 4.51

40.41 11.74 8.04 6.71 6.03 5.63 5.36 5.17 5.02 4.91 4.82 4.75

43.12 12.44 8.48 7.05 6.33 5.90 5.61 5.40 5.24 5.12 5.03 4.95

45.40 13.03 8.85 7.35 6.58 6.12 5.82 5.60 5.43 5.30 5.20 5.12

47.36 13.54 9.18 7.60 6.80 6.32 6.00 5.77 5.59 5.46 5.35 5.27

49.07 13.99 9.46 7.83 6.99 6.49 6.16 5.92 5.74 5.60 5.49 5.39

50.59 14.39 9.72 8.03 7.17 6.65 6.30 6.05 5.87 5.72 5.61 5.51

51.96 14.75 9.95 8.21 7.32 6.79 6.43 6.18 5.98 5.83 5.71 5.61

Refer to Example 11.4, in which you compared the average attention spans for students given three different “meal” treatments in the morning: no breakfast, a light breakfast, or a full breakfast. The ANOVA F-test in Example 11.5 indicated a signiﬁcant difference in the population means. Use Tukey’s method for paired comparisons to determine which of the three population means differ from the others. 苶 For this example, there are k 3 treatment means, with s 兹MSE 2.436. Tukey’s method can be used, with each of the three samples containing nt 5 measurements and (n k) 12 degrees of freedom. Consult Table 11 in Appendix I to ﬁnd q.05(k, df ) q.05(3, 12) 3.77 and calculate the “yardstick” as s 2.436 v q.05(3, 12) 3.77 4.11 兹n苶t 兹5苶 Solution

冢冣

冢

冣

The three treatment means are arranged in order from the smallest, 9.4, to the largest, 14.0, in Figure 11.8. The next step is to check the difference between every pair of means. The only difference that exceeds v 4.11 is the difference between no breakfast and a light breakfast. These two treatments are thus declared signiﬁcantly different. You cannot declare a difference between the other two pairs of treatments. To indicate this fact visually, Figure 11.8 shows a line under those pairs of means that are not signiﬁcantly different. FIGU R E 1 1 . 8

Ranked means for Example 11.7

● None 9.4

Full 13.0

Light 14.0

The results here may seem confusing. However, it usually helps to think of ranking the means and interpreting nonsigniﬁcant differences as our inability to distinctly rank those means underlined by the same line. For this example, the light breakfast deﬁnitely ranked higher than no breakfast, but the full breakfast could not be ranked higher than no breakfast, or lower than the light breakfast. The probability that we make at least one error among the three comparisons is at most a .05.

11.6 RANKING POPULATION MEANS

If zero is not in the interval, there is evidence of a difference between the two methods.

FIGU R E 1 1 . 9

MINITAB output for Example 11.7

❍

465

Most computer programs provide an option to perform paired comparisons, including Tukey’s method. The MINITAB output in Figure 11.9 shows its form of Tukey’s test, which differs slightly from the method we have presented. The three intervals that you see in the printout marked “Lower” and “Upper” represent the difference in the two sample means plus or minus the yardstick v. If the interval contains the value 0, the two means are judged to be not signiﬁcantly different. You can see that only means 1 and 2 (none versus light) show a signiﬁcant difference. ● Tukey's 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Meal Individual confidence level = 97.94% Meal = 1 subtracted from: Meal Lower 2 0.493 3 -0.507

Center 4.600 3.600

Upper 8.707 7.707

-----+---------+---------+---------+---(-----------*-----------) (----------*-----------) -----+---------+---------+---------+----3.5 0.0 3.5 7.0

Meal = 2 subtracted from: Meal Lower 3 -5.107

Center -1.000

Upper 3.107

-----+---------+---------+---------+---(-----------*-----------) -----+---------+---------+---------+----3.5 0.0 3.5 7.0

As you study two more experimental designs in the next sections of this chapter, remember that, once you have found a factor to be signiﬁcant, you should use Tukey’s method or another method of paired comparisons to ﬁnd out exactly where the differences lie!

11.6

EXERCISES

BASIC TECHNIQUES 11.19 Suppose you wish to use Tukey’s method of paired comparisons to rank a set of population means. In addition to the analysis of variance assumptions, what other property must the treatment means satisfy? 11.20 Consult Tables 11(a) and 11(b) in Appendix I and ﬁnd the values of qa(k, df ) for these cases: a. a .05, k 5, df 7 b. a .05, k 3, df 10 c. a .01, k 4, df 8 d. a .01, k 7, df 5 11.21 If the sample size for each treatment is nt and

if s 2 is based on 12 df, ﬁnd v in these cases: a. a .05, k 4, nt 5 b. a .01, k 6, nt 8

11.22 An independent random sampling design was

used to compare the means of six treatments based on

samples of four observations per treatment. The pooled estimator of s 2 is 9.12, and the sample means follow:

x苶1 101.6 x苶2 98.4 x苶3 112.3 x苶5 104.2 x苶6 113.8 x苶4 92.9 a. Give the value of v that you would use to make pairwise comparisons of the treatment means for a .05. b. Rank the treatment means using pairwise comparisons. APPLICATIONS 11.23 Swamp Sites, again Refer to Exercise 11.13 and data set EX1113. Rank the mean leaf growth for the four locations. Use a .01. 11.24 Calcium Refer to Exercise 11.15 and data set

EX1115. The paired comparisons option in MINITAB generated the output provided here. What do these results tell you about the differences in the population

466

❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

means? Does this conﬁrm your conclusions in Exercise 11.15? MINITAB output for Exercise 11.24 Tukey's 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Method Individual confidence level = 97.94% Method = 1 subtracted from: Method 2 3 Method 2 3

Lower Center Upper -0.0014377 -0.0008400 -0.0002423 -0.0001777 0.0004200 0.0010177 --------+---------+---------+---------+ (-----*-----) (-----*-----) --------+---------+---------+---------+ -0.0010 0.0000 0.0010 0.0020

Method = 2 subtracted from: Method 3 Method 3

Lower Center Upper 0.0006623 0.0012600 0.0018577 --------+---------+---------+---------+ (-----*-----) --------+---------+---------+---------+ -0.0010 0.0000 0.0010 0.0020

11.25 Glucose Tolerance Physicians

depend on laboratory test results when managing medical problems such as diabetes or epilepsy. In a uniformity test for glucose tolerance, three different laboratories were each sent nt 5 identical blood samples from a person who had drunk 50 milligrams (mg) of glucose dissolved in water. The laboratory results (in mg/dl) are listed here:

EX1125

Lab 1

Lab 2

Lab 3

120.1 110.7 108.9 104.2 100.4

98.3 112.1 107.7 107.9 99.2

103.0 108.5 101.1 110.0 105.4

a. Do the data indicate a difference in the average readings for the three laboratories? b. Use Tukey’s method for paired comparisons to rank the three treatment means. Use a .05.

11.7

11.26 The Cost of Lumber, continued The analy-

sis of variance F-test in Exercise 11.17 (and data set EX1117) determined that there was indeed a difference in the average cost of lumber for the four states. The following information from Exercise 11.17 is given in the table: x苶1 242.2 x苶2 214.8 x苶3 231.6 x苶4 248.6

Sample Means

MSE Error df : ni : k:

41.25 16 5 4

Use Tukey’s method for paired comparisons to determine which means differ signiﬁcantly from the others at the a .01 level. 11.27 GRE Scores The Graduate Record

Examination (GRE) scores were recorded for students admitted to three different graduate programs at a local university.

EX1127

Graduate Program 1

532 548 619 509 627

670 590 640 710 690

502 607 549 524 542

a. Do these data provide sufficient evidence to indicate a difference in the mean GRE scores for applicants admitted to the three programs? b. Find a 95% confidence interval for the difference in mean GRE scores for programs 1 and 2. c. If you find a significant difference in the average GRE scores for the three programs, use Tukey’s method for paired comparisons to determine which means differ significantly from the others. Use a .05.

THE RANDOMIZED BLOCK DESIGN: A TWO-WAY CLASSIFICATION The completely randomized design introduced in Section 11.4 is a generalization of the two independent samples design presented in Section 10.4. It is meant to be used when the experimental units are quite similar or homogeneous in their makeup and when there is only one factor—the treatment—that might inﬂuence the response. Any other variation in the response is due to random variation or experimental error.

11.8 THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN

❍

467

Sometimes it is clear to the researcher that the experimental units are not homogeneous. Experimental subjects or animals, agricultural fields, days of the week, and other experimental units often add their own variability to the response. Although the researcher is not really interested in this source of variation, but rather in some treatment he chooses to apply, he may be able to increase the information by isolating this source of variation using the randomized block design—a direct extension of the matched pairs or paired-difference design in Section 10.5. In a randomized block design, the experimenter is interested in comparing k treatment means. The design uses blocks of k experimental units that are relatively similar, or homogeneous, with one unit within each block randomly assigned to each treatment. If the randomized block design involves k treatments within each of b blocks, then the total number of observations in the experiment is n bk. A production supervisor wants to compare the mean times for assembly-line operators to assemble an item using one of three methods: A, B, or C. Expecting variation in assembly times from operator to operator, the supervisor uses a randomized block design to compare the three methods. Five assembly-line operators are selected to serve as blocks, and each is assigned to assemble the item three times, once for each of the three methods. Since the sequence in which the operator uses the three methods may be important (fatigue or increasing dexterity may be factors affecting the response), each operator should be assigned a random sequencing of the three methods. For example, operator 1 might be assigned to perform method C first, followed by A and B. Operator 2 might perform method A first, then C and B. To compare four different teaching methods, a group of students might be divided into blocks of size 4, so that the groups are most nearly matched according to academic achievement. To compare the average costs for three different cellular phone companies, costs might be compared at each of three usage levels: low, medium, and high. To compare the average yields for three species of fruit trees when a variation in yield is expected because of the field in which the trees are planted, a researcher uses five fields. She divides each field into three plots on which the three species of fruit trees are planted. Matching or blocking can take place in many different ways. Comparisons of treatments are often made within blocks of time, within blocks of people, or within similar external environments. The purpose of blocking is to remove or isolate the block-to-block variability that might otherwise hide the effect of the treatments. You will find more examples of the use of the randomized block design in the exercises at the end of the next section.

b blocks k treatments n bk

11.8

THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN The randomized block design identiﬁes two factors: treatments and blocks—both of which affect the response.

Partitioning the Total Variation in the Experiment Let xij be the response when the ith treatment (i 1, 2, . . . , k) is applied in the j th block ( j 1, 2, . . . , b). The total variation in the n bk observations is (Sxij)2 Total SS S(xij 苶x )2 Sx 2ij n

468 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

This is partitioned into three (rather than two) parts in such a way that Total SS SSB SST SSE where • • •

SSB (sum of squares for blocks) measures the variation among the block means. SST (sum of squares for treatments) measures the variation among the treatment means. SSE (sum of squares for error) measures the variation of the differences among the treatment observations within blocks, which measures the experimental error.

The calculational formulas for the four sums of squares are similar in form to those you used for the completely randomized design in Section 11.5. Although you can simplify your work by using a computer program to calculate these sums of squares, the formulas are given next. CALCULATING THE SUMS OF SQUARES FOR A RANDOMIZED BLOCK DESIGN, k TREATMENTS IN b BLOCKS G2 CM n where G S xij Total of all n bk observations Total SS Sx 2ij CM (Sum of squares of all x-values) CM

Total SS SST SSB SSE

T 2i

SST Sb CM B 2j

SSB Sk CM SSE Total SS SST SSB with Ti Total of all observations receiving treatment i, i 1, 2, . . . , k Bj Total of all observations in block j, j 1, 2, . . . , b Each of the three sources of variation, when divided by the appropriate degrees of freedom, provides an estimate of the variation in the experiment. Since Total SS involves n bk squared observations, its degrees of freedom are df (n 1). Similarly, SST involves k squared totals, and its degrees of freedom are df (k 1), while SSB involves b squared totals and has (b 1) degrees of freedom. Finally, since the degrees of freedom are additive, the remaining degrees of freedom associated with SSE can be shown algebraically to be df (b 1)(k 1). These three sources of variation and their respective degrees of freedom are combined to form the mean squares as MS SS/df, and the total variation in the experiment is then displayed in an analysis of variance (or ANOVA) table as shown here:

11.8 THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN

11.8

TABLE 11.3

469

ANOVA TABLE FOR A RANDOMIZED BLOCK DESIGN, k TREATMENTS AND b BLOCKS

Degrees of freedom are additive.

EXAMPLE

❍

Source

Treatments Blocks Error

k1 b1 (b 1)(k 1)

SST SSB SSE

MST SST/(k 1) MSB SSB/(b 1) MSE SSE/(b 1)(k 1)

MST/MSE MSB/MSE

Total

n 1 bk 1

The cellular phone industry is involved in a ﬁerce battle for customers, with each company devising its own complex pricing plan to lure customers. Since the cost of a cell phone minute varies drastically depending on the number of minutes per month used by the customer, a consumer watchdog group decided to compare the average costs for four cellular phone companies using three different usage levels as blocks. The monthly costs (in dollars) computed by the cell phone companies for peak-time callers at low (20 minutes per month), middle (150 minutes per month), and high (1000 minutes per month) usage levels are given in Table 11.3. Construct the analysis of variance table for this experiment.

●

Monthly Phone Costs of Four Companies at Three Usage Levels Company Usage Level

Blocks contain experimental units that are relatively the same.

Totals

Low Middle High

27 68 308

24 76 326

31 65 312

23 67 300

B1 105 B2 276 B3 1246

Totals

T1 403

T2 426

T3 408

T4 390

G 1627

Solution The experiment is designed as a randomized block design with b 3 usage levels (blocks) and k 4 companies (treatments), so there are n bk 12 observations and G 1627. Then

G2 16272 CM 220,594.0833 n 12 Total SS (272 242 3002) CM 189,798.9167 4032 3902 SST CM 222.25 3 1052 2762 12462 SSB CM 189,335.1667 4 and by subtraction, SSE Total SS SST SSB 241.5 These four sources of variation, their degrees of freedom, sums of squares, and mean squares are shown in the shaded area of the analysis of variance table, generated by MINITAB and given in Figure 11.10. You will ﬁnd instructions for generating this output in the section “My MINITAB ” at the end of this chapter.

470 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

FI GU R E 1 1 . 1 0

MINITAB output for Example 11.8

● Two-way ANOVA: Dollars versus Usage, Company Source Usage Company Error Total

DF 2 3 6 11

S = 6.344

SS 189335 222 242 189799

MS 94667.6 74.1 40.3

R-Sq = 99.87%

F 2351.99 1.84

P 0.000 0.240

R-Sq(adj) = 99.77%

Notice that the MINITAB ANOVA table shows two different F statistics and p-values. It will not surprise you to know that these statistics are used to test hypotheses concerning the equality of both the treatment and block means.

Testing the Equality of the Treatment and Block Means The mean squares in the analysis of variance table can be used to test the null hypotheses H0 : No difference among the k treatment means or H0 : No difference among the b block means versus the alternative hypothesis Ha : At least one of the means is different from at least one other using a theoretical argument similar to the one we used for the completely randomized design. •

Remember that s 2 is the common variance for the observations in all bk block-treatment combinations. The quantity SSE MSE (b 1)(k 1)

•

is an unbiased estimate of s 2, whether or not H0 is true. The two mean squares, MST and MSB, estimate s 2 only if H0 is true and tend to be unusually large if H0 is false and either the treatment or block means are different. The test statistics MST F and MSE

MSB F MSE

are used to test the equality of treatment and block means, respectively. Both statistics tend to be larger than usual if H0 is false. Hence, you can reject H0 for large values of F, using right-tailed critical values of the F distribution with the appropriate degrees of freedom (see Table 6 in Appendix I) or computer-generated p-values to draw statistical conclusions about the equality of the population means. As an alternative, you can use the F Probabilities applet to ﬁnd either critical values of F or p-values.

11.8 THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN

❍

471

TESTS FOR A RANDOMIZED BLOCK DESIGN For comparing treatment means: 1. Null hypothesis: H0 : The treatment means are equal 2. Alternative hypothesis: Ha : At least two of the treatment means differ 3. Test statistic: F MST/MSE, where F is based on df1 (k 1) and df2 (b 1)(k 1) 4. Rejection region: Reject if F Fa, where Fa lies in the upper tail of the F distribution (see the ﬁgure), or when the p-value a For comparing block means: 1. Null hypothesis: H0 : The block means are equal 2. Alternative hypothesis: Ha : At least two of the block means differ 3. Test statistic: F MSB/MSE, where F is based on df1 (b 1) and df2 (b 1)(k 1) 4. Rejection region: Reject if F Fa, where Fa lies in the upper tail of the F distribution (see the ﬁgure), or when the p-value a f(F)

EXAMPLE

11.9

Fα

Do the data in Example 11.8 provide sufficient evidence to indicate a difference in the average monthly cell phone cost depending on the company the customer uses? Solution The cell phone companies represent the treatments in this randomized block design, and the differences in their average monthly costs are of primary interest to the researcher. To test

H0 : No difference in the average cost among companies versus the alternative that the average cost is different for at least one of the four companies, you use the analysis of variance F statistic, calculated as MST 74.1 F 1.84 MSE 40.3 and shown in the column marked “F” and the row marked “Company” in Figure 11.10. The exact p-value for this statistical test is also given in Figure 11.10 as .240, which is too large to allow rejection of H0. The results do not show a signiﬁcant difference in the treatment means. That is, there is insufficient evidence to indicate a difference in the average monthly costs for the four companies.

472 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

The researcher in Example 11.9 was fairly certain in using a randomized block design that there would be a significant difference in the block means—that is, a significant difference in the average monthly costs depending on the usage level. This suspicion is justified by looking at the test of equality of block means. Notice that the observed test statistic is F 2351.99 with P .000, showing a highly significant difference, as expected, in the block means.

Identifying Differences in the Treatment and Block Means Once the overall F-test for equality of the treatment or block means has been performed, what more can you do to identify the nature of any differences you have found? As in Section 11.5, you can use Tukey’s method of paired comparisons to determine which pairs of treatment or block means are significantly different from one another. However, if the F-test does not indicate a significant difference in the means, there is no reason to use Tukey’s procedure. If you have a special interest in a particular pair of treatment or block means, you can estimate the difference using a (1 a)100% confidence interval.† The formulas for these procedures, shown next, follow a pattern similar to the formulas for the completely randomized design. Remember that MSE always provides an unbiased estimator of s 2 and uses information from the entire set of measurements. Hence, it is the best available estimator of s 2, regardless of what test or estimation procedure you are using. You will again use Degrees of freedom for Tukey’s test and for confidence intervals are error df.

s 2 MSE

with df (b 1)(k 1)

to estimate s in comparing the treatment and block means. 2

COMPARING TREATMENT AND BLOCK MEANS Tukey’s yardstick for comparing block means: s v qa(b, df ) 兹苶k

冢冣

Tukey’s yardstick for comparing treatment means: s v qa(k, df ) 兹苶b

冢冣

(1 a)100% conﬁdence interval for the difference in two block means: 1 1 (B 苶i 苶 Bj) ta/2 s 2 k k

冢莦冣冪莦

where 苶 Bi is the average of all observations in block i (1 a)100% conﬁdence interval for the difference in two treatment means: 1 1 苶i 苶 Tj ) ta/2 s 2 (T b b

冢莦冣冪莦

where 苶 Ti is the average of all observations in treatment i.

†

You cannot construct a conﬁdence interval for a single mean unless the blocks have been randomly selected from among the population of all blocks. The procedure for constructing intervals for single means is beyond the scope of this book.

11.8 THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN

❍

473

Note: The values qa(*, df ) from Table 11 in Appendix I, ta/2 from Table 4 in Appendix I, and s 2 MSE all depend on df (b 1)(k 1) degrees of freedom.

EXAMPLE

11.10

Identify the nature of any differences you found in the average monthly cell phone costs from Example 11.8. Since the F-test did not show any significant differences in the average costs for the four companies, there is no reason to use Tukey’s method of paired comparisons. Suppose, however, that you are an executive for company B and your major competitor is company C. Can you claim a significant difference in the two average costs? Using a 95% confidence interval, you can calculate

Solution

冪MSE 莦冢莦b冣

苶2 苶 T3) t.025 (T You cannot form a conﬁdence interval or test an hypothesis about a single treatment mean in a randomized block design!

冢3 3冣 2.447冪40.3 莦冢3莦冣 426

408

6 12.68 so the difference between the two average costs is estimated as between $6.68 and $18.68. Since 0 is contained in the interval, you do not have evidence to indicate a signiﬁcant difference in your average costs. Sorry!

Some Cautionary Comments on Blocking Here are some important points to remember: •

•

A randomized block design should not be used when treatments and blocks both correspond to experimental factors of interest to the researcher. In designating one factor as a block, you may assume that the effect of the treatment will be the same, regardless of which block you are using. If this is not the case, the two factors—blocks and treatments—are said to interact, and your analysis could lead to incorrect conclusions regarding the relationship between the treatments and the response. When an interaction is suspected between two factors, you should analyze the data as a factorial experiment, which is introduced in the next section. Remember that blocking may not always be beneficial. When SSB is removed from SSE, the number of degrees of freedom associated with SSE gets smaller. For blocking to be beneficial, the information gained by isolating the block variation must outweigh the loss of degrees of freedom for error. Usually, though, if you suspect that the experimental units are not homogeneous and you can group the units into blocks, it pays to use the randomized block design! Finally, remember that you cannot construct confidence intervals for individual treatment means unless it is reasonable to assume that the b blocks have been randomly selected from a population of blocks. If you construct such an interval, the sample treatment mean will be biased by the positive and negative effects that the blocks have on the response.

474 ❍

11.8

CHAPTER 11 THE ANALYSIS OF VARIANCE

EXERCISES

BASIC TECHNIQUES 11.28 A randomized block design was used to compare the means of three treatments within six blocks. Construct an ANOVA table showing the sources of variation and their respective degrees of freedom. 11.29 Suppose that the analysis of variance calculations for Exercise 11.28 are SST 11.4, SSB 17.1, and Total SS 42.7. Complete the ANOVA table, showing all sums of squares, mean squares, and pertinent F-values. 11.30 Do the data of Exercise 11.28 provide sufficient evidence to indicate differences among the treatment means? Test using a .05. 11.31 Refer to Exercise 11.28. Find a 95% conﬁ-

dence interval for the difference between a pair of treatment means A and B if x苶A 21.9 and 苶xB 24.2. 11.32 Do the data of Exercise 11.28 provide sufficient evidence to indicate that blocking increased the amount of information in the experiment about the treatment means? Justify your answer. 11.33 The data that follow are observations

collected from an experiment that compared four treatments, A, B, C, and D, within each of three blocks, using a randomized block design.

EX1133

Treatment Block

Total

1 2 3

6 4 12

10 9 15

8 5 14

9 7 14

33 25 55

Total

113

a. Do the data present sufficient evidence to indicate differences among the treatment means? Test using a .05. b. Do the data present sufficient evidence to indicate differences among the block means? Test using a .05. c. Rank the four treatment means using Tukey’s method of paired comparisons with a .01. d. Find a 95% conﬁdence interval for the difference in means for treatments A and B. e. Does it appear that the use of a randomized block design for this experiment was justiﬁed? Explain.

11.34 The data shown here are observations collected from an experiment that compared three treatments, A, B, and C, within each of ﬁve blocks, using a randomized block design:

EX1134

Block Treatment

Total

A B C

2.1 3.4 3.0

2.6 3.8 3.6

1.9 3.6 3.2

3.2 4.1 3.9

2.7 3.9 3.9

12.5 18.8 17.6

Total

8.5

10.0

8.7

11.2

10.5

48.9

MINITAB output for Exercise 11.34 Two-way ANOVA: Response versus Trts, Blocks Source Trts Blocks Error Total

DF 2 4 8 14

S = 0.1673

SS 4.476 1.796 0.224 6.496

MS 2.238 0.449 0.028

R-Sq = 96.55%

F 79.93 16.04

P 0.000 0.001

R-Sq(adj) = 93.97%

Use the MINITAB ouput to analyze the experiment. Investigate possible differences in the block and/or treatment means and, if any differences exist, use an appropriate method to speciﬁcally identify where the differences lie. Has blocking been effective in this experiment? Present your results in the form of a report. 11.35 The partially completed ANOVA table for a randomized block design is presented here: Source

Treatments Blocks Error

Total

a. b. c. d. e.

14.2 18.9 41.9

How many blocks are involved in the design? How many observations are in each treatment total? How many observations are in each block total? Fill in the blanks in the ANOVA table. Do the data present sufficient evidence to indicate differences among the treatment means? Test using a .05. f. Do the data present sufficient evidence to indicate differences among the block means? Test using a .05.

11.8 THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN

APPLICATIONS 11.36 Gas Mileage A study was conducted

to compare automobile gasoline mileage for three formulations of gasoline. A was a non-leaded 87 octane formulation, B was a non-leaded 91 octane formulation, and C was a non-leaded 87 octane formulation with 15% ethanol. Four automobiles, all of the same make and model, were used in the experiment, and each formulation was tested in each automobile. Using each formulation in the same automobile has the effect of eliminating (blocking out) automobile-toautomobile variability. The data (in miles per gallon) follow. Automobile

A B C

475

Illustration for Exercise 11.37

EX1136

Formulation

❍

25.7 27.2 26.1

27.0 28.1 27.5

27.3 27.9 26.8

26.1 27.7 27.8

a. Do the data provide sufficient evidence to indicate a difference in mean mileage per gallon for the three gasoline formulations? b. Is there evidence of a difference in mean mileage for the four automobiles? c. Suppose that prior to looking at the data, you had decided to compare the mean mileage per gallon for formulations A and B. Find a 90% confidence interval for this difference. d. Use an appropriate method to identify the pairwise differences, if any, in the average mileages for the three formulations. 11.37 Water Resistance in Textiles An experiment was conducted to compare the EX1137 effects of four different chemicals, A, B, C, and D, in producing water resistance in textiles. A strip of material, randomly selected from a bolt, was cut into four pieces, and the four pieces were randomly assigned to receive one of the four chemicals, A, B, C, or D. This process was replicated three times, thus producing a randomized block design. The design, with moistureresistance measurements, is as shown in the figure (low readings indicate low moisture penetration). Analyze the experiment using a method appropriate for this randomized block design. Identify the blocks and treatments, and investigate any possible differences in treatment means. If any differences exist, use an appropriate method to specifically identify where the differences lie. What are the practical implications for the chemical producers? Has blocking been effective in this experiment? Present your results in the form of a report.

Blocks (bolt samples) 1 C 9.9 A 10.1 B 11.4 D 12.1

2 D 13.4 B 12.9 A 12.2 C 12.3

3 B 12.7 D 12.9 C 11.4 A 11.9

11.38 Glare in Rearview Mirrors An experiment was conducted to compare the glare characteristics of four types of automobile rearview mirrors. Forty drivers were randomly selected to participate in the experiment. Each driver was exposed to the glare produced by a headlight located 30 feet behind the rear window of the experimental automobile. The driver then rated the glare produced by the rearview mirror on a scale of 1 (low) to 10 (high). Each of the four mirrors was tested by each driver; the mirrors were assigned to a driver in random order. An analysis of variance of the data produced this ANOVA table: Source Mirrors Drivers Error Total

46.98 8.42 638.61

a. Fill in the blanks in the ANOVA table. b. Do the data present sufficient evidence to indicate differences in the mean glare ratings of the four rearview mirrors? Calculate the approximate p-value and use it to make your decision. c. Do the data present sufficient evidence to indicate that the level of glare perceived by the drivers varied from driver to driver? Use the p-value approach. d. Based on the results of part b, what are the practical implications of this experiment for the manufacturers of the rearview mirrors? 11.39 Slash Pine Seedings An experiment was conducted to determine the effects of three methods of soil preparation on the ﬁrst-year growth of slash pine seedlings. Four locations (state forest lands) were selected, and each location was divided into three plots. Since it was felt that soil fertility within a location was more homogeneous than between locations, a randomized block design was employed using locations

EX1139

476 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

as blocks. The methods of soil preparation were A (no preparation), B (light fertilization), and C (burning). Each soil preparation was randomly applied to a plot within each location. On each plot, the same number of seedlings were planted and the average ﬁrst-year growth of the seedlings was recorded on each plot. Use the MINITAB printout to answer the questions. Location Soil Preparation

A B C

11 15 10

13 17 15

16 20 13

10 12 10

a. Conduct an analysis of variance. Do the data provide evidence to indicate a difference in the mean growths for the three soil preparations? b. Is there evidence to indicate a difference in mean rates of growth for the four locations? c. Use Tukey’s method of paired comparisons to rank the mean growths for the three soil preparations. Use a .01. d. Use a 95% conﬁdence interval to estimate the difference in mean growths for methods A and B. MINITAB output for Exercise 11.39 Two-way ANOVA: Growth versus Soil Prep, Location Source Soil Prep Location Error Total S = 1.374

Soil Prep 1 2 3

Location 1 2 3 4

DF 2 3 6 11

SS 38.000 61.667 11.333 111.000

R-Sq = 89.79%

Mean 12.5 16.0 12.0

Mean 12.0000 15.0000 16.3333 10.6667

MS 19.0000 20.5556 1.8889

F 10.06 10.88

P 0.012 0.008

R-Sq(adj) = 81.28%

Individual 95% CIs For Mean Based on Pooled StDev ---------+---------+---------+---------+-(-------*-------) (-------*-------) (-------*-------) ---------+---------+---------+---------+12.0 14.0 16.0 18.0 Individual 95% CIs For Mean Based on Pooled StDev ------+---------+---------+---------+----(-------*-------) (-------*-------) (------*-------) (-------*------) -----+---------+---------+---------+----10.0 12.5 15.0 17.5

results are given in the table. Use the MINITAB printout to answer the questions. Dogs 1

A 1342 B 1608 C 1881

C 1698 B 1387 A 1140

B 1296 A 1029 C 1549

A 1150 C 1579 B 1319

a. How many degrees of freedom are associated with SSE? b. Do the data present sufficient evidence to indicate a difference in the mean uptakes of calcium for the three levels of digitalis? c. Use Tukey’s method of paired comparisons with a .01 to rank the mean calcium uptakes for the three levels of digitalis. d. Do the data indicate a difference in the mean uptakes of calcium for the four heart muscles? e. Use Tukey’s method of paired comparisons with a .01 to rank the mean calcium uptakes for the heart muscles of the four dogs used in the experiment. Are these results of any practical value to the researcher? f. Give the standard error of the difference between the mean calcium uptakes for two levels of digitalis. g. Find a 95% conﬁdence interval for the difference in mean responses between treatments A and B. MINITAB output for Exercise 11.40 Two-way ANOVA: Uptake versus Digitalis, Dog Source Digitalis Dog Error Total S = 31.86

Digitalis 1 2 3

DF 2 3 6 11

R-Sq = 99.13%

Mean 1165.25 1402.50 1676.75

11.40 Digitalis and Calcium Uptake A

study was conducted to compare the effects of three levels of digitalis on the levels of calcium in the heart muscles of dogs. Because general level of calcium uptake varies from one animal to another, the tissue for a heart muscle was regarded as a block, and comparisons of the three digitalis levels (treatments) were made within a given animal. The calcium uptakes for the three levels of digitalis, A, B, and C, were compared based on the heart muscles of four dogs and the

EX1140

Dog 1 2 3 4

SS 542177 173415 6090 703682

Mean 1610.33 1408.33 1291.33 1349.33

MS 262089 57805 1015

F 258.24 56.96

P 0.000 0.000

R-Sq(adj) = 98.41%

Individual 95% CIs For Mean Based on Pooled StDev -----+---------+---------+---------+---(--*-) (--*-) (--*-) -----+---------+---------+---------+---1200 1350 1500 1650 Individual 95% CIs For Mean Based on Pooled StDev ------+---------+---------+---------+--(---*---) (--*---) (---*--) (--*---) ------+---------+---------+---------+--1320 1440 1560 1680

11.41 Bidding on Construction Jobs A

building contractor employs three construction engineers, A, B, and C, to estimate and bid on jobs.

EX1141

11.8 THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN

To determine whether one tends to be a more conservative (or liberal) estimator than the others, the contractor selects four projected construction jobs and has each estimator independently estimate the cost (in dollars per square foot) of each job. The data are shown in the table: Construction Job Estimator A B C Total

Total

35.10 37.45 36.30

34.50 34.60 35.10

29.25 33.10 32.45

31.60 34.40 32.90

130.45 139.55 136.75

108.85

104.20

94.80

98.90

406.75

Analyze the experiment using the appropriate methods. Identify the blocks and treatments, and investigate any possible differences in treatment means. If any differences exist, use an appropriate method to speciﬁcally identify where the differences lie. Has blocking been effective in this experiment? What are the practical implications of the experiment? Present your results in the form of a report. 11.42 “In Good Hands” The cost of auto-

mobile insurance varies by location, ages of the drivers, and type of coverage. The following are estimates for the annual 2006–2007 premium for a single male, licensed for 6–8 years, who drives a Honda Accord 12,600 to 15,000 miles per year and has no violations or accidents. These estimates are provided by the California Department of Insurance for the year 2006–2007 on its website (http://www.insurance .ca.gov).2

EX1142

Insurance Company Location

21st Century

Riverside $1870 San Bernardino 2064 Hollywood 3542 Long Beach 2228

Allstate

AAA

$2250 2286 3773 2617

$2154 2316 3235 2681

Fireman’s State Fund Farm $2324 2005 3360 3279

$3053 3151 3883 3396

Source: www.insurance.ca.gov

a. What type of design was used in collecting these data? b. Is there sufficient evidence to indicate that insurance premiums for the same type of coverage differs from company to company? c. Is there sufficient evidence to indicate that insurance premiums vary from location to location? d. Use Tukey’s procedure to determine which insurance companies listed here differ from others in the premiums they charge for this typical client. Use a .05. e. Summarize your ﬁndings.

❍

477

11.43 Warehouse Shopping Warehouse

stores such as Costco and Sam’s Club are the shopping choice of many Americans because of the low cost associated with bulk shopping. When a new warehouse grocery store called WinCo Foods was opened in Moreno Valley, California, an advertising mailer claimed that they were the area’s “low price leader.”3 They compared their prices with those of four other grocery stores for a number of items purchased on the same day. A partial list of the items and their prices is given in the following table.

EX1143

Stores Items

WinCo Albertsons Ralphs

Salad mix, 1 lb. bag Hillshire Farm® Smoked Sausage, 16 oz. Kellogg’s Raisin Bran®, 25.5 oz. Kraft® Philadelphia® Cream Cheese, 8 oz. Kraft® Ranch Dressing, 16 oz. Langers® Apple Juice, 128 oz. Dial® Bar Soap, Gold, 8–4.5 oz. Jif® Peanut Butter, Creamy, 28 oz.

Stater Food-4Bros Less

0.88

1.99

1.79

1.89

0.98

2.48

4.29

2.50

3.00

3.68

2.48

4.99

4.69

3.49

3.38

1.18

1.50

1.99

1.89

1.97

1.58

3.89

2.69

2.50

1.68

1.98

4.99

4.59

3.79

2.98

3.48

5.79

4.19

3.99

4.58

2.58

4.89

3.99

3.89

2.68

a. What are the blocks and treatments in this experiment? b. Do the data provide evidence to indicate that there are signiﬁcant differences in prices from store to store? Support your answer statistically using the ANOVA printout that follows. c. Are there signiﬁcant differences from block to block? Was blocking effective? d. The advertisement includes the following statement: “Though this list is not intended to represent a typical weekly grocery order or a random list of grocery items, WinCo continues to be the area’s low price leader.” How might this statement affect the reliability of your conclusions in part b? Two-way ANOVA: Price versus Item, Store Source Item Store Error Total

DF 7 4 28 39

SS 38.2360 16.6644 7.8862 62.7866

MS 5.46228 4.16610 0.28165

S = 0.5307 R-Sq = 87.44%

F 19.39 14.79

P 0.000 0.000

R-Sq(adj) = 82.51%

478 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

11.44 Warehouse Shopping, continued Refer to Exercise 11.43. The printout that follows provides the average costs of the selected items for the k 5 stores. Store

Mean

Albertsons Food-4-Less Ralphs Stater Bros WinCo

4.04125 2.74125 3.30375 3.05500 2.08000

a. What is the appropriate value of q.05(k, df ) for testing for differences among stores?

冪莦

MSE b. What is the value of v q.05(k, df ) ? b c. Use Tukey’s pairwise comparison test among stores used to determine which stores differ signiﬁcantly in average prices of the selected items.

THE a b FACTORIAL EXPERIMENT: A TWO-WAY CLASSIFICATION

11.9

Suppose the manager of a manufacturing plant suspects that the output (in number of units produced per shift) of a production line depends on two factors: • •

Which of two supervisors is in charge of the line Which of three shifts—day, swing, or night—is being measured

That is, the manager is interested in two factors: “supervisor” at two levels and “shift” at three levels. Can you use a randomized block design, designating one of the two factors as a block factor? In order to do this, you would need to assume that the effect of the two supervisors is the same, regardless of which shift you are considering. This may not be the case; maybe the ﬁrst supervisor is most effective in the morning, and the second is more effective at night. You cannot generalize and say that one supervisor is better than the other or that the output of one particular shift is best. You need to investigate not only the average output for the two supervisors and the average output for the three shifts, but also the interaction or relationship between the two factors. Consider two different examples that show the effect of interaction on the responses in this situation. EXAMPLE

TABLE 11.4

11.11

Suppose that the two supervisors are each observed on three randomly selected days for each of the three different shifts. The average outputs for the three shifts are shown in Table 11.4 for each of the supervisors. Look at the relationship between the two factors in the line chart for these means, shown in Figure 11.11. Notice that supervisor 2 always produces a higher output, regardless of the shift. The two factors behave independently; that is, the output is always about 100 units higher for supervisor 2, no matter which shift you look at.

●

Average Outputs for Two Supervisors on Three Shifts Shift Supervisor

Day

Swing

Night

1 2

487 602

498 602

550 637

11.9 THE a b FACTORIAL EXPERIMENT: A TWO-WAY CLASSIFICATION

FIGU R E 1 1 . 1 1

Interaction plot for means in Table 11.4

❍

479

●

Interaction Plot (data means) for Response 650

Supervisor 1 2

625

Mean

600 575 550 525 500 Day

Swing Shift

Night

Now consider another set of data for the same situation, shown in Table 11.5. There is a deﬁnite difference in the results, depending on which shift you look at, and the interaction can be seen in the crossed lines of the chart in Figure 11.12. TABLE 11.5

●

Average Outputs for Two Supervisors on Three Shifts Shift

Interaction plot for means in Table 11.5

Day

Swing

Night

1 2

602 487

498 602

450 657

●

Interaction Plot (data means) for Response Supervisor 1 2

650

600 Mean

FIGU R E 1 1 . 1 2

Supervisor

550

500

450 Day

Swing Shift

Night

480 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

When the effect of one factor on the response changes, depending on the level at which the other factor is measured, the two factors are said to interact.

11.10

This situation is an example of a factorial experiment in which there are a total of 2 3 possible combinations of the levels for the two factors. These 2 3 6 combinations form the treatments, and the experiment is called a 2 3 factorial experiment. This type of experiment can actually be used to investigate the effects of three or more factors on a response and to explore the interactions between the factors. However, we conﬁne our discussion to two factors and their interaction. When you compare treatment means for a factorial experiment (or for any other experiment), you will need more than one observation per treatment. For example, if you obtain two observations for each of the factor combinations of a complete factorial experiment, you have two replications of the experiment. In the next section on the analysis of variance for a factorial experiment, you can assume that each treatment or combination of factor levels is replicated the same number of times r.

THE ANALYSIS OF VARIANCE FOR AN a b FACTORIAL EXPERIMENT An analysis of variance for a two-factor factorial experiment replicated r times follows the same pattern as the previous designs. If the letters A and B are used to identify the two factors, the total variation in the experiment Total SS S(x 苶x )2 Sx2 CM is partitioned into four parts in such a way that Total SS SSA SSB SS(AB) SSE where • • • •

SSA (sum of squares for factor A) measures the variation among the factor A means. SSB (sum of squares for factor B) measures the variation among the factor B means. SS(AB) (sum of squares for interaction) measures the variation among the different combinations of factor levels. SSE (sum of squares for error) measures the variation of the differences among the observations within each combination of factor levels—the experimental error.

Sums of squares SSA and SSB are often called the main effect sums of squares, to distinguish them from the interaction sum of squares. Although you can simplify your work by using a computer program to calculate these sums of squares, the calculational formulas are given next. You can assume that there are: • • • •

a levels of factor A b levels of factor B r replications of each of the ab factor combinations A total of n abr observations

11.10 THE ANALYSIS OF VARIANCE FOR AN a b FACTORIAL EXPERIMENT

❍

481

CALCULATING THE SUMS OF SQUARES FOR A TWO-FACTOR FACTORIAL EXPERIMENT G2 CM n

Total SS Sx 2 CM

A2 SSA Si CM br

B2j

SSB Sar CM

(AB)2ij

SS(AB) Sr CM SSA SSB where G Sum of all n abr observations Ai Total of all observations at the ith level of factor A, i 1, 2, . . . , a Bj Total of all observations at the jth level of factor B, j 1, 2, . . . , b (AB)ij Total of the r observations at the ith level of factor A and the jth level of factor B

Each of the ﬁve sources of variation, when divided by the appropriate degrees of freedom, provides an estimate of the variation in the experiment. These estimates are called mean squares—MS SS/df—and are displayed along with their respective sums of squares and df in the analysis of variance (or ANOVA) table.

ANOVA TABLE FOR r REPLICATIONS OF A TWO-FACTOR FACTORIAL EXPERIMENT: FACTOR A AT a LEVELS AND FACTOR B AT b LEVELS Source

SSA

SSA MSA a1

MSA MSE

SSB

SSB MSB b1

MSB MSE

(a 1)(b 1)

SS(AB)

SS(AB) MS(AB) (a 1) (b 1)

MS(AB) MSE

Error

ab (r 1)

SSE

SSE MSE ab(r 1)

Total

abr 1

Total SS

Finally, the equality of means for various levels of the factor combinations (the interaction effect) and for the levels of both main effects, A and B, can be tested using the ANOVA F-tests, as shown next.

482 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

TESTS FOR A FACTORIAL EXPERIMENT •

For interaction:

1. Null hypothesis: H0 : Factors A and B do not interact 2. Alternative hypothesis: Ha : Factors A and B interact 3. Test statistic: F MS(AB)/MSE, where F is based on df1 (a 1)(b 1) and df2 ab(r 1) 4. Rejection region: Reject H0 when F Fa, where Fa lies in the upper tail of the F distribution (see the ﬁgure), or when the p-value a •

For main effects, factor A:

1. Null hypothesis: H0 : There are no differences among the factor A means 2. Alternative hypothesis: Ha : At least two of the factor a means differ 3. Test statistic: F MSA/MSE, where F is based on df1 (a 1) and df2 ab(r 1) 4. Rejection region: Reject H0 when F Fa (see the ﬁgure) or when the p-value a •

For main effects, factor B:

1. Null hypothesis: H0 : There are no differences among the factor B means 2. Alternative hypothesis: Ha : At least two of the factor B means differ 3. Test statistic: F MSB/MSE, where F is based on df1 (b 1) and df2 ab(r 1) 4. Rejection region: Reject H0 when F Fa (see the ﬁgure) or when the p-value a

f(F)

EXAMPLE

11.12

Fα

Table 11.6 shows the original data used to generate Table 11.5 in Example 11.11. That is, the two supervisors were each observed on three randomly selected days for each of the three different shifts, and the production outputs were recorded. Analyze these data using the appropriate analysis of variance procedure.

11.10 THE ANALYSIS OF VARIANCE FOR AN a b FACTORIAL EXPERIMENT

TABLE 11.6

●

❍

483

Outputs for Two Supervisors on Three Shifts Shift Supervisor

Day

Swing

Night

571 610 625

480 474 540

470 430 450

480 516 465

625 600 581

630 680 661

The computer output in Figure 11.13 was generated using the two-way analysis of variance procedure in the MINITAB software package. You can verify the quantities in the ANOVA table using the calculational formulas presented earlier, or you may choose just to use the results and interpret their meaning. Solution

FIGU R E 1 1 . 1 3

MINITAB output for Example 11.12

● Two-way ANOVA: Output versus Supervisor, Shift Source Supervisor Shift Interaction Error Total S = 26.83

Supervisor 1 2

Shift Day Swing Night

If the interaction is not signiﬁcant, test each of the factors individually.

DF 1 2 2 12 17

SS 19208 247 81127 8640 109222

R-Sq = 92.09%

Mean 516.667 582.000

Mean 544.5 550.0 553.5

MS 19208.0 123.5 40563.5 720.0

F 26.68 0.17 56.34

P 0.000 0.844 0.000

R-Sq(adj) = 88.79%

Individual 95% CIs For Mean Based on Pooled StDev ----+---------+---------+---------+----(-------*------) (-------*-------) ----+---------+---------+---------+----510 540 570 600 Individual 95% CIs For Mean Based on Pooled StDev ---+---------+---------+---------+-------(---------------*---------------) (---------------*---------------) (---------------*---------------) ---+---------+---------+---------+-------525 540 555 570

At this point, you have undoubtedly discovered the familiar pattern in testing the signiﬁcance of the various experimental factors with the F statistic and its p-value. The small p-value (P .000) in the row marked “Supervisor” means that there is sufficient evidence to declare a difference in the mean levels for factor A—that is, a difference in mean outputs per supervisor. This fact is visually apparent in the nonoverlapping conﬁdence intervals for the supervisor means shown in the printout. But this is overshadowed by the fact that there is strong evidence (P .000) of an interaction between factors A and B. This means that the average output for a given shift depends on the supervisor on duty. You saw this effect clearly in Figure 11.11. The three largest mean outputs occur when supervisor 1 is on the day shift and when supervisor 2 is on either the swing or night shift. As a practical result, the manager should schedule supervisor 1 for the day shift and supervisor 2 for the night shift.

484 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

If the interaction effect is significant, the differences in the treatment means can be further studied, not by comparing the means for factor A or B individually but rather by looking at comparisons for the 2 3 (AB) factor-level combinations. If the interaction effect is not significant, then the significance of the main effect means should be investigated, first with the overall F-test and next with Tukey’s method for paired comparisons and/or specific confidence intervals. Remember that these analysis of variance procedures always use s 2 MSE as the best estimator of s 2 with degrees of freedom equal to df ab(r 1). For example, using Tukey’s yardstick to compare the average outputs for the two supervisors on each of the three shifts, you could calculate 苶 s 0 兹72 v q.05(6, 12) 4.75 73.59 兹3苶兹苶r

冢冣

冢

冣

Since all three pairs of means—602 and 487 on the day shift, 498 and 602 on the swing shift, and 450 and 657 on the night shift—differ by more than v, our practical conclusions have been conﬁrmed statistically.

11.10

EXERCISES

BASIC TECHNIQUES 11.45 Suppose you were to conduct a two-factor factorial experiment, factor A at four levels and factor B at ﬁve levels, with three replications per treatment. a. How many treatments are involved in the experiment? b. How many observations are involved? c. List the sources of variation and their respective degrees of freedom. 11.46 The analysis of variance table for a 3 4 fac-

torial experiment, with factor A at three levels and factor B at four levels, and with two observations per treatment, is shown here: Source

Total

2 3 6 12

5.3 9.1 24.5

43.7

a. Fill in the missing items in the table. b. Do the data provide sufficient evidence to indicate that factors A and B interact? Test using a .05. What are the practical implications of your answer? c. Do the data provide sufficient evidence to indicate that factors A and B affect the response variable x? Explain.

11.47 Refer to Exercise 11.46. The means of two of the factor-level combinations—say, A1B1 and A2B1— are x苶1 8.3 and 苶x2 6.3, respectively. Find a 95% conﬁdence interval for the difference between the two corresponding population means. 11.48 The table gives data for a 3 3 facto-

rial experiment, with two replications per treatment:

EX1148

Levels of Factor A Levels of Factor B 1 2 3

5, 7 8, 7 14, 11

9, 7 12, 13 8, 9

4, 6 7, 10 12, 15

a. Perform an analysis of variance for the data, and present the results in an analysis of variance table. b. What do we mean when we say that factors A and B interact? c. Do the data provide sufficient evidence to indicate interaction between factors A and B? Test using a .05. d. Find the approximate p-value for the test in part c. e. What are the practical implications of your results in part c? Explain your results using a line graph similar to the one in Figure 11.11. 11.49 2 2 Factorial The table gives data for a 2 2 factorial experiment, with four replications per treatment.

EX1149

11.10 THE ANALYSIS OF VARIANCE FOR AN a b FACTORIAL EXPERIMENT

2.1, 2.7, 2.4, 2.5

3.7, 3.2, 3.0, 3.5

3.1, 3.6, 3.4, 3.9

2.9, 2.7, 2.2, 2.5

11.50 Demand for Diamonds A chain of

jewelry stores conducted an experiment to investigate the effect of price and location on the demand for its diamonds. Six small-town stores were selected for the study, as well as six stores located in large suburban malls. Two stores in each of these locations were assigned to each of three item percentage markups. The percentage gain (or loss) in sales for each store was recorded at the end of 1 month. The data are shown in the accompanying table.

EX1150

a. The accompanying graph was generated by MINITAB. Verify that the four points that connect the two lines are the means of the four observations within each factor-level combination. What does the graph tell you about the interaction between factors A and B? MINITAB interaction plot for Exercise 11.49

Markup Location

Small towns

10 4

3 7

10 24

Suburban malls

14 18

8 3

4 3

Interaction Plot (data means) for Response Factor A 1 2

3.50

Mean

3.25

3.00

2.75

2.50

2 Factor B

b. Use the MINITAB output to test for a signiﬁcant interaction between A and B. Does this conﬁrm your conclusions in part a? MINITAB output for Exercise 11.49 Two-way ANOVA: Response versus Factor A, Factor B Source Factor A Factor B Interaction Error Total S = 0.3007

DF 1 1 1 12 15

SS 0.0000 0.0900 3.4225 1.0850 4.5975

R-Sq = 76.40%

485

APPLICATIONS

Levels of Factor A Levels of Factor B

❍

MS 0.00000 0.09000 3.42250 0.09042

F 0.00 1.00 37.85

P 1.000 0.338 0.000

R-Sq(adj) = 70.50%

c. Considering your results in part b, how can you explain the fact that neither of the main effects is significant? d. If a significant interaction is found, is it necessary to test for significant main effect differences? Explain. e. Write a short paragraph summarizing the results of this experiment.

a. Do the data provide sufficient evidence to indicate an interaction between markup and location? Test using a .05. b. What are the practical implications of your test in part a? c. Draw a line graph similar to Figure 11.11 to help visualize the results of this experiment. Summarize the results. d. Find a 95% conﬁdence interval for the difference in mean change in sales for stores in small towns versus those in suburban malls if the stores are using price markup 3. 11.51 Terrain Visualization A study was conducted to determine the effect of two factors on terrain visualization training for soldiers.4 During the training programs, participants viewed contour maps of various terrains and then were permitted to view a computer reconstruction of the terrain as it would appear from a speciﬁed angle. The two factors investigated in the experiment were the participants’ spatial abilities (abilities to visualize in three dimensions) and the viewing procedures (active or passive). Active participation permitted participants to view the computer-generated reconstructions of the terrain from any and all angles. Passive participation gave the participants a set of preselected reconstructions of the terrain. Participants were tested according to spatial ability, and from the test scores 20 were categorized as possessing high spatial ability, 20 medium, and 20 low. Then 10 participants within each of these groups were assigned to each of the two training modes, active or passive. The

486 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

accompanying tables are the ANOVA table computed the researchers and the table of the treatment means. Source

Main effects: Training condition Ability Interaction: Training condition Ability Within cells

1 2

2 54

MINITAB output for Exercise 11.52 Two-way ANOVA: Cost versus City, Distance

Error df

Source City Distance Interaction Error Total

103.7009 760.5889

54 54

3.66 26.87

.0610 .0005

S = 5.737

124.9905 28.3015

4.42

.0167

DF 2 3 6 12 23

SS 201.33 1873.33 303.67 395.00 2773.33

R-Sq = 85.76%

Distance 1 2 3 4

MS 100.667 624.444 50.611 32.917

F 3.06 18.97 1.54

P 0.084 0.000 0.247

R-Sq(adj) = 72.70%

Individual 95% CIs For Mean Based on Pooled StDev ------+---------+---------+---------+--(-----+------) (-----+-----) (------+-----) (-----+-----) ------+---------+---------+---------+--10 20 30 40

Mean 32.1667 19.1667 11.8333 9.5000

Training Condition Spatial Ability

Active

Passive

High Medium Low

17.895 5.031 1.728

9.508 5.648 1.610

MINITAB plots for Exercise 11.52 Interaction Plot (data means) for Cost 45

Note: Maximum score 36.

City Chicago Houston NY

40 35 30 Mean

a. Explain how the authors arrived at the degrees of freedom shown in the ANOVA table. b. Are the F-values correct? c. Interpret the test results. What are their practical implications? d. Use Table 6 in Appendix I to approximate the pvalues for the F statistics shown in the ANOVA table.

25 20 15 10 1

Distance

Source: H.F. Barsam and Z.M. Simutis, “Computer-Based Graphics for Terrain Visualization Training,” Human Factors, no. 26, 1984. Copyright 1984 by the Human Factors Society, Inc. Reproduced by permission.

Main Effects Plot (data means) for Cost City

Distance

11.52 The Cost of Flying In an attempt to determine what factors affect airfares, a researcher recorded a weighted average of the costs per mile for two airports in each of three major U.S. cities for each of four different travel distances.5 The results are shown in the table.

EX1152

City

Cost

Distance

New York

Houston

Chicago

300 miles 301–750 miles 751–1500 miles 1500 miles

40, 48 19, 26 10, 14 9, 10

20, 26 15, 17 10, 13 8, 11

19, 40 14, 24 9, 15 7, 12

Use the MINITAB output to analyze the experiment with the appropriate method. Identify the two factors, and investigate any possible effect due to their interaction or the main effects. What are the practical implications of this experiment? Explain your conclusions in the form of a report.

Chicago

Houston

11.53 Fourth-Grade Test Scores A local school board was interested in comparing test scores on a standarized reading test for fourth-grade students in their district. They selected a random sample of ﬁve male and ﬁve female fourth grade students at each of four different elementary schools in the district and recorded the test scores. The results are shown in the table below.

EX1153

11.11 REVISITING THE ANALYSIS OF VARIANCE ASSUMPTIONS

Gender

School 1

School 2

School 3

School 4

Male

631 566 620 542 560

642 710 649 596 660

651 611 755 693 620

350 565 543 509 494

Female

669 644 600 610 559

722 769 723 649 766

709 545 657 722 711

505 498 474 470 463

a. What type of experimental design is this? What are the experimental units? What are the factors and levels of interest to the school board? b. Perform the appropriate analysis of variance for this experiment. c. Do the data indicate that effect of gender on the average test score is different depending on the student’s school? Test the appropriate hypothesis using a .05. d. Plot the average scores using an interaction plot. How would you describe the effect of gender and school on the average test scores? e. Do the data indicate that either of the main effects is signiﬁcant? If the main effect is signiﬁcant, use Tukey’s method of paired comparisons to examine the differences in detail. Use a .01. 11.54 Management Training An experiment was conducted to investigate the effect of management training on the decision-making abilities of supervisors in a large corporation. Sixteen supervisors were selected, and eight were randomly chosen to receive managerial training. Four trained and four untrained supervisors were then randomly selected to function in a situation in which a standard problem arose. The other eight supervisors were presented with an emergency situation in which standard procedures could not be used. The response was a management behavior rating for each supervisor as assessed by a rating scheme devised by the experimenter.

11.11

❍

487

a. What are the experimental units in this experiment? b. What are the two factors considered in the experiment? c. What are the levels of each factor? d. How many treatments are there in the experiment? e. What type of experimental design has been used? 11.55 Management Training, continued

Refer to Exercise 11.54. The data for this experiment are shown in the table.

EX1155

Training (A) Situation (B)

Trained

Not Trained

Totals

Standard

85 91 80 78

53 49 38 45

519

Emergency

76 67 82 71

40 52 46 39

473

630

362

992

Totals

a. Construct the ANOVA table for this experiment. b. Is there a significant interaction between the presence or absence of training and the type of decision-making situation? Test at the 5% level of significance. c. Do the data indicate a significant difference in behavior ratings for the two types of situations at the 5% level of significance? d. Do behavior ratings differ significantly for the two types of training categories at the 5% level of significance. e. Plot the average scores using an interaction plot. How would you describe the effect of training and emergency situation on the decision-making abilities of the supervisors?

REVISITING THE ANALYSIS OF VARIANCE ASSUMPTIONS In Section 11.3, you learned that the assumptions and test procedures for the analysis of variance are similar to those required for the t and F-tests in Chapter 10—namely, that observations within a treatment group must be normally distributed with common variance s 2. You also learned that the analysis of variance procedures are fairly

488 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

robust when the sample sizes are equal and the data are fairly mound-shaped. If this is the case, one way to protect yourself from inaccurate conclusions is to try when possible to select samples of equal sizes! There are some quick and simple ways to check the data for violation of assumptions. Look ﬁrst at the type of response variable you are measuring. You might immediately see a problem with either the normality or common variance assumption. It may be that the data you have collected cannot be measured quantitatively. For example, many responses, such as product preferences, can be ranked only as “A is better than B” or “C is the least preferable.” Data that are qualitative cannot have a normal distribution. If the response variable is discrete and can assume only three values— say, 0, 1, or 2—then it is again unreasonable to assume that the response variable is normally distributed. Suppose that the response variable is binomial—say, the proportion p of people who favor a particular type of investment. Although binomial data can be approximately mound-shaped under certain conditions, they violate the equal variance assumption. The variance of a sample proportion is pq p(1 p) s 2 n n so that the variance changes depending on the value of p. As the treatment means change, the value of p changes and so does the variance s 2. A similar situation occurs when the response variable is a Poisson random variable—say, the number of industrial accidents per month in a manufacturing plant. Since the variance of a Poisson random variable is s 2 m, the variance changes exactly as the treatment mean changes. If you cannot see any ﬂagrant violations in the type of data being measured, look at the range of the data within each treatment group. If these ranges are nearly the same, then the common variance assumption is probably reasonable. To check for normality, you might make a quick dotplot or stem and leaf plot for a particular treatment group. However, quite often you do not have enough measurements to obtain a reasonable plot. If you are using a computer program to analyze your experiment, there are some valuable diagnostic tools you can use. These procedures are too complicated to be performed using hand calculations, but they are easy to use when the computer does all the work!

Residual Plots In the analysis of variance, the total variation in the data is partitioned into several parts, depending on the factors identiﬁed as important to the researcher. Once the effects of these sources of variation have been removed, the “leftover” variability in each observation is called the residual for that data point. These residuals represent experimental error, the basic variability in the experiment, and should have an approximately normal distribution with a mean of 0 and the same variation for each treatment group. Most computer packages will provide options for plotting these residuals: •

The normal probability plot of residuals is a graph that plots the residuals for each observation against the expected value of that residual had it come from a normal distribution. If the residuals are approximately normal, the plot will closely resemble a straight line, sloping upward to the right.

11.11 REVISITING THE ANALYSIS OF VARIANCE ASSUMPTIONS

•

11.13

FIGU R E 1 1 . 1 4

The data from Example 11.4 involving the attention spans of three groups of elementary students were analyzed using MINITAB. The graphs in Figure 11.14, generated by MINITAB, are the normal probability plot and the residuals versus ﬁt plot for this experiment. Look at the straight-line pattern in the normal probability plot, which indicates a normal distribution in the residuals. In the other plot, the residuals are plotted against the estimated expected values, which are the sample averages for each of the three treatments in the completely randomized design. The random scatter around the horizontal “zero error line” and the constant spread indicate no violations in the constant variance assumption. ● Residuals versus the Fitted Values (response is Span)

Normal Probability Plot of the Residuals (response is Span)

Percent

MINITAB diagnostic plots for Example 11.13

489

The plot of residuals versus ﬁt or residuals versus variables is a graph that plots the residuals against the expected value of that observation using the experimental design we have used. If no assumptions have been violated and there are no “leftover” sources of variation other than experimental error, this plot should show a random scatter of points around the horizontal “zero error line” for each treatment group, with approximately the same vertical spread.

95 90

3 2

80 70 60 50 40 30 20

Residual

EXAMPLE

❍

1 0

10 5

2 3

EXAMPLE

TABLE 11.7

11.14

5.0

2.5

0.0 Residual

2.5

5.0

12 Fitted Value

A company plans to promote a new product by using one of three advertising campaigns. To investigate the extent of product recognition from these three campaigns, 15 market areas were selected and ﬁve were randomly assigned to each advertising plan. At the end of the ad campaigns, random samples of 400 adults were selected in each area and the proportions who were familiar with the new product were recorded, as in Table 11.7. Have any of the analysis of variance assumptions been violated in this experiment? ●

Proportions of Product Recognition for Three Advertising Campaigns Campaign 1

Campaign 2

Campaign 3

.33 .29 .21 .32 .25

.28 .41 .34 .39 .27

.21 .30 .26 .33 .31

Solution The experiment is designed as a completely randomized design, but the response variable is a binomial sample proportion. This indicates that both the normality and the common variance assumptions might be invalid. Look at the normal probability plot of the residuals and the plot of residuals versus ﬁt generated as an option in the MINITAB analysis of variance procedure and shown in Figure 11.15. The

490 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

curved pattern in the normal probability plot indicates that the residuals do not have a normal distribution. In the residual versus ﬁt plot, you can see three vertical lines of residuals, one for each of the three ad campaigns. Notice that two of the lines (campaigns 1 and 3) are close together and have similar spread. However, the third line (campaign 2) is farther to the right, which indicates a larger sample proportion and consequently a larger variance in this group. Both analysis of variance assumptions are suspect in this experiment.

FI GU R E 1 1 . 1 5

MINITAB diagnostic plots for Example 11.14

● Residuals versus the Fitted Values (response is Proportion)

Normal Probability Plot of the Residuals (response is Proportion) 0.08

0.06 0.04

80 70 60 50 40 30 20

Residual

Percent

95 90

0.00 0.02 0.04

0.06

5 1

0.02

0.08 0.10

0.05

0.00 Residual

0.05

0.10

0.15

0.28

0.29

0.30

0.31 Fitted Value

0.32

0.33

0.34

What can you do when the ANOVA assumptions are not satisﬁed? The constant variance assumption can often be remedied by transforming the response measurements. That is, instead of using the original measurements, you might use their square roots, logarithms, or some other function of the response. Transformations that tend to stabilize the variance of the response also tend to make their distributions more nearly normal. When nothing can be done to even approximately satisfy the ANOVA assumptions or if the data are rankings, you should use nonparametric testing and estimation procedures, presented in Chapter 15. We have mentioned these procedures before; they are almost as powerful in detecting treatment differences as the tests presented in this chapter when the data are normally distributed. When the parametric ANOVA assumptions are violated, the nonparametric tests are generally more powerful.

11.12

A BRIEF SUMMARY We presented three different experimental designs in this chapter, each of which can be analyzed using the analysis of variance procedure. The objective of the analysis of variance is to detect differences in the mean responses for experimental units that have received different treatments—that is, different combinations of the experimental factor levels. Once an overall test of the differences is performed, the nature of these differences (if any exist) can be explored using methods of paired comparisons and/or interval estimation procedures. The three designs presented in this chapter represent only a brief introduction to the subject of analyzing designed experiments. Designs are available for experiments that involve several design variables, as well as more than two treatment factors and other more complex designs. Remember that design variables are factors whose effect you want to control and hence remove from experimental error, whereas treatment

CHAPTER REVIEW

❍

491

variables are factors whose effect you want to investigate. If your experiment is properly designed, you will be able to analyze it using the analysis of variance. Experiments in which the levels of a variable are measured experimentally rather than controlled or preselected ahead of time may be analyzed using linear or multiple regression analysis—the subject of Chapters 12 and 13.

CHAPTER REVIEW Key Concepts and Formulas I.

Experimental Designs

1. Experimental units, factors, levels, treatments, response variables. 2. Assumptions: Observations within each treatment group must be normally distributed with a common variance s 2. 3. One-way classification—completely randomized design: Independent random samples are selected from each of k populations. 4. Two-way classification—randomized block design: k treatments are compared within b relatively homogeneous groups of experimental units called blocks. 5. Two-way classification—a b factorial experiment: Two factors, A and B, are compared at several levels. Each factor–level combination is replicated r times to allow for the investigation of an interaction between the two factors. II. Analysis of Variance

1. The total variation in the experiment is divided into variation (sums of squares) explained by the various experimental factors and variation due to experimental error (unexplained). 2. If there is an effect due to a particular factor, its mean square (MS SS/df ) is usually large and F MS(factor)/MSE is large. 3. Test statistics for the various experimental factors are based on F statistics, with appropriate degrees of freedom (df2 Error degrees of freedom).

III. Interpreting an Analysis of Variance

1. For the completely randomized and randomized block design, each factor is tested for significance. 2. For the factorial experiment, first test for a significant interaction. If the interaction is significant, main effects need not be tested. The nature of the differences in the factor–level combinations should be further examined. 3. If a significant difference in the population means is found, Tukey’s method of pairwise comparisons or a similar method can be used to further identify the nature of the differences. 4. If you have a special interest in one population mean or the difference between two population means, you can use a confidence interval estimate. (For a randomized block design, confidence intervals do not provide unbiased estimates for single population means.) IV. Checking the Analysis of Variance Assumptions

1. To check for normality, use the normal probability plot for the residuals. The residuals should exhibit a straight-line pattern, increasing upwards toward the right. 2. To check for equality of variance, use the residuals versus ﬁt plot. The plot should exhibit a random scatter, with the same vertical spread around the horizontal “zero error line.”

492 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

Analysis of Variance Procedures The statistical procedures used to perform the analysis of variance for the three different experimental designs in this chapter are found in a MINITAB submenu by choosing Stat 씮 ANOVA. You will see choices for One-way, One-way (Unstacked), and Two-way that will generate Dialog boxes used for the completely randomized, randomized block, and factorial designs, respectively. You must properly store the data and then choose the columns corresponding to the necessary factors in the experiment. We will display some of the Dialog boxes and Session window outputs for the examples in this chapter, beginning with a one-way classification—the completely randomized breakfast study in Example 11.4. First, enter the 15 recorded attention spans in column C1 of a MINITAB worksheet and name them “Span.” Next, enter the integers 1, 2, and 3 into a second column C2 to identify the meal assignment (treatment) for each observation. You can let MINITAB set this pattern for you using Calc 씮 Make Patterned Data 씮 Simple Set of Numbers and entering the appropriate numbers, as shown in Figure 11.16. Then use Stat 씮 ANOVA 씮 One-way to generate the Dialog box in Figure 11.17.† You must select the column of observations for the “Response” box and the column of treatment indicators for the “Factor” box. Then you have several options. Under Comparisons, you can select “Tukey’s family error rate” (which has a default level of 5%) to obtain paired comparisons output. Under Graphs, you can select individual value plots and/or box plots to compare the three meal assignments, and you can generate residual plots (use “Normal plot of residuals” and/or “Residuals versus fits”) to verify the validity of the ANOVA assumptions. Click OK from the main dialog box to obtain the output in Figure 11.3 in the text. The Stat 씮 ANOVA 씮 Two-way command can be used for both the randomized block and the factorial designs. You must first enter all of the observations into a single column and then integers or descriptive names to indicate either of these cases: • •

The block and treatment for each of the measurements in a randomized block design The levels of factors A and B for the factorial experiment.

MINITAB will recognize a number of replications within each factor-level combination in the factorial experiment and will break out the sum of squares for interaction (as long as you do not check the box “Fit additive model”). Since these two designs involve the same sequence of commands, we will use the data from Example 11.12 to generate the analysis of variance for the factorial experiment. The data are entered into the worksheet in Figure 11.18. See if you can use the Calc 씮 Make Patterned Data 씮 Simple Set of Numbers to enter the data into columns C2–C3. Once the data have been entered, use Stat 씮 ANOVA 씮 Two-way to generate the Dialog box in Figure 11.19. Choose “Output” for the “Response” box, and “Supervisor” and “Shift” for the “Row factor” and “Column factor,” respectively. You may choose to display the main effect means along with 95% conﬁdence intervals by checking “Display means,” and you may select residual plots if you wish. Click OK to obtain the ANOVA printout in Figure 11.13. †

If you had entered each of the three samples into separate columns, the proper command would have been Stat 씮 ANOVA 씮 One-way (Unstacked).

MY MINITAB

FIGU R E 1 1 . 1 6

●

FIGU R E 1 1 . 1 7

●

❍

493

494 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

FI GU R E 1 1 . 1 8

●

Data Display Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

FI GU R E 1 1 . 1 9

Output Supervisor 571 1 610 1 625 1 480 2 516 2 465 2 480 1 474 1 540 1 625 2 600 2 581 2 470 1 430 1 450 1 630 2 680 2 661 2

Shift 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3

●

Since the interaction between supervisors and shifts is highly signiﬁcant, you may want to explore the nature of this interaction by plotting the average output for each supervisor at each of the three shifts. Use Stat 씮 ANOVA 씮 Interactions Plot and choose the appropriate response and factor variables. The plot is generated by MINITAB and shown in Figure 11.20. You can see the strong difference in the behaviors of the mean outputs for the two supervisors, indicating a strong interaction between the two factors.

SUPPLEMENTARY EXERCISES

FIGU R E 1 1 . 2 0

❍

495

●

Supplementary Exercises 11.56 Reaction Times vs. Stimuli Twenty-

seven people participated in an experiment to compare the effects of ﬁve different stimuli on reaction time. The experiment was run using a completely randomized design, and, regardless of the results of the analysis of variance, the experimenters wanted to compare stimuli A and D. The results of the experiment are given here. Use the MINITAB printout to complete the exercise.

EX1156

MINITAB output for Exercise 11.56 One-way ANOVA: Time versus Stimulus Source Stimulus Error Total

DF 4 22 26

S = 0.1611

SS 1.2118 0.5711 1.7830

Level A B C D E

R-Sq = 67.97%

N 4 7 6 5 5

Pooled StDev =

Stimulus A B C D E

Reaction Time (sec) .8 .7 1.2 1.0 .6

.6 .8 1.0 .9 .4

.6 .5 .9 .9 .4

.5 .5 1.2 1.1 .7

.6 1.3 .7 .3

.9 .7 .8

Total

Mean

2.5 4.7 6.4 4.6 2.4

.625 .671 1.067 .920 .480

Mean 0.6250 0.6714 1.0667 0.9200 0.4800 0.1611

MS 0.3030 0.0260

F 11.67

P 0.000

R-Sq(adj) = 62.14%

StDev 0.1258 0.1496 0.1966 0.1483 0.1643

Individual 95% CIs For Mean Based on Pooled StDev -------+---------+---------+---------+-(------*------) (----*----) (-----*----) (-----*-----) (-----*-----) -------+---------+---------+---------+-0.50 0.75 1.00 1.25

a. Conduct an analysis of variance and test for a difference in the mean reaction times due to the ﬁve stimuli. b. Compare stimuli A and D to see if there is a difference in mean reaction times.

496 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

11.57 Refer to Exercise 11.56. Use this MINITAB output to identify the differences in the treatment means. MINITAB output for Exercise 11.57 Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Stimulus Individual confidence level = 99.29% Stimulus = A subtracted from: Stimulus B C D E

Lower -0.2535 0.1328 -0.0260 -0.4660

Center 0.0464 0.4417 0.2950 -0.1450

Upper 0.3463 0.7505 0.6160 0.1760

--------+---------+---------+---------+(-----*-----) (-----*-----) (-----*-----) (-----*-----) --------+---------+---------+---------+-0.50 0.00 0.50 1.00

Upper 0.6615 0.5288 0.0888

--------+---------+---------+---------+(-----*-----) (-----*-----) (-----*-----) --------+---------+---------+---------+-0.50 0.00 0.50 1.00

11.59 Reaction Times II The experiment in

Exercise 11.56 might have been conducted more effectively using a randomized block design with people as blocks, since you would expect mean reaction time to vary from one person to another. Hence, four people were used in a new experiment, and each person was subjected to each of the ﬁve stimuli in a random order. The reaction times (in seconds) are listed here:

EX1159

Stimulus = B subtracted from: Stimulus C D E

Lower 0.1290 -0.0316 -0.4716

Center 0.3952 0.2486 -0.1914

Stimulus = C subtracted from: Stimulus D E

Lower -0.4364 -0.8764

Center -0.1467 -0.5867

Upper 0.1431 -0.2969

--------+---------+---------+---------+(-----*-----) (-----*-----) --------+---------+---------+---------+-0.50 0.00 0.50 1.00

Stimulus = D subtracted from: Stimulus E

Lower -0.7426

Center -0.4400

Upper -0.1374

Stimulus Subject

1 2 3 4

.7 .6 .9 .6

.8 .6 1.0 .8

1.0 1.1 1.2 .9

1.0 1.0 1.1 1.0

.5 .6 .6 .4

--------+---------+---------+---------+(-----*-----) --------+---------+---------+---------+-0.50 0.00 0.50 1.00

MINITAB output for Exercise 11.59

11.58 Refer to Exercise 11.56. What do the normal probability plot and the residuals versus ﬁt plot tell you about the validity of your analysis of variance results? MINITAB diagnostic plots for Exercise 11.58

Two-way ANOVA: Time versus Subject, Stimulus Source Subject Stimulus Error Total

DF 3 4 12 19

S = 0.08416

R-Sq = 91.60%

Normal Probability Plot of the Residuals (response is Time)

Stimulus A B C D E

Percent

95 90 80 70 60 50 40 30 20 10 5 1

0.4

0.3

0.2

0.1

0.0 Residual

0.1

0.2

0.3

0.4

Residuals versus the Fitted Values (response is Time) 0.3 0.2

Residual

0.0 0.1 0.2 0.3 0.4

0.5

0.6

0.7 0.8 Fitted Value

0.9

1.0

1.1

MS 0.046667 0.196750 0.007083

F 6.59 27.78

P 0.007 0.000

R-Sq(adj) = 86.70%

Individual 95% CIs For Mean Based on Pooled StDev ---------+---------+---------+---------+(----*----) (----*----) (---*----) (---*----) (---*----) ---------+---------+---------+---------+0.60 0.80 1.00 1.20

a. Use the MINITAB printout to analyze the data and test for differences in treatment means. b. Use Tukey’s method of paired comparisons to identify the significant pairwise differences in the stimuli. c. Does it appear that blocking was effective in this experiment? 11.60 Heart Rate and Exercise An experiment was conducted to examine the effect of age on heart rate when a person is subjected to a specific amount of exercise. Ten male subjects were randomly selected from four age groups: 10–19, 20–39, 40–59, and 60–69. Each subject walked on a treadmill at a fixed grade for a period of 12 minutes, and the increase in heart rate, the difference before and after exercise, was recorded (in beats per minute):

EX1160

0.1

Mean 0.700 0.800 1.050 1.025 0.525

SS 0.140 0.787 0.085 1.012

SUPPLEMENTARY EXERCISES

Total

10–19

20–39

40–59

60–69

29 33 26 27 39 35 33 29 36 22

24 27 33 31 21 28 24 34 21 32

37 25 22 33 28 26 30 34 27 33

28 29 34 36 21 20 25 24 33 32

309

275

295

282

Use an appropriate computer program to answer these questions: a. Do the data provide sufficient evidence to indicate a difference in mean increase in heart rate among the four age groups? Test by using a .05. b. Find a 90% conﬁdence interval for the difference in mean increase in heart rate between age groups 10–19 and 60–69. c. Find a 90% conﬁdence interval for the mean increase in heart rate for the age group 20–39. d. Approximately how many people would you need in each group if you wanted to be able to estimate a group mean correct to within two beats per minute with probability equal to .95? 11.61 Learning to Sell A company wished

to study the effects of four training programs on the sales abilities of their sales personnel. Thirtytwo people were randomly divided into four groups of equal size, and each group was then subjected to one of the different sales training programs. Because there were some dropouts during the training programs due to illness, vacations, and so on, the number of trainees completing the programs varied from group to group. At the end of the training programs, each salesperson was randomly assigned a sales area from a group of sales areas that were judged to have equivalent sales potentials. The sales made by each of the four groups of salespeople during the ﬁrst week after completing the training program are listed in the table:

EX1161

Training Program

Total

78 84 86 92 69 73

99 86 90 93 94 85 97 91

74 87 80 83 78

81 63 71 65 86 79 73 70

482

735

402

588

❍

497

Analyze the experiment using the appropriate method. Identify the treatments or factors of interest to the researcher and investigate any signiﬁcant effects. What are the practical implications of this experiment? Write a paragraph explaining the results of your analysis. 11.62 4 2 Factorial Suppose you were to conduct a two-factor factorial experiment, factor A at four levels and factor B at two levels, with r replications per treatment. a. How many treatments are involved in the experiment?

b. How many observations are involved? c. List the sources of variation and their respective degrees of freedom. 11.63 2 3 Factorial The analysis of variance table for a 2 3 factorial experiment, factor A at two levels and factor B at three levels, with ﬁve observations per treatment, is shown in the table. Source

A B AB Error

1.14 2.58 .49

Total

8.41

a. Do the data provide sufficient evidence to indicate an interaction between factors A and B? Test using a .05. What are the practical implications of your answer? b. Give the approximate p-value for the test in part a. c. Do the data provide sufficient evidence to indicate that factor A affects the response? Test using a .05. d. Do the data provide sufficient evidence to indicate that factor B affects the response? Test using a .05. 11.64 Refer to Exercise 11.63. The means of all observations, at the factor A levels A1 and A2 are 苶x1 3.7 and 苶x2 1.4, respectively. Find a 95% confidence interval for the difference in mean response for factor levels A1 and A2. 11.65 The Whitefly in California The whitefly, which causes defoliation of shrubs and trees and a reduction in salable crop yields, has emerged as a pest in Southern California. In a study to determine factors that affect the life cycle of the whitefly, an experiment was conducted in which whiteflies were placed on two different types of plants at three

EX1165

498 ❍

CHAPTER 11 THE ANALYSIS OF VARIANCE

different temperatures. The observation of interest was the total number of eggs laid by caged females under one of the six possible treatment combinations. Each treatment combination was run using ﬁve cages. Temperature Plant

70°F

77°F

82°F

Cotton

37 21 36 43 31

34 54 40 42 16

46 32 41 36 38

Cucumber

50 53 25 37 48

59 53 31 69 51

43 62 71 49 59

S = 11.09

SS 1512.30 487.47 111.20 2952.40 5063.37

R-Sq = 41.69%

1.65 1.70 1.40 2.10

1.72 1.85 1.75 1.95

1.50 1.46 1.38 1.65

1.60 1.80 1.55 2.00

11.67 America’s Market Basket Exercise 10.40 examined an advertisement for Albertsons, a supermarket chain in the western United States. The advertiser claims that Albertsons has consistently had lower prices than four other full-service supermarkets. As part of a survey conducted by an “independent market basket price-checking company,” the average weekly total based on the prices of approximately 95 items is given for ﬁve different supermarket chains recorded during 4 consecutive weeks.6

MS 1512.30 243.73 55.60 123.02

F 12.29 1.98 0.45

P 0.002 0.160 0.642

R-Sq(adj) = 29.54%

Albertsons Ralphs

a. What type of experimental design has been used? b. Do the data provide sufficient evidence to indicate that the effect of temperature on the number of eggs laid is different depending on the type of plant? Use the MINITAB printout to test the appropriate hypothesis. c. Plot the treatment means for cotton as a function of temperature. Plot the treatment means for cucumber as a function of temperature. Comment on the similarity or difference in these two plots. d. Find the mean number of eggs laid on cotton and cucumber based on 15 observations each. Calculate a 95% conﬁdence interval for the difference in the underlying population means.

Week 1 $254.26 Week 2 240.62 Week 3 231.90 Week 4 234.13

11.66 Pollution from Chemical Plants

Four chemical plants, producing the same product and owned by the same company, discharge effluents into streams in the vicinity of their locations. To check on the extent of the pollution created by the effluents and to determine whether this varies from plant to plant, the company collected random samples of liquid waste, ﬁve specimens for each of the four plants. The data are shown in the table:

$256.03 255.65 255.12 261.18

Vons

Alpha Beta Lucky

$267.92 251.55 245.89 254.12

$260.71 251.80 246.77 249.45

$258.84 242.14 246.80 248.99

a. What type of design has been used in this experiment? b. Conduct an analysis of variance for the data. c. Is there sufficient evidence to indicate that there is a difference in the average weekly totals for the five supermarkets? Use a .05. d. Use Tukey’s method for paired comparisons to determine which of the means are significantly different from each other. Use a .05. 11.68 Yield of Wheat The yields of wheat (in bushels per acre) were compared for five different varieties, A, B, C, D, and E, at six different locations. Each variety was randomly assigned to a plot at each location. The results of the experiment are shown in the accompanying table, along with a MINITAB printout of the analysis of variance. Analyze the experiment using the appropriate method. Identify the treatments or factors of interest to the researcher and investigate any effects that exist. Use the diagnostic plots to comment on the validity of the analysis of

EX1168 EX1166

1.37 2.05 1.65 1.88

EX1167

Two-way ANOVA: Eggs versus Plant, Temperature DF 1 2 2 24 29

Polluting Effluents (lb/gal of waste)

A B C D

a. Do the data provide sufficient evidence to indicate a difference in the mean amounts of effluents discharged by the four plants? b. If the maximum mean discharge of effluents is 1.5 lb/gal, do the data provide sufficient evidence to indicate that the limit is exceeded at plant A? c. Estimate the difference in the mean discharge of effluents between plants A and D, using a 95% conﬁdence interval.

MINITAB output for Exercise 11.65

Source Plant Temperature Interaction Error Total

Plant

SUPPLEMENTARY EXERCISES

variance assumptions. What are the practical implications of this experiment? Write a paragraph explaining the results of your analysis. Location Variety A B C D E

35.3 30.7 38.2 34.9 32.4

31.0 32.2 33.4 36.1 28.9

32.7 31.4 33.6 35.2 29.2

36.8 31.7 37.1 38.3 30.7

37.2 35.0 37.3 40.2 33.9

33.1 32.7 38.2 36.0 32.1

MINITAB output for Exercise 11.68

DF 4 5 20 29

S = 1.384

SS 142.670 68.142 38.303 249.142

R-Sq = 84.62%

MS 35.6675 13.6283 1.9165

Physical Activity

F 18.61 7.11

P 0.000 0.001

Normal Probability Plot of the Residuals (response is Yield) 99

Percent

95 90 80 70 60 50 40 30 20 10 5

0 Residual

Residuals versus the Fitted Values (response is Yield) 2

1 Residual

Same

Less

50.1 47.2 49.7 50.4

45.7 44.2 46.8 44.9

40.9 41.3 39.2 40.9

Females

41.2 39.8 41.5 38.2

37.2 39.4 38.6 37.8

36.5 35.0 37.2 35.4

a. Is this a factorial experiment or a randomized block design? Explain. b. Is there a significant interaction between levels of physical activity and gender? Are there significant differences between males and females? Levels of physical activity? c. If the interaction is significant, use Tukey’s pairwise procedure to investigate differences among the six cell means. Comment on the results found using this procedure. Use a .05.

MINITAB diagnostic plots for Exercise 11.68

More Males

R-Sq(adj) = 77.69%

Individual 95% CIs For Mean Based on Pooled StDev Mean +---------+---------+---------+--------34.3500 (-----*-----) 32.2833 (----*-----) 36.3000 (-----*----) 36.7833 (-----*-----) 31.2000 (-----*-----) +---------+---------+---------+--------30.0 32.0 34.0 36.0

Varieties A B C D E

499

to assess cardiorespiratory fitness levels in youth aged 12 to 19 years.7 Attaining fitness standards is a common prerequisite for entry into occupations such as law enforcement, firefighting, and the military, as well as other jobs that involve physically demanding labor. Estimated maximum oxygen uptake (VO2max) was used to measure a person’s cardiorespiratory level. The focus of our study investigates the relationship between levels of physical activity (more than others, same as others, or less than others) and gender on VO2max. The data that follows are based on this study.

Two-way ANOVA: Yield versus Varieties, Location Source Varieties Locations Error Total

❍

11.70 In a study of starting salaries of assistant professors,8 ﬁve male assistant professors and ﬁve female assistant professors at each of three types of institutions granting doctoral degrees were polled and their initial starting salaries were recorded under the condition of anonymity. The results of the survey in $1000 are given in the following table.

EX1170

Gender

Public Universities

Private/Independent

Church-Related

$57.3 57.9 56.5 76.5 62.0

$85.8 75.2 66.9 73.0 73.0

$78.9 69.3 69.7 58.2 61.2

47.4 56.7 69.0 63.2 65.3

62.1 69.1 66.5 61.8 76.7

60.4 62.1 59.8 71.9 61.6

Males

2 30

34 36 Fitted Value

Females

11.69 Physical Fitness Researchers Russell R. Pate and colleagues analyzed the results of the National Health and Nutrition Examination Survey

EX1169

Source: Based on “Average Salary for Men and Women Faculty by Category, Affiliation, and Academic Rank, 2005–2006.”

CHAPTER 11 THE ANALYSIS OF VARIANCE

a. What type of design was used in collecting these data? b. Use an analysis of variance to test if there are significant differences in gender, in type of institution, and to test for a significant interaction of gender type of institution. c. Find a 95% confidence interval estimate for the difference in starting salaries for male assistant professors and female assistant professors. Interpret this interval in terms of a gender difference in starting salaries. d. Use Tukey’s procedure to investigate differences in assistant professor salaries for the three types of institutions. Use a .01. e. Summarize the results of your analysis. 11.71 Pottery in the United Kingdom An article in Archaeometry involved an analysis of 26 samples of Romano-British pottery, found at four different kiln sites in the United Kingdom.9 Since one site only yielded two samples, consider the samples found at the other three sites. The samples were analyzed to determine their chemical composition and the percentage of iron oxide is shown below.

EX1171

Llanederyn 7.00 7.08 7.09 6.37 7.06 6.26 4.26

5.78 5.49 6.92 6.13 6.64 6.69 6.44

Island Thorns

Ashley Rails

1.28 2.39 1.50 1.88 1.51

1.12 1.14 .92 2.74 1.64

Dallas

63 67 60 71

66 67 68 75

San Philadelphia Francisco 61 64 60 73

64 60 61 73

a. What type of experimental design was used in this article? If the design used is a randomized block design, what are the blocks and what are the treatments? b. Conduct an analysis of variance for the data. c. Are there signiﬁcant differences in the average satisfaction scores for the four wireless providers considered here? d. Are there signiﬁcant differences in the average satisfaction scores for the four cities? 11.73 Cell Phones, continued Refer to Exer-

cise 11.72. The diagnostic plots for this experiment are shown below. Does it appear that any of the analysis of variance assumptions have been violated? Explain. Normal Probability Plot of the Residuals (response is Score) 99 95 90 80 70 60 50 40 30 20 10 5 1

a. What type of experimental design is this? b. Use an analysis of variance to determine if there is a difference in the average percentage of iron oxide at the three sites. Use a .01.

0 Residual

Residuals versus the Fitted Values (response is Score) 3 2 1 Residual

c. If you have access to a computer program, generate the diagnostic plots for this experiment. Does it appear that any of the analysis of variance assumptions have been violated? Explain.

Chicago AT&T Wireless Cingular Wireless Sprint Verizon Wireless

Percent

500 ❍

0 1 2

11.72 Cell Phones How satisﬁed are you

with your current mobile-phone service provider? Surveys done by Consumer Reports indicate that there is a high level of dissatisfaction among consumers, resulting in high customer turnover rates.10 The following table shows the overall satisfaction scores, based on a maximum score of 100, for four wireless providers in four different cities.

EX1172

3 4 60

68 70 Fitted Value

11.74 Professor’s Salaries II Each year, the American Association of University Professors reports on salaries of academic professors at universities

EX1174

CASE STUDY

and colleges in the United States.8 The following data (in thousands of dollars), adapted from this report, are based on samples of n 10 in each of three professorial ranks, for both male and female professors. Rank Gender Male

Female

Assistant Professor

Associate Professor

$64.4 62.2 64.2 64.9 67.5

$70.0 77.7 77.1 76.0 70.1

$74.4 77.2 76.3 78.8 73.1

$109.4 111.3 112.5 111.6 118.3

$110.5 104.4 106.3 106.9 109.9

56.6 57.6 53.5 64.4 62.6

59.0 58.6 54.9 62.9 59.8

65.4 71.9 65.9 67.9 73.6

66.3 74.6 73.0 69.4 71.0

110.3 97.0 91.5 103.5 95.6

100.9 102.8 102.0 96.7 97.8

501

a. Identify the design used in this survey. b. Use the appropriate analysis of variance for these data. c. Do the data indicate that the salary at the different ranks vary by gender? d. If there is no interaction, determine whether there are differences in salaries by rank, and whether there are differences by gender. Discuss your results.

Full Professor

$63.9 63.9 64.8 68.3 67.5

❍

e. Plot the average salaries using an interaction plot. If the main effect of ranks is signiﬁcant, use Tukey’s method of pairwise comparisons to determine if there are signiﬁcant differences among the ranks. Use a .01.

Source: Based on “Average Salary for Men and Women Faculty by Category, Affiliation, and Academic Rank, 2005–2006.”

CASE STUDY Tickets

“A Fine Mess” Do you risk a parking ticket by parking where you shouldn’t or forgetting how much time you have left on the parking meter? Do the fines associated with various parking infractions vary depending on the city in which you receive a parking ticket? To look at this issue, the fines imposed for overtime parking, parking in a red zone, and parking next to a fire hydrant were recorded for 13 cities in southern California.11 City Long Beach Bakersfield Orange San Bernardino Riverside San Luis Obispo Beverly Hills Palm Springs Laguna Beach Del Mar Los Angeles San Diego Newport Beach

Overtime Parking $17 17 22 20 21 8 23 22 22 25 20 35 32

Red Zone

Fire Hydrant

$30 33 30 30 30 20 38 28 22 40 55 60 42

$30 33 32 78 30 75 30 46 32 55 30 60 30

Source: From “A Fine Mess,” by R. McGarvey, Avenues, July/August 1994. Reprinted by permission of the author.

1. Identify the design used for the data collection in this case study. 2. Analyze the data using the appropriate analysis. What can you say about the variation among the cities in this study? Among fines for the three types of violations? Can Tukey’s procedure be of use in further delineating any significant differences you may find? Would confidence interval estimates be useful in your analysis? 3. Summarize the results of your analysis of these data.

Linear Regression and Correlation GENERAL OBJECTIVES In this chapter, we consider the situation in which the mean value of a random variable y is related to another variable x. By measuring both y and x for each experimental unit, thereby generating bivariate data, you can use the information provided by x to estimate the average value of y and to predict values of y for preassigned values of x.

CHAPTER INDEX ● Analysis of variance for linear regression (12.4) ● Correlation analysis (12.8) ● Diagnostic tools for checking the regression assumptions (12.6)

● Estimation and prediction using the ﬁtted line (12.7) ● The method of least squares (12.3) ● A simple linear probabilistic model (12.2) ● Testing the usefulness of the linear regression model: inferences about b, the ANOVA F-test, and r 2 (12.5)

How Do I Make Sure That My Calculations Are Correct?

Is Your Car “Made in the U.S.A.”? The phrase “made in the U.S.A.” has become a battle cry in the past few years as American workers try to protect their jobs from overseas competition. In the case study at the end of this chapter, we explore the changing attitudes of American consumers toward automobiles made outside the United States, using a simple linear regression analysis.

502

12.2 A SIMPLE LINEAR PROBABILISTIC MODEL

12.1

❍

503

INTRODUCTION High school seniors, freshmen entering college, their parents, and a university administration are concerned about the academic achievement of a student after he or she has enrolled in a university. Can you estimate or predict a student’s grade point average (GPA) at the end of the freshman year before the student enrolls in the university? At ﬁrst glance this might seem like a difficult problem. However, you would expect highly motivated students who have graduated with a high class rank from a high school with superior academic standards to achieve a high GPA at the end of the college freshman year. On the other hand, students who lack motivation or who have achieved only moderate success in high school are not expected to do so well. You would expect the college achievement of a student to be a function of several variables: • • • •

Rank in high school class High school’s overall rating High school GPA SAT scores

This problem is of a fairly general nature. You are interested in a random variable y (college GPA) that is related to a number of independent variables. The objective is to create a prediction equation that expresses y as a function of these independent variables. Then, if you can measure the independent variables, you can substitute these values into the prediction equation and obtain the prediction for y—the student’s college GPA in our example. But which variables should you use as predictors? How strong is their relationship to y? How do you construct a good prediction equation for y as a function of the selected predictor variables? We will answer these questions in the next two chapters. In this chapter, we restrict our attention to the simple problem of predicting y as a linear function of a single predictor variable x. This problem was originally addressed in Chapter 3 in the discussion of bivariate data. Remember that we used the equation of a straight line to describe the relationship between x and y and we described the strength of the relationship using the correlation coefficient r. We rely on some of these results as we revisit the subject of linear regression and correlation.

12.2

A SIMPLE LINEAR PROBABILISTIC MODEL Consider the problem of trying to predict the value of a response y based on the value of an independent variable x. The best-ﬁtting line of Chapter 3, y a bx was based on a sample of n bivariate observations drawn from a larger population of measurements. The line that describes the relationship between y and x in the population is similar to, but not the same as, the best-ﬁtting line from the sample. How can you construct a population model to describe the relationship between a random variable y and a related independent variable x? You begin by assuming that the variable of interest, y, is linearly related to an independent variable x. To describe the linear relationship, you can use the deterministic model y a bx

504 ❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

where a is the y-intercept—the value of y when x 0—and b is the slope of the line, deﬁned as the change in y for a one-unit change in x, as shown in Figure 12.1. This model describes a deterministic relationship between the variable of interest y, sometimes called the response variable, and the independent variable x, often called the predictor variable. That is, the linear equation determines an exact value of y when the value of x is given. Is this a realistic model for an experimental situation? Consider the following example.

FI GU R E 1 2 . 1

The y-intercept and slope for a line

●

Slope = β

slope change in y for a 1-unit change in x y-intercept value of y when x 0

y-intercept = α 0

Table 12.1 displays the mathematics achievement test scores for a random sample of n 10 college freshmen, along with their ﬁnal calculus grades. A bivariate plot of these scores and grades is given in Figure 12.2. You can use the Building a Scatterplot applet to refresh your memory as to how this plot is drawn. Notice that the points do not lie exactly on a line but rather seem to be deviations about an underlying line. A simple way to modify the deterministic model is to add a random error component to explain the deviations of the points about the line. A particular response y is described using the probabilistic model y a bx e

TABLE 12.1

●

Mathematics Achievement Test Scores and Final Calculus Grades for College Freshmen

Student

Mathematics Achievement Test Score

Final Calculus Grade

1 2 3 4 5 6 7 8 9 10

39 43 21 64 57 47 28 75 34 52

65 78 52 82 92 89 73 98 56 75

12.2 A SIMPLE LINEAR PROBABILISTIC MODEL

FIGU R E 1 2 . 2

Scatterplot of the data in Table 12.1

❍

505

● 100

Grade

50 20

50 Score

The ﬁrst part of the equation, a bx—called the line of means—describes the average value of y for a given value of x. The error component e allows each individual response y to deviate from the line of means by a small amount. In order to use this probabilistic model for making inferences, you need to be more speciﬁc about this “small amount,” e. ASSUMPTIONS ABOUT THE RANDOM ERROR e Assume that the values of e satisfy these conditions: • • •

Are independent in the probabilistic sense Have a mean of 0 and a common variance equal to s 2 Have a normal probability distribution

These assumptions about the random error e are shown in Figure 12.3 for three ﬁxed values of x—say, x1, x2, and x3. Notice the similarity between these assumptions and the assumptions necessary for the tests in Chapters 10 and 11. We will revisit these assumptions later in this chapter and provide some diagnostic tools for you to use in checking their validity.

FIGU R E 1 2 . 3

Linear probabilistic model

●

506 ❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

Remember that this model is created for a population of measurements that is generally unknown to you. However, you can use sample information to estimate the values of a and b, which are the coefficients of the line of means, E( y) a bx. These estimates are used to form the best-ﬁtting line for a given set of data, called the least squares line or regression line. We review how to calculate the intercept and the slope of this line in the next section.

THE METHOD OF LEAST SQUARES

12.3

The statistical procedure for finding the best-fitting line for a set of bivariate data does mathematically what you do visually when you move a ruler until you think you have minimized the vertical distances, or deviations, from the ruler to a set of points. The formula for the best-fitting line is yˆ a bx

slope coefficient of x y-intercept constant term

FI GU R E 1 2 . 4

Method of Least Squares applet

where a and b are the estimates of the intercept and slope parameters a and b, respectively. The ﬁtted line for the data in Table 12.1 is shown in the Method of Least Squares applet, Figure 12.4. The red vertical lines (light blue in Figure 12.4) drawn from the prediction line to each point (xi, yi ) represent the deviations of the points from the line. ●

To minimize the distances from the points to the fitted line, you can use the principle of least squares. PRINCIPLE OF LEAST SQUARES The line that minimizes the sum of squares of the deviations of the observed values of y from those predicted is the best-fitting line. The sum of squared deviations is commonly called the sum of squares for error (SSE) and defined as SSE S( yi yî)2 S( yi a bxi)2

12.3 THE METHOD OF LEAST SQUARES

❍

507

Look at the regression line and the data points in Figure 12.4. SSE is the sum of the squared distances represented by the area of the yellow squares (light blue in Figure 12.4). Finding the values of a and b, the estimates of a and b, uses differential calculus, which is beyond the scope of this text. Rather than derive their values, we will simply present formulas for calculating the values of a and b—called the least-squares estimators of a and b. We will use notation that is based on the sums of squares for the variables in the regression problem, which are similar in form to the sums of squares used in Chapter 11. These formulas look different from the formulas presented in Chapter 3, but they are in fact algebraically identical! You should use the data entry method for your scientiﬁc calculator to enter the sample data. • •

•

If your calculator has only a one-variable statistics function, you can still save some time in ﬁnding the necessary sums and sums of squares. If your calculator has a two-variable statistics function, or if you have a graphing calculator, the calculator will automatically store all of the sums and sums of squares as well as the values of a, b, and the correlation coefficient r. Make sure you consult your calculator manual to ﬁnd the easiest way to obtain the least squares estimators.

LEAST-SQUARES ESTIMATORS OF a AND b S b xy and Sxx

a 苶y bx苶

where the quantities Sxy and Sxx are deﬁned as (Sxi )(Syi) Sxy S(xi 苶x )( yi 苶y) Sxi yi n and (S xi)2 Sxx S(xi 苶x )2 Sx 2i n Notice that the sum of squares of the x-values is found using the computing formula given in Section 2.3 and the sum of the cross-products is the numerator of the covariance deﬁned in Section 3.4. EXAMPLE

12.1

Find the least-squares prediction line for the calculus grade data in Table 12.1. Use the data in Table 12.2 and the data entry method in your scientiﬁc calculator to ﬁnd the following sums of squares:

Solution

(Sxi)2 (460)2 23,634 2474 Sxx Sx 2i n 10 (Sxi)(Syi) (460)(760) Sxy Sxi yi 36,854 1894 n 10 Syi 760 76 苶y n 1 0

Sxi 460 苶x n 10 46

508 ❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

TABLE 12.2

●

Calculations for the Data in Table 12.1 xi

x i2

xi yi

y i2

65 78 52 82 92 89 73 98 56 75

39 43 21 64 57 47 28 75 34 52

1521 1849 441 4096 3249 2209 784 5625 1156 2704

2535 3354 1092 5248 5244 4183 2044 7350 1904 3900

4225 6084 2704 6724 8464 7921 5329 9604 3136 5625

760

460

23,634

36,854

59,816

Sum

Then S 1894 b xy .76556 Sxx 2474 You can predict y for a given value of x by substituting x into the equation to ﬁnd yˆ.

and a 苶y bx苶 76 (.76556)(46) 40.78424

The least-squares regression line is then yˆ a bx 40.78424 .76556x The graph of this line is shown in Figure 12.4. It can now be used to predict y for a given value of x—either by referring to Figure 12.4 or by substituting the proper value of x into the equation. For example, if a freshman scored x 50 on the achievement test, the student’s predicted calculus grade is (using full decimal accuracy) yˆ a b(50) 40.78424 (.76556)(50) 79.06

How Do I Make Sure That My Calculations Are Correct? • • • •

Be careful of rounding errors. Carry at least six significant figures, and round off only in reporting the end result. Use a scientific or graphing calculator to do all the work for you. Most of these calculators will calculate the values for a and b if you enter the data properly. Use a computer software program if you have access to one. Always plot the data and graph the line. If the line does not fit through the points, you have probably made a mistake!

You can use the Method of Least Squares applet to ﬁnd the values of a and b that determine the best ﬁtting line, yˆ a bx. The horizontal line that you see is the line y 苶y. Use your mouse to drag the line and watch the yellow squares change size. The object is to make SSE—the total area of the yellow squares (light blue in Figure 12.4)—as small as possible. The value of SSE is the red portion of the bar on the left of the applet (dark blue in Figure 12.4) marked SSE . When you think that you have minimized SSE, click the button and see how well you did!

12.4 AN ANALYSIS OF VARIANCE FOR LINEAR REGRESSION

❍

509

AN ANALYSIS OF VARIANCE FOR LINEAR REGRESSION

12.4

In Chapter 11, you used the analysis of variance procedures to divide the total variation in the experiment into portions attributed to various factors of interest to the experimenter. In a regression analysis, the response y is related to the independent variable x. Hence, the total variation in the response variable y, given by (Syi )2 Total SS Syy S(yi 苶y )2 Sy 2i n is divided into two portions: • •

SSR (sum of squares for regression) measures the amount of variation explained by using the regression line with one independent variable x SSE (sum of squares for error) measures the “residual” variation in the data that is not explained by the independent variable x

so that Total SS SSR SSE For a particular value of the response yi, you can visualize this breakdown in the variation using the vertical distances illustrated in Figure 12.5. You can see that SSR is the sum of the squared deviations of the differences between the estimated response without using x ( y苶 ) and the estimated response using x (the regression line, yˆ); SSE is the sum of the squared differences between the regression line ( yˆ) and the point y.

Deviations from the ﬁtted line

●

100

SSE

{ }

80 Grade

FIGU R E 1 2 . 5

SSR

^y = 40.7842 + 0.76556x

50 20

Score

It is not too hard to show algebraically that SSR S( yˆi 苶yi)2 S(a bxi 苶y )2 S( 苶y bx苶 bxi 苶 y )2 b2S(xi 苶x )2 S 2 (S y)2 xy Sxx x Sxx Sxx

冢冣

Since Total SS SSR SSE, you can complete the partition by calculating (S y)2 SSE Total SS SSR Syy x Sxx

510 ❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

Remember from Chapter 11 that each of the various sources of variation, when divided by the appropriate degrees of freedom, provides an estimate of the variation in the experiment. These estimates are called mean squares—MS SS/df—and are displayed in an ANOVA table. In examining the degrees of freedom associated with each of these sums of squares, notice that the total degrees of freedom for n measurements is (n 1). Since estimating the regression line, yˆ a bxi y苶 bx苶 bxi , involves estimating one additional parameter b, there is one degree of freedom associated with SSR, leaving (n 2) degrees of freedom with SSE. As with all ANOVA tables we have discussed, the mean square for error, SSE MSE s 2 n2 is an unbiased estimator of the underlying variance s 2. The analysis of variance table is shown in Table 12.3.

TABLE 12.3

●

Analysis of Variance for Linear Regression Source

Regression

(S y)2 x S xx

MSR

Error

(S y)2 Syy x Sxx

MSE

Total

Syy

For the data in Table 12.1, you can calculate (Syi)2 (760)2 Total SS Syy Sy 2i 59,816 2056 n 10 (S y)2 (1894)2 1449.9741 SSR x S xx 2474 so that SSE Total SS SSR 2056 1449.9741 606.0259 and 606.0259 SSE MSE 75.7532 8 n2 The analysis of variance table, part of the linear regression output generated by MINITAB, is the lower shaded section in the printout in Figure 12.6. The ﬁrst two lines give the equation of the least-squares line, yˆ 40.8 .766x. The least-squares estimates a and b are given with greater accuracy in the column labeled “Coef.” You can ﬁnd instructions for generating this output in the section “My MINITAB ” at the end of this chapter.

12.4 AN ANALYSIS OF VARIANCE FOR LINEAR REGRESSION

FIGU R E 1 2 . 6

●

MINITAB output for the data of Table 12.1

❍

511

Regression Analysis: y versus x The regression equation is y = 40.8 + 0.766 x Predictor Constant x

Coef 40.784 0.7656

S = 8.70363

SE Coef 8.507 0.1750

R-Sq = 70.5%

Analysis of Variance Source DF Regression 1 Residual Error 8 Total 9

T 4.79 4.38

P 0.001 0.002

R-Sq(adj) = 66.8%

SS 1450.0 606.0 2056.0

MS 1450.0 75.8

F 19.14

P 0.002

The MINITAB output also gives some information about the variation in the experiment. Each of the least-squares estimates, a and b, has an associated standard error, labeled “SE Coef ” in Figure 12.6. In the middle of the printout, you will ﬁnd the best 苶兹75.753 苶2 苶 8.70363—which measures the unbiased estimate of s—S 兹MSE residual error, the unexplained or “leftover” variation in the experiment. It will not surprise you to know that the t and F statistics and their p-values found in the printout are used to test statistical hypotheses. We explain these entries in the next section.

Look for a and b in the column called “Coef.”

EXERCISES

12.4

BASIC TECHNIQUES

y 2x 1 by graphing the points corresponding to x 0, 1, and 2. Give the y-intercept and slope for the line. How is this line related to the line y 2x 1 of Exercise 12.1?

a. Use the data entry method on your scientific or graphing calculator to enter the n 5 observations. Find the sums of squares and cross-products, Sxx, Sxy, and Syy. b. Find the least-squares line for the data. c. Plot the five points and graph the line in part b. Does the line appear to provide a good fit to the data points? d. Construct the ANOVA table for the linear regression.

12.3 Give the equation and graph for a line with

12.7 Six points have these coordinates:

12.1 Graph the line corresponding to the equation

y 2x 1 by graphing the points corresponding to x 0, 1, and 2. Give the y-intercept and slope for the line. 12.2 Graph the line corresponding to the equation

y-intercept equal to 3 and slope equal to 1. 12.4 Give the equation and graph for a line with

y-intercept equal to 3 and slope equal to 1. 12.5 What is the difference between deterministic

and probabilistic mathematical models? 12.6 You are given ﬁve points with these coordi-

nates: x

5.6

4.6

4.5

3.7

3.2

2.7

a. Find the least-squares line for the data. b. Plot the six points and graph the line. Does the line appear to provide a good ﬁt to the data points? c. Use the least-squares line to predict the value of y when x 3.5. d. Fill in the missing entries in the MINITAB analysis of variance table.

512 ❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

MINITAB ANOVA table for Exercise 12.7 Analysis of Variance Source Regression Residual Error Total

DF * * *

SS *** 0.1429 5.5750

MS 5.4321 ***

APPLICATIONS 12.8 Professor Asimov Professor Isaac Asimov was one of the most proliﬁc writers of all time. Prior to his death, he wrote nearly 500 books during a 40-year career. In fact, as his career progressed, he became even more productive in terms of the number of books written within a given period of time.1 The data give the time in months required to write his books in increments of 100: Number of Books, x

100

200

300

400

490

Time in Months, y

237

350

419

465

507

a. Assume that the number of books x and the time in months y are linearly related. Find the least-squares line relating y to x. b. Plot the time as a function of the number of books written using a scatterplot, and graph the leastsquares line on the same paper. Does it seem to provide a good ﬁt to the data points? c. Construct the ANOVA table for the linear regression. 12.9 A Chemical Experiment Using a

chemical procedure called differential pulse polarography, a chemist measured the peak current generated (in microamperes) when a solution containing a given amount of nickel (in parts per billion) is added to a buffer :2

c. Plot the points and the ﬁtted line. Does the assumption of a linear relationship appear to be reasonable? d. Use the regression line to predict the peak current generated when a solution containing 100 ppb of nickel is added to the buffer. e. Construct the ANOVA table for the linear regression. 12.10 Sleep Deprivation A study was

conducted to determine the effects of sleep deprivation on people’s ability to solve problems without sleep. A total of 10 subjects participated in the study, two at each of ﬁve sleep deprivation levels—8, 12, 16, 20, and 24 hours. After his or her speciﬁed sleep deprivation period, each subject was administered a set of simple addition problems, and the number of errors was recorded. These results were obtained:

EX1210

Number of Errors, y

8, 6

6, 10

8, 14

Number of Hours without Sleep, x

Number of Errors, y

14, 12

16, 12

Number of Hours without Sleep, x

a. How many pairs of observations are in the experiment? b. What are the total number of degrees of freedom? c. Complete the MINITAB printout. MINITAB output for Exercise 12.10

EX1209

x Ni (ppb) 19.1 38.2 57.3 76.2 95 114 131 150 170

y Peak Current (mA) .095 .174 .256 .348 .429 .500 .580 .651 .722

a. Use the data entry method for your calculator to calculate the preliminary sums of squares and crossproducts, Sxx, Syy, and Sxy. b. Calculate the least-squares regression line.

Regression Analysis: y versus x The regression equation is y = 3.00 + 0.475 x Predictor Constant x

Coef 3.000 ***

S = 2.24165

R-Sq = 64.2%

Analysis of Variance Source DF Regression ** Residual Error ** Total **

SE Coef 2.127 0.1253

SS 72.200 *** ***

T 1.41 3.79

P 0.196 0.005

R-Sq(adj) = 59.8% MS 72.200 5.025

F 14.37

P 0.005

d. What is the least-squares prediction equation? e. Use the prediction equation to predict the number of errors for a person who has not slept for 10 hours.

EX1211

12.11 Achievement Tests The Academic Performance Index (API) is a measure of

12.4 AN ANALYSIS OF VARIANCE FOR LINEAR REGRESSION

school achievement based on the results of the Stanford 9 Achievement test. Scores range from 200 to 1000, with 800 considered a long-range goal for schools. The following table shows the API for eight elementary schools in Riverside County, California, along with the percent of students at that school who are considered English Language Learners (ELL).3 School API ELL

588 58

659 22

710 14

657 30

669 11

641 26

557 39

743 6

a. Which of the two variables is the independent variable and which is the dependent variable? Explain your choice. b. Use a scatterplot to plot the data. Is the assumption of a linear relationship between x and y reasonable? c. Assuming that x and y are linearly related, calculate the least-squares regression line. d. Plot the line on the scatterplot in part b. Does the line ﬁt through the data points? 12.12 How Long Is It? How good are you at estimating? To test a subject’s ability to estimate sizes, he was shown 10 different objects and asked to estimate their length or diameter. The object was then measured, and the results were recorded in the table below.

EX1212

Object Pencil Dinner plate Book 1 Cell phone Photograph Toy Belt Clothespin Book 2 Calculator

Estimated (inches)

Actual (inches)

7.00 9.50 7.50 4.00 14.50 3.75 42.00 2.75 10.00 3.50

6.00 10.25 6.75 4.25 15.75 5.00 41.50 3.75 9.25 4.75

a. Find the least-squares regression line for predicting the actual measurement as a function of the estimated measurement. b. Plot the points and the ﬁtted line. Does the assumption of a linear relationship appear to be reasonable? 12.13 Test Interviews Of two personnel evaluation techniques available, the ﬁrst requires a two-hour test interview while the second

EX1213

❍

513

can be completed in less than an hour. The scores for each of the 15 individuals who took both tests are given in the next table. Applicant

Test 1 (x)

Test 2(y)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

75 89 60 71 92 105 55 87 73 77 84 91 75 82 76

38 56 35 45 59 70 31 52 48 41 51 58 45 49 47

a. Construct a scatterplot for the data. Does the assumption of linearity appear to be reasonable? b. Find the least-squares line for the data. c. Use the regression line to predict the score on the second test for an applicant who scored 85 on Test 1. 12.14 Test Interviews, continued Refer to Exer-

cise 12.13. Construct the ANOVA table for the linear regression relating y, the score on Test 2, to x, the score on Test 1. 12.15 Armspan and Height Leonardo da Vinci (1452–1519) drew a sketch of a man, indicating that a person’s armspan (measuring across the back with your arms outstretched to make a “T”) is roughly equal to the person’s height. To test this claim, we measured eight people with the following results:

EX1215

Person

Armspan (inches) Height (inches)

68 69

62.25 62

65 65

69.5 70

Person

Armspan (inches) Height (inches)

68 67

69 67

62 63

60.25 62

514 ❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

b. If da Vinci is correct, and a person’s armspan is roughly the same as the person’s height, what should the slope of the regression line be? c. Calculate the regression line for predicting height based on a person’s armspan. Does the value of the slope b conﬁrm your conclusions in part b? d. If a person has an armspan of 62 inches, what would you predict the person’s height to be? 12.16 Strawberries The following data

were obtained in an experiment relating the dependent variable, y (texture of strawberries), with x (coded storage temperature).

EX1216

a. Draw a scatterplot for armspan and height. Use the same scale on both the horizontal and vertical axes. Describe the relationship between the two variables.

12.5

4.0

3.5

2.0

0.5

0.0

a. Find the least-squares line for the data. b. Plot the data points and graph the least-squares line as a check on your calculations. c. Construct the ANOVA table.

TESTING THE USEFULNESS OF THE LINEAR REGRESSION MODEL In considering linear regression, you may ask two questions: • •

Is the independent variable x useful in predicting the response variable y? If so, how well does it work?

This section examines several statistical tests and measures that will help you reach some answers. Once you have determined that the model is working, you can then use the model for predicting the response y for a given value of x.

Inferences Concerning b, the Slope of the Line of Means Is the least-squares regression line useful? That is, is the regression equation that uses information provided by x substantially better than the simple predictor y苶 that does not rely on x? If the independent variable x is not useful in the population model y a bx e, then the value of y does not change for different values of x. The only way that this happens for all values of x is when the slope b of the line of means equals 0. This would indicate that the relationship between y and x is not linear, so that the initial question about the usefulness of the independent variable x can be restated as: Is there a linear relationship between x and y? You can answer this question by using either a test of hypothesis or a conﬁdence interval for b. Both of these procedures are based on the sampling distribution of b,

12.5 TESTING THE USEFULNESS OF THE LINEAR REGRESSION MODEL

❍

515

the sample estimator of the slope b. It can be shown that, if the assumptions about the random error e are valid, then the estimator b has a normal distribution in repeated sampling with mean E(b) b and standard error given by SE

冪莦 s2 Sxx

where s 2 is the variance of the random error e. Since the value of s 2 is estimated with s 2 MSE, you can base inferences on the statistic given by bb t 苶 SE/S苶兹M xx which has a t distribution with df (n 2), the degrees of freedom associated with MSE. TEST OF HYPOTHESIS CONCERNING THE SLOPE OF A LINE 1. Null hypothesis: H0 : b b0 2. Alternative hypothesis: One-Tailed Test

Two-Tailed Test

Ha : b b0 (or b b0)

Ha : b b0

b b0 3. Test statistic: t MSE/S苶兹苶 xx When the assumptions given in Section 12.2 are satisﬁed, the test statistic will have a Student’s t distribution with (n 2) degrees of freedom. 4. Rejection region: Reject H0 when One-Tailed Test

Two-Tailed Test

t ta (or t ta when the alternative hypothesis is Ha : b b0)

t ta/2

or when p-value a

α 0

tα

α/2

α/2 –tα/2

tα/2

The values of ta and ta/2 can be found using Table 4 in Appendix I or the t Probabilities applet. Use the values of t corresponding to (n 2) degrees of freedom.

516 ❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

EXAMPLE

12.2

Determine whether there is a signiﬁcant linear relationship between the calculus grades and test scores listed in Table 12.1. Test at the 5% level of signiﬁcance. Solution

The hypotheses to be tested are

H0 : b 0

versus

Ha : b 0

and the observed value of the test statistic is calculated as b0 .7656 0 t 4.38 苶苶苶2 SE/Sxx 5.753苶 /2474 兹M 兹7 with (n 2) 8 degrees of freedom. With a .05, you can reject H0 when t 2.306 or t 2.306. Since the observed value of the test statistic falls into the rejection region, H0 is rejected and you can conclude that there is a signiﬁcant linear relationship between the calculus grades and the test scores for the population of college freshmen.

You can use the t-Test for the Slope applet shown in Figure 12.7 to ﬁnd p-values or rejection regions for this test. You must ﬁrst calculate the standard error 苶苶, SE 兹MSE/S xx type its value into the box marked “Std Error,” and press “Enter.” FI GU R E 1 2 . 7

t-Test for the Slope applet

●

•

If you enter the value of b into the formula at the top of the applet and press “Enter,” the applet will calculate the test statistic and its one- or two-tailed p-value. If you enter the signiﬁcance level a in the box marked “prob:” and select the “Area to Right” or “Two Tails” option in the drop-down list, the applet will calculate the positive value of t necessary for rejecting H0. (You could also use the Student’s t Probabilities applet to ﬁnd the critical values.)

What is the p-value for the test performed in Example 12.2? Does this p-value conﬁrm our conclusions? Another way to make inferences about the value of b is to construct a conﬁdence interval for b and examine the range of possible values for b.

12.5 TESTING THE USEFULNESS OF THE LINEAR REGRESSION MODEL

❍

517

A (1 a)100% CONFIDENCE INTERVAL FOR b b ta/2(SE) where ta/2 is based on (n 2) degrees of freedom and MSE s2 SE Sxx Sxx

冪莦冪莦

EXAMPLE

12.3

Find a 95% conﬁdence interval estimate of the slope b for the calculus grade data in Table 12.1. Solution

Substituting previously calculated values into

冪莦

MSE b t.025 Sxx you have

冪莦莦

75.7532 .766 2.306 2474 .766 .404

The resulting 95% confidence interval is .362 to 1.170. Since the interval does not contain 0, you can conclude that the true value of b is not 0, and you can reject the null hypothesis H0 : b 0 in favor of Ha : b 0, a conclusion that agrees with the findings in Example 12.2. Furthermore, the confidence interval estimate indicates that there is an increase from as little as .4 to as much as 1.2 points in a calculus test score for each 1-point increase in the achievement test score. If you are using computer software to perform the regression analysis, you will find the t statistic and its p-value on the printout. Look at the MINITAB regression analysis printout reproduced in Figure 12.8. In the second portion of the printout, you will find the least-squares estimates a (“Constant”) and b (“x”) in the column marked “Coef,” their standard errors (“SE Coef”), the calculated value of the t statistic (“T”) used for testing the hypothesis that the parameter equals 0, and its p-value (“P”). The t-test for significant regression, H0 : b 0, has a p-value of P .002, and the null hypothesis is rejected, as in Example 12.2. Does this agree with the p-value found using the t-Test for the Slope applet in Figure 12.7? In any event, there is a significant linear relationship between x and y.

FIGU R E 1 2 . 8

MINITAB output for the calculus grade data

● Regression Analysis: y versus x The regression equation is y = 40.8 + 0.766 x Predictor Constant x

Look for the standard error of b in the column marked “SE Coef.”

S = 8.70363

Coef 40.784 0.7656

SE Coef 8.507 0.1750

R-Sq = 70.5%

Analysis of Variance Source DF Regression 1 Residual Error 8 Total 9

SS 1450.0 606.0 2056.0

T 4.79 4.38

P 0.001 0.002

R-Sq(adj) = 66.8% MS 1450.0 75.8

F 19.14

P 0.002

518

❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

The Analysis of Variance F-Test The analysis of variance portion of the printout in Figure 12.8 shows an F statistic given by ANOVA F-tests are always one-tailed (upper-tail).

MSR F 19.14 MSE with 1 numerator degree of freedom and (n 2) 8 denominator degrees of freedom. This is an equivalent test statistic that can also be used for testing the hypothesis H0 : b 0. Notice that, within rounding error, the value of F is equal to t 2 with the identical p-value. In this case, if you use ﬁve-decimal-place accuracy prior to rounding, you ﬁnd that t 2 (.76556/.17498)2 (4.37513)2 19.14175 ⬇ 19.14 F as given in the printout. This is no accident and results from the fact that the square of a t statistic with df degrees of freedom has the same distribution as an F-statistic with 1 numerator and df denominator degrees of freedom. The F-test is a more general test of the usefulness of the model and can be used when the model has more than one independent variable.

Measuring the Strength of the Relationship: The Coefficient of Determination How well does the regression model ﬁt? To answer this question, you can use a measure related to the correlation coefficient r, introduced in Chapter 3. Remember that Sxy sy r x sx sy 兹S苶 xx Syy

On computer printouts, r 2 is often given as a percentage rather than a proportion.

for 1 r 1

where sxy, sx, and sy were deﬁned in Chapter 3 and the various sums of squares were deﬁned in Section 12.4. The sum of squares for regression, SSR, in the analysis of variance measures the portion of the total variation, Total SS Syy, that can be explained by the regression of y on x. The remaining portion, SSE, is the “unexplained” variation attributed to random error. One way to measure the strength of the relationship between the response variable y and the predictor variable x is to calculate the coefficient of determination—the proportion of the total variation that is explained by the linear regression of y on x. For the calculus grade data, this proportion is equal to SSR 1450 .705 or 70.5% Total SS 2056 (S y)2 Since Total SS Syy and SSR x, you can write Sxx 2 Sxy SSR (S y)2 x r 2 Total SS Sxx Syy 苶兹S xx Syy

冢

冣

Therefore, the coefficient of determination, which was calculated as SSR/Total SS, is simply the square of the correlation coefficient r. It is the entry labeled “R-Sq” in Figure 12.8. Remember that the analysis of variance table isolates the variation due to regression (SSR) from the total variation in the experiment. Doing so reduces the amount of random variation in the experiment, now measured by SSE rather than Total SS. In this context, the coefficient of determination, r2, can be deﬁned as follows:

12.5 TESTING THE USEFULNESS OF THE LINEAR REGRESSION MODEL

❍

519

The coefficient of determination r 2 can be interpreted as the percent reduction in the total variation in the experiment obtained by using the regression line yˆ a bx, instead of ignoring x and using the sample mean 苶y to predict the response variable y. Definition

r 2 is called “R-Sq” on the MINITAB printout.

For the calculus grade data, a reduction of r 2 .705 or 70.5% is substantial. The regression model is working very well!

Interpreting the Results of a Significant Regression Once you have performed the t-test or F-test to determine the signiﬁcance of the linear regression, you must interpret your results carefully. The slope b of the line of means is estimated based on data from only a particular region of observation. Even if you do not reject the null hypothesis that the slope of the line equals 0, it does not necessarily mean that y and x are unrelated. It may be that you have committed a Type II error—falsely declaring that the slope is 0 and that x and y are unrelated.

Fitting the Wrong Model It may happen that y and x are perfectly related in a nonlinear way, as shown in Figure 12.9. Here are three possibilities: FIGU R E 1 2 . 9

Curvilinear relationship

● y e1

Lin

Line 2

• • •

If observations were taken only within the interval b x c, the relationship would appear to be linear with a positive slope. If observations were taken only within the interval d x f, the relationship would appear to be linear with a negative slope. If the observations were taken over the interval c x d, the line would be ﬁtted with a slope close to 0, indicating no linear relationship between y and x.

For the example shown in Figure 12.9, no straight line accurately describes the true relationship between x and y, which is really a curvilinear relationship. In this case, we have chosen the wrong model to describe the relationship. Sometimes this type of mistake can be detected using residual plots, the subject of Section 12.7.

It is dangerous to try to predict values of y outside of the range of the ﬁtted data.

Extrapolation One serious problem is to apply the results of a linear regression analysis to values of x that are not included within the range of the ﬁtted data. This is called extrapolation and can lead to serious errors in prediction, as shown for line 1 in Figure 12.9.

❍

520

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

Prediction results would be good over the interval b x c but would seriously overestimate the values of y for x c.

Causality When there is a signiﬁcant regression of y and x, it is tempting to conclude that x causes y. However, it is possible that one or more unknown variables that you have not even measured and that are not included in the analysis may be causing the observed relationship. In general, the statistician reports the results of an analysis but leaves conclusions concerning causality to scientists and investigators who are experts in these areas. These experts are better prepared to make such decisions!

EXERCISES

12.5

BASIC TECHNIQUES 12.17 Refer to Exercise 12.6. The data are reproduced below. x

a. Do the data present sufficient evidence to indicate that y and x are linearly related? Test the hypothesis that b 0 at the 5% level of significance. b. Use the ANOVA table from Exercise 12.6 to calculate F MSR/MSE. Verify that the square of the t statistic used in part a is equal to F. c. Compare the two-tailed critical value for the t-test in part a with the critical value for F with a .05. What is the relationship between the critical values? 12.18 Refer to Exercise 12.17. Find a 95% confidence interval for the slope of the line. What does the phrase “95% confident” mean? 12.19 Refer to Exercise 12.7. The data, along with

the MINITAB analysis of variance table are reproduced below. x

5.6

4.6

4.5

3.7

3.2

2.7

MINITAB ANOVA table for Exercise 12.19 Regression Analysis: y versus x Analysis of Variance Source DF Regression 1 Residual Error 4 Total 5

SS 5.4321 0.1429 5.5750

MS 5.4321 0.0357

F 152.10

P 0.000

a. Do the data provide sufficient evidence to indicate that y and x are linearly related? Use the informa-

tion in the MINITAB printout to answer this question at the 1% level of signiﬁcance. b. Calculate the coefficient of determination r 2. What information does this value give about the usefulness of the linear model? APPLICATIONS 12.20 Air Pollution An experiment was

designed to compare several different types of air pollution monitors.4 The monitor was set up, and then exposed to different concentrations of ozone, ranging between 15 and 230 parts per million (ppm) for periods of 8–72 hours. Filters on the monitor were then analyzed, and the amount (in micrograms) of sodium nitrate (NO3) recorded by the monitor was measured. The results for one type of monitor are given in the table.

EX1220

Ozone, x (ppm/hr)

1.3

1.7

2.2

2.7

2.9

NO3, y (mg)

2.44

5.21

6.07

8.98

10.82

12.16

a. Find the least-squares regression line relating the monitor’s response to the ozone concentration. b. Do the data provide sufficient evidence to indicate that there is a linear relationship between the ozone concentration and the amount of sodium nitrate detected? c. Calculate r 2. What does this value tell you about the effectiveness of the linear regression analysis? 12.21 The Cost of Flying How is the cost of a plane ﬂight related to the length of the trip? The table shows the average round-trip coach airfare paid by customers of American Airlines on each of 18 heavily traveled U.S. air routes.5

EX1221

12.5 TESTING THE USEFULNESS OF THE LINEAR REGRESSION MODEL

Route

Distance (miles)

Cost

Dallas–Austin Houston–Dallas Chicago–Detroit Chicago–St. Louis Chicago–Cleveland Chicago–Atlanta New York–Miami New York–San Juan New York–Chicago Chicago–Denver Dallas–Salt Lake New York–Dallas Chicago–Seattle Los Angeles–Chicago Los Angeles–Atlanta New York–Los Angeles Los Angeles–Honolulu New York–San Francisco

178 232 238 262 301 593 1092 1608 714 901 1005 1374 1736 1757 1946 2463 2556 2574

$125 123 148 136 129 162 224 264 287 256 365 459 424 361 309 444 323 513

❍

521

the model provides a good ﬁt for the data? Are there any assumptions that may have been violated in ﬁtting the linear model? 12.23 Refer to the sleep deprivation experiment described in Exercise 12.10 and data set EX1210. The data and the MINITAB printout are reproduced here. Number of Errors, y

8, 6

6, 10

8, 14

Number of Hours without Sleep, x

Number of Errors, y

14, 12

16, 12

Number of Hours without Sleep, x

MINITAB output for Exercise 12.23 Regression Analysis: y versus x The regression equation is y = 3.00 + 0.475 x

a. If you want to estimate the cost of a ﬂight based on the distance traveled, which variable is the response variable and which is the independent predictor variable? b. Assume that there is a linear relationship between cost and distance. Calculate the least-squares regression line describing cost as a linear function of distance. c. Plot the data points and the regression line. Does it appear that the line ﬁts the data? d. Use the appropriate statistical tests and measures to explain the usefulness of the regression model for predicting cost. 12.22 Professor Asimov, continued Refer to the

data in Exercise 12.8, relating x, the number of books written by Professor Isaac Asimov, to y, the number of months he took to write his books (in increments of 100). The data are reproduced below. Number of Books, x

100

200

300

400

490

Time in Months, y

237

350

419

465

507

a. Do the data support the hypothesis that b 0? Use the p-value approach, bounding the p-value using Table 4 of Appendix I or ﬁnding the exact p-value using the t-Test for the Slope applet. Explain your conclusions in practical terms. b. Use the ANOVA table in Exercise 12.8, part c, to calculate the coefficient of determination r 2. What percentage reduction in the total variation is achieved by using the linear regression model? c. Plot the data or refer to the plot in Exercise 12.8, part b. Do the results of parts a and b indicate that

Predictor Constant x S = 2.24165

Coef 3.000 0.4750

SE Coef 2.127 0.1253

R-Sq = 64.2%

T 1.41 3.79

P 0.196 0.005

R-Sq(adj) = 59.8%

Analysis of Variance Source Regression Residual Error Total

DF 1 8 9

SS 72.200 40.200 112.400

MS 72.200 5.025

F 14.37

P 0.005

a. Do the data present sufficient evidence to indicate that the number of errors is linearly related to the number of hours without sleep? Identify the two test statistics in the printout that can be used to answer this question. b. Would you expect the relationship between y and x to be linear if x varied over a wider range (say, x 4 to x 48)? c. How do you describe the strength of the relationship between y and x? d. What is the best estimate of the common population variance s 2? e. Find a 95% conﬁdence interval for the slope of the line. 12.24 Strawberries II The following data (Exercise 12.16 and data set EX1216) were obtained in an experiment relating the dependent variable, y (texture of strawberries), with x (coded storage temperature). Use the information from Exercise 12.16 to answer the following questions: x

4.0

3.5

2.0

0.5

0.0

a. What is the best estimate of s 2, the variance of the random error ?

522

❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

b. Do the data indicate that texture and storage temperature are linearly related? Use a .05. c. Calculate the coefficient of determination, r 2. d. Of what value is the linear model in increasing the accuracy of prediction as compared to the predictor, y苶? 12.25 Laptops and Learning In Exercise

1.61 we described an informal experiment conducted at McNair Academic High School in Jersey City, New Jersey. Two freshman algebra classes were studied, one of which used laptop computers at school and at home, while the other class did not. In each class, students were given a survey at the beginning and end of the semester, measuring his or her technological level. The scores were recorded for the end of semester survey (x) and the ﬁnal examination (y) for the laptop group.6 The data and the MINITAB printout are shown here.

EX1225

Student

Posttest

Final Exam

Student

Posttest

Final Exam

1 2 3 4 5 6 7 8 9 10

100 96 88 100 100 96 80 68 92 96

98 97 88 100 100 78 68 47 90 94

11 12 13 14 15 16 17 18 19 20

88 92 68 84 84 88 72 88 72 88

84 93 57 84 81 83 84 93 57 83

Regression Analysis: y versus x The regression equation is y = -26.8 + 1.26 x Predictor Constant x S = 7.61912

Coef -26.82 1.2617

SE Coef 14.76 0.1685

R-Sq = 75.7%

Analysis of Variance Source DF Regression 1 Residual Error 18 Total 19

12.6

SS 3254.0 1044.9 4299.0

T -1.82 7.49

P 0.086 0.000

R-Sq(adj) = 74.3% MS 3254.0 58.1

F 56.05

P 0.000

a. Construct a scatterplot for the data. Does the assumption of linearity appear to be reasonable? b. What is the equation of the regression line used for predicting final exam score as a function of the posttest score? c. Do the data present sufficient evidence to indicate that final exam score is linearly related to the posttest score? Use a .01. d. Find a 99% confidence interval for the slope of the regression line. 12.26 Laptops and Learning, continued Refer to Exercise 12.25. a. Use the MINITAB printout to find the value of the coefficient of determination, r 2. Show that r 2 SSR/Total SS. b. What percentage reduction in the total variation is achieved by using the linear regression model? 12.27 Armspan and Height II In Exercise 12.15 (data set EX1215), we measured the armspan and height of eight people with the following results: Person

Armspan (inches) Height (inches)

68 69

62.25 62

65 65

69.5 70

Person

Armspan (inches) Height (inches)

68 67

69 67

62 63

60.25 62

a. Does the data provide sufficient evidence to indicate that there is a linear relationship between armspan and height? Test at the 5% level of significance. b. Construct a 95% confidence interval for the slope of the line of means, b. c. If Leonardo da Vinci is correct, and a person’s armspan is roughly the same as the person’s height, the slope of the regression line is approximately equal to 1. Is this supposition confirmed by the confidence interval constructed in part b? Explain.

DIAGNOSTIC TOOLS FOR CHECKING THE REGRESSION ASSUMPTIONS Even though you have determined—using the t-test for the slope (or the ANOVA F-test) and the value of r 2—that x is useful in predicting the value of y, the results of a regression analysis are valid only when the data satisfy the necessary regression assumptions.

12.6 DIAGNOSTIC TOOLS FOR CHECKING THE REGRESSION ASSUMPTIONS

❍

523

REGRESSION ASSUMPTIONS •

The relationship between y and x must be linear, given by the model y a bx e The values of the random error term e (1) are independent, (2) have a mean of 0 and a common variance s 2, independent of x, and (3) are normally distributed.

•

Since these assumptions are quite similar to those presented in Chapter 11 for an analysis of variance, it should not surprise you to ﬁnd that the diagnostic tools for checking these assumptions are the same as those we used in that chapter. These tools involve the analysis of the residual error, the unexplained variation in each observation once the variation explained by the regression model has been removed.

Dependent Error Terms The error terms are often dependent when the observations are collected at regular time intervals. When this is the case, the observations make up a time series whose error terms are correlated. This in turn causes bias in the estimates of model parameters. Time series data should be analyzed using time series methods. You will ﬁnd a discussion of time-series analysis in the text Statistics for Management and Economics, 7th edition, by Mendenhall, Beaver, and Beaver.

Residual Plots The other regression assumptions can be checked using residual plots, which are fairly complicated to construct by hand but easy to use once a computer has graphed them for you! In simple linear regression, you can use the plot of residuals versus fit to check for a constant variance as well as to make sure that the linear model is in fact adequate. This plot should be free of any patterns. It should appear as a random scatter of points about 0 on the vertical axis with approximately the same vertical spread for all values of yˆ. One property of the residuals is that they sum to 0 and therefore have a sample mean of 0. The plot of the residuals versus fit for the calculus grade example is shown in Figure 12.10. There are no apparent patterns in this residual plot, which indicates that the model assumptions appear to be satisfied for these data. Plot of the residuals versus yˆ for Example 12.1

● Residuals versus Fitted Value (response is y) 15

10 Residual

FIGU R E 1 2 . 1 0

0 5 10 60

80 Fitted Value

100

524

❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

Recall from Chapter 11 that the normal probability plot is a graph that plots the residuals against the expected value of that residual if it had come from a normal distribution. When the residuals are normally distributed or approximately so, the plot should appear as a straight line, sloping upward. The normal probability plot for the residuals in Example 12.1 is given in Figure 12.11. With the exception of the fourth and ﬁfth plotted points, the remaining points appear to lie approximately on a straight line. This plot is not unusual and does not indicate underlying nonnormality. The most serious violations of the normality assumption usually appear in the tails of the distribution because this is where the normal distribution differs most from other types of distributions with a similar mean and measure of spread. Hence, curvature in either or both of the two ends of the normal probability plot is indicative of nonnormality.

Residuals vs. ﬁts ⇔ random scatter Normal plot ⇔ straight line, sloping up

FI GU R E 1 2 . 1 1

Normal probability plot of residuals for Example 12.1

● Normal Probability Plot of the Residuals (response is y) 99 95

Percent

90 80 70 60 50 40 30 20 10 5 1 20

12.6

0 Residual

EXERCISES

BASIC TECHNIQUES 12.28 What diagnostic plot can you use to determine whether the data satisfy the normality assumption? What should the plot look like for normal residuals?

appear that any regression assumptions have been violated? Explain. MINITAB output for Exercise 12.31

12.29 What diagnostic plot can you use to determine whether the incorrect model has been used? What should the plot look like if the correct model has been used?

whether the assumption of equal variance has been violated? What should the plot look like when the variances are equal for all values of x? 12.31 Refer to the data in Exercise 12.7. The normal probability plot and the residuals versus ﬁtted values plots generated by MINITAB are shown here. Does it

99 95 90

Percent

12.30 What diagnostic plot can you use to determine

Normal Probability Plot of the Residuals (response is y)

80 70 60 50 40 30 20 10 5 1 0.4

0.3

0.2

0.1

0.0 Residual

01.

02.

0.3

0.4

12.6 DIAGNOSTIC TOOLS FOR CHECKING THE REGRESSION ASSUMPTIONS

❍

525

Isaac Asimov are related to the number of months y he took to write them. A plot of the data is shown.

Residuals versus the Fitted Value (response is y)

0.1

500

0.0

450 y(Time in Months)

Residual

0.2

0.1 0.2 0.3 2.5

3.0

3.5

4.0 Fitted Value

4.5

5.0

5.5

400 350 300 250 200 100

200

300 x(Number of Books)

400

500

APPLICATIONS 12.32 Air Pollution Refer to Exercise 12.20, in

which an air pollution monitor’s response to ozone was recorded for several different concentrations of ozone. Use the MINITAB residual plots to comment on the validity of the regression assumptions. MINITAB output for Exercise 12.32 Residuals versus the Fitted Value (response is NO3)

a. Can you see any pattern other than a linear relationship in the original plot? b. The value of r 2 for these data is .959. What does this tell you about the ﬁt of the regression line? c. Look at the accompanying diagnostic plots for these data. Do you see any pattern in the residuals? Does this suggest that the relationship between number of months and number of books written is something other than linear?

0.50 Residuals versus the Fitted Values (response is y)

Residual

0.25 20 0.00 10 Residual

0.25

0.50 2

0 10 20

Fitted Value

30 250

Normal Probability Plot of the Residuals (response is NO3)

300

350

400 Fitted Value

450

500

550

99 Normal Probability Plot of the Residuals (response is y) 99

80 70 60 50 40 30 20

95 90

Percent

95 90

10 5 1 1.0

0.5

0.0 Residual

0.5

1.0

12.33 Professor Asimov, again Refer to Exer-

cise 12.8, in which the number of books x written by

80 70 60 50 40 30 20 10 5 1 50

0 Residual

526

❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

12.34 Laptops and Learning, again Refer to the

data given in Exercise 12.25. The MINITAB printout is reproduced here. Regression Analysis: y versus x The regression equation is y = -26.8 + 1.26 x Predictor Constant x S = 7.61912

Coef -26.82 1.2617

SE Coef 14.76 0.1685

R-Sq = 75.7%

Analysis of Variance Source DF Regression 1 Residual Error 18 Total 19

T -1.82 7.49

P 0.086 0.000

R-Sq(adj) = 74.3%

SS 3254.0 1044.9 4299.0

MS 3254.0 58.1

F 56.05

P 0.000

a. What assumptions must be made about the distribution of the random error, ? b. What is the best estimate of s 2, the variance of the random error, ? c. Use the diagnostic plots for these data to comment on the validity of the regression assumptions. Normal Probability Plot of the Residuals (response is y) 99

Percent

95 90 80 70 60 50 40 30 20

12.35 HDTVs In Exercise 3.19, Consumer

Reports gave the prices for the top 10 LCD high deﬁnition TVs (HDTVs) in the 30- to 40-inch category: Does the price of an LCD TV depend on the size of the screen? The table below shows the ten costs again, along with the screen size in inches.7

EX1235

Brand

Price

Size

JVC LT-40FH96 Sony Bravia KDL-V32XBR1 Sony Bravia KDL-V40XBR1 Toshiba 37HLX95 Sharp Aquos LC-32DA5U Sony Bravia KLV-S32A10 Panasonic Viera TC-32LX50 JVC LT-37X776 LG 37LP1D Samsung LN-R328W

$2900 1800 2600 3000 1300 1500 1350 2000 2200 1200

40 32 40 37 32 32 32 37 37 32

Does the price of an HDTV depend on the size of the screen? Suppose we assume that the relationship between x and y is linear, and perform a linear regression, resulting in a value of r 2 .787. a. What does the value of r 2 tell you about the strength of the relationship between price and screen size? b. The residual plot for this data, generated by MINITAB, is shown below. Does this plot reveal any outliers in the data set? If so, which point is the outlier? Residuals versus the Fitted Values (response is Price)

800

5 1

600 10

0 Residual

400 Residual

Residuals versus the Fitted Values (response is y)

200 0

200 400

10 Residual

1500

1750

2000

2250 Fitted Value

2500

2750

3000

20 60

80 Fitted Value

100

c. Plot the values of x and y using a scatterplot. Does this plot conﬁrm your suspicions in part b? Which HDTV does the outlier represent? Is this a faulty measurement that should be removed from the data set? Explain.

12.7 ESTIMATION AND PREDICTION USING THE FITTED LINE

12.7

❍

527

ESTIMATION AND PREDICTION USING THE FITTED LINE Now that you have • •

tested the ﬁtted regression line, yˆ a bx, to make sure that it is useful for prediction and used the diagnostic tools to make sure that none of the regression assumptions have been violated

you are ready to use the line for one of its two purposes: • •

Estimating the average value of y for a given value of x Predicting a particular value of y for a given value of x

The sample of n pairs of observations have been chosen from a population in which the average value of y is related to the value of the predictor variable x by the line of means, E( y) a bx an unknown line, shown as a broken line in Figure 12.12. Remember that for a ﬁxed value of x—say, x0—the particular values of y deviate from the line of means. These values of y are assumed to have a normal distribution with mean equal to a bx0 and variance s 2, as shown in Figure 12.12.

FIGU R E 1 2 . 1 2

Distribution of y for x x0

● y

Line of means E(y) = α + βx

x = x0

Since the computed values of a and b vary from sample to sample, each new sample produces a different regression line yˆ a bx, which can be used either to estimate the line of means or to predict a particular value of y. Figure 12.13 shows one of the possible conﬁgurations of the ﬁtted line (blue), the unknown line of means (gray), and a particular value of y (the blue dot).

528

❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

FI GU R E 1 2 . 1 3

Error in estimating E (y) and in predicting y

●

Actual value of y you are attempting to predict bx a+ y= x α+β E(y) =

Error of estimating E(y)

Predicted value of y x0

How far will our estimator yˆ a bx0 be from the quantity to be estimated or predicted? This depends, as always, on the variability in our estimator, measured by its standard error. It can be shown that yˆ a bx0 the estimated value of y when x x0, is an unbiased estimator of the line of means, a bx0, and that yˆ is normally distributed with the standard error of yˆ estimated by SE(yˆ)

冪莦冢莦莦莦冣 1 (x0 x苶 )2 MSE Sxx n

Estimation and testing are based on the statistic yˆ E( y) t SE( yˆ) which has a t distribution with (n 2) degrees of freedom. To form a (1 a)100% confidence interval for the average value of y when x x0, measured by the line of means, a bx0, you can use the usual form for a confidence interval based on the t distribution: For a given value of x, the prediction interval is always wider than the confidence interval.

yˆ ta/2SE( yˆ) If you choose to predict a particular value of y when x x0, however, there is some additional error in the prediction because of the deviation of y from the line of means. If you examine Figure 12.13, you can see that the error in prediction has two components: • •

The error in using the ﬁtted line to estimate the line of means The error caused by the deviation of y from the line of means, measured by s2

The variance of the difference between y and yˆ is the sum of these two variances and forms the basis for the standard error of ( y yˆ) used for prediction: SE( y yˆ)

冪莦冤莦莦莦冥莦 1 (x0 x苶 )2 MSE 1 n Sxx

and the (1 a)100% prediction interval is formed as yˆ ta/2SE( y yˆ)

12.7 ESTIMATION AND PREDICTION USING THE FITTED LINE

❍

529

(1 a)100% CONFIDENCE AND PREDICTION INTERVALS •

For estimating the average value of y when x x0: yˆ ta/2

•

冪莦冤莦莦莦冥 1 (x0 x苶 )2 MSE Sxx n

For predicting a particular value of y when x x0: yˆ ta/2

冪莦冤莦莦莦冥 1 (x0 x苶 )2 MSE 1 n Sx x

where ta/2 is the value of t with (n 2) degrees of freedom and area a/2 to its right. EXAMPLE

12.4

Use the information in Example 12.1 to estimate the average calculus grade for students whose achievement score is 50, with a 95% conﬁdence interval. The point estimate of E(y兩x0 50), the average calculus grade for students whose achievement score is 50, is

Solution

yˆ 40.78424 .76556(50) 79.06 The standard error of yˆ is

冤莦莦冥冪莦冤莦莦莦冥冪莦莦

1 (x0 x苶 )2 (50 46)2 1 MSE 75.7532 2.840 n 2474 Sxx 10 and the 95% conﬁdence interval is 79.06 2.306(2.840) 79.06 6.55 Our results indicate that the average calculus grade for students who score 50 on the achievement test will lie between 72.51 and 85.61.

EXAMPLE

12.5

A student took the achievement test and scored 50 but has not yet taken the calculus test. Using the information in Example 12.1, predict the calculus grade for this student with a 95% prediction interval. The predicted value of y is yˆ 79.06, as in Example 12.4. However, the error in prediction is measured by SE( y yˆ), and the 95% prediction interval is

Solution

冤莦莦莦冥冪莦莦

79.06 2.306

(50 46)2 1 75.7532 1 2474 10

79.06 2.306(9.155) 79.06 21.11 or from 57.95 to 100.17. The prediction interval is wider than the conﬁdence interval in Example 12.4 because of the extra variability in predicting the actual value of the response y.

530

❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

One particular point on the line of means is often of interest to experimenters, the y-intercept a—the average value of y when x0 0. EXAMPLE

12.6

Prior to fitting a line to the calculus grade-achievement score data, you may have thought that a score of 0 on the achievement test would predict a grade of 0 on the calculus test. This implies that we should fit a model with a equal to 0. Do the data support the hypothesis of a 0 intercept? You can answer this question by constructing a 95% confidence interval for the y-intercept a, which is the average value of y when x 0. The estimate of a is Solution

yˆ 40.784 .76556(0) 40.784 a and the 95% conﬁdence interval is

冪莦冤莦莦莦冥 (0 46) 1 40.784 2.306 75.7532 冤 10 莦冪莦莦 247莦 4 冥 1 (x0 x苶 )2 MSE n Sxx

yˆ ta/2

40.784 19.617 or from 21.167 to 60.401, an interval that does not contain the value a 0. Hence, it is unlikely that the y-intercept is 0. You should include a nonzero intercept in the model y a bx e. For this special situation in which you are interested in testing or estimating the y-intercept a for the line of means, the inferences involve the sample estimate a. The test for a 0 intercept is given in Figure 12.14 in the shaded line labeled “Constant.” The coefficient given as 40.784 is a, with standard error given in the column labeled “SE Coef ” as 8.507, which agrees with the value calculated in Example 12.6. The value of t 4.79 is found by dividing a by its standard error with p-value .001. FI GU R E 1 2 . 1 4

Portion of the MINITAB output for Example 12.6

●

Predictor Constant x

Coef 40.784 0.7656

SE Coef 8.507 0.1750

T 4.79 4.38

P 0.001 0.002

You can see that it is quite time-consuming to calculate these estimation and prediction intervals by hand. Moreover, it is difficult to maintain accuracy in your calculations. Fortunately, computer programs can perform these calculations for you. The MINITAB regression command provides an option for either estimation or prediction when you specify the necessary value(s) of x. The printout in Figure 12.15 gives the values of yˆ 79.06 labeled “Fit,” the standard error of yˆ, SE( yˆ), labeled “SE Fit,” the conﬁdence interval for the average value of y when x 50, labeled “95.0% CI,” and the prediction interval for y when x 50, labeled “95.0% PI.” FI GU R E 1 2 . 1 5

MINITAB option for estimation and prediction

●

Predicted Values for New Observations New Obs Fit SE Fit 95% CI 1 79.06 2.84 (72.51, 85.61) Values of Predictors for New Observations New Obs x 1 50.0

95% PI (57.95, 100.17)

12.7 ESTIMATION AND PREDICTION USING THE FITTED LINE

❍

531

The confidence bands and prediction bands generated by MINITAB for the calculus grades data are shown in Figure 12.16. Notice that in general the confidence bands are narrower than the prediction bands for every value of the achievement test score x. Certainly you would expect predictions for an individual value to be much more variable than estimates of the average value. Also notice that the bands seem to get wider as the value of x0 gets farther from the mean 苶x. This is because the standard errors used in the confidence and prediction intervals contain the term (x0 苶x )2, which gets larger as the two values diverge. In practice, this means that estimation and prediction are more accurate when x0 is near the center of the range of the x-values. You can locate the calculated confidence and prediction intervals when x 50 in Figure 12.16. FIGU R E 1 2 . 1 6

●

Conﬁdence and prediction intervals for the data in Table 12.1

Fitted Line Plot y 40.78 0.7656 x Regression 95% CI 95% PI

120 110

S R-Sq R-Sq(adj)

100

8.70363 70.5% 66.8%

Grade

90 80 70 60 50 40 30 20

12.7

50 Score

EXERCISES

BASIC TECHNIQUES 12.36 Refer to Exercise 12.6.

a. Estimate the average value of y when x 1, using a 90% conﬁdence interval. b. Find a 90% prediction interval for some value of y to be observed in the future when x 1. 12.37 Refer to Exercise 12.7. Portions of the MINITAB printout are shown here.

12.38 What to Buy? A marketing research

Regression Analysis: y versus x

SE Coef 0.1759 0.04518

experiment was conducted to study the relationship between the length of time necessary for a buyer to reach a decision and the number of alternative package designs of a product presented. Brand names were eliminated from the packages to reduce the effects of brand preferences. The buyers made their selections using the manufacturer’s product descriptions on the packages as the only buying guide. The length of time necessary to reach a decision was

EX1238

The regression equation is y = 6.00 - 0.557 x Coef 6.0000 -0.55714

a. Find a 95% confidence interval for the average value of y when x 2. b. Find a 95% prediction interval for some value of y to be observed in the future when x 2. c. The last line in the third section of the printout indicates a problem with one of the fitted values. What value of x corresponds to the fitted value yˆ 1.5429? What problem has the MINITAB program detected? APPLICATIONS

MINITAB output for Exercise 12.37

Predictor Constant x

T 34.10 -12.33

P 0.000 0.000

Predicted Values for New Observations New Obs Fit SE Fit 95.0% CI 95.0% PI 1 4.8857 0.1027 (4.6006, 5.1708) (4.2886, 5.4829) 2 1.5429 0.2174 (0.9392, 2.1466) (0.7430, 2.3427) X X denotes a point that is an outlier in the predictors. Values of Predictors for New Observations New Obs x 1 2.00 2 8.00

532

❍

CHAPTER 12 LINEAR REGRESSION AND CORRELATION

recorded for 15 participants in the marketing research study.

MINITAB output for Exercise 12.39 Regression Analysis: y versus x

Length of Decision Time, y (sec) 5, 8, 8, 7, 9 7, 9, 8, 9, 10 10, 11, 10, 12, 9

The regression equation is y = 251206 + 27.4 x

Number of Alternatives, x

Predictor Constant x

a. Find the least-squares line appropriate for these data. b. Plot the points and graph the line as a check on your calculations. c. Calculate s 2. d. Do the data present sufficient evidence to indicate that the length of decision time is linearly related to the number of alternative package designs? (Test at the a .05 level of signiﬁcance.) e. Find the approximate p-value for the test and interpret its value. f. If they are available, examine the diagnostic plots to check the validity of the regression assumptions. g. Estimate the average length of time necessary to reach a decision when three alternatives are presented, using a 95% conﬁdence interval. 12.39 Housing Prices If you try to rent an

apartment or buy a house, you ﬁnd that real estate representatives establish apartment rents and house prices on the basis of square footage of heated ﬂoor space. The data in the table give the square footages and sales prices of n 12 houses randomly selected from those sold in a small city. Use the MINITAB printout to answer the questions.

S = 1792.72

Coef 251206 27.406

SE Coef 3389 1.828

R-Sq = 95.7%

T 74.13 14.99

P 0.000 0.000

R-Sq(adj) = 95.3%

Predicted Values for New Observations New Obs Fit SE Fit 95.0% CI 1 299989 526 (298817, 301161) 2 306018 602 (304676, 307360)

95.0% PI (295826, 304151) (301804, 310232)

Values of Predictors for New Observations New Obs x 1 1780 2 2000

a. Can you see any pattern other than a linear relationship in the original plot? b. The value of r 2 for these data is .957. What does this tell you about the ﬁt of the regression line? c. Look at the accompanying diagnostic plots for these data. Do you see any pattern in the residuals? Does this suggest that the relationship between price and square feet is something other than linear? MINITAB output for Exercise 12.39

EX1239

1460 2108 1743 1499 1864 2391

Price, y

Square Feet, x

Price, y

$288,700 309,300 301,400 291,100 302,400 314,900

1977 1610 1530 1759 1821 2216

$305,400 297,000 292,400 298,200 304,300 311,700

95 90

Percent

Square Feet, x

Normal Probability Plot of the Residuals (response is y) 99

80 70 60 50 40 30 20 10 5 1 5000 4000 3000 2000 1000 0 Residual

1000

2000

3000

4000

Plot of data for Exercise 12.39 Residuals versus the Fitted Values (response is y)

315,000 3000 310,000

1000 Residual

y (Price)

2000 305,000 300,000 295,000

0 1000 2000

290,000 3000 1500

1600

1700

1800 1900 2000 x (Square Feet)

2100

2200

2300

2400

290,000

295,000

300,000

305,000 Fitted Value

310,000

315,000

320,000

12.8 CORRELATION ANALYSIS

12.40 Housing Prices II Refer to Exercise 12.39

and data set EX1239. a. Estimate the average increase in the price for an increase of 1 square foot for houses sold in the city. Use a 99% confidence interval. Interpret your estimate. b. A real estate salesperson needs to estimate the average sales price of houses with a total of 2000 square feet of heated space. Use a 95% confidence interval and interpret your estimate. c. Calculate the price per square foot for each house and then calculate the sample mean. Why is this estimate of the average cost per square foot not equal to the answer in part a? Should it be? Explain. d. Suppose that a house with 1780 square feet of heated floor space is offered for sale. Construct a 95% prediction interval for the price at which the house will sell. 12.41 Strawberries III The following data (Exer-

cises 12.16 and 12.24) were obtained in an experiment relating the dependent variable, y (texture of strawberries), with x (coded storage temperature). x

4.0

3.5

2.0

0.5

0.0

a. Estimate the expected strawberry texture for a coded storage temperature of x 1. Use a 99% conﬁdence interval. b. Predict the particular value of y when x 1 with a 99% prediction interval. c. At what value of x will the width of the prediction interval for a particular value of y be a minimum, assuming n remains ﬁxed? 12.42 Tom Brady The number of passes completed and the total number of passing yards for Tom Brady, quarterback for the New England Patriots, were recorded for the 16 regular games in the 2006 football season.8 Week 6 was a bye and no data was reported.

EX1242

12.8

Week

Completions</

Introduction to Probability and Statistics 13th - Mendenhall, Beaver - PDFCOFFEE.COM (2024)