Introduction to probability and statistics

Area TABLE z Areas under the Normal Curve, pages 688–689 z 00 01 02 03 04 05 06 07 08 09 Ϫ3.4 Ϫ3.3 Ϫ3.2 Ϫ3.1 Ϫ3.0 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 0006 0008 0011 0003 0004 0005 0008 0011 0003 0004 0005 0007 0010 0002 0003 0005 0007 0010 Ϫ2.9 Ϫ2.8 Ϫ2.7 Ϫ2.6 Ϫ2.5 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 Ϫ2.4 Ϫ2.3 Ϫ2.2 Ϫ2.1 Ϫ2.0 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 Ϫ1.9 Ϫ1.8 Ϫ1.7 Ϫ1.6 Ϫ1.5 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 Ϫ1.4 Ϫ1.3 Ϫ1.2 Ϫ1.1 Ϫ1.0 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 Ϫ0.9 Ϫ0.8 Ϫ0.7 Ϫ0.6 Ϫ0.5 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 Ϫ0.4 Ϫ0.3 Ϫ0.2 Ϫ0.1 Ϫ0.0 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 (continued) z 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 9994 9996 9997 9989 9992 9995 9996 9997 9990 9993 9995 9996 9997 9990 9993 9995 9997 9998 a ta TABLE Critical Values of t page 691 df t.100 t.050 t.025 t.010 t.005 df 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 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 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 ϱ 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 ϱ SOURCE: From “Table of Percentage Points of the t-Distribution,” Biometrika 32 (1941):300 Reproduced by permission of the Biometrika Trustees 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? 27 79 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 to Calculate Binomial Probabilities? 190 How Do I Calculate Poisson Probabilities Using the Formula? 198 How Do I Use Table to Calculate Poisson Probabilities? 199 How Do I Use Table 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 xෆ? 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 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 Figure 2.4 How Extreme Values Affect the Mean and Median applet Figure 2.9 Why Divide n Ϫ 1? Figure 2.19 Building a Box Plot applet CHAPTER Figure 3.6 Building a Scatterplot applet Figure 3.9 Exploring Correlation applet Figure 3.12 How a Line Works applet CHAPTER Figure 4.6 Tossing Dice applet Figure 4.16 Flipping Fair Coins applet Figure 4.17 Flipping Weighted Coins applet CHAPTER Figure 8.10 Interpreting Confidence Intervals applet CHAPTER 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 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 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 Figure 7.7 Central Limit Theorem applet Figure 7.10 Normal Probabilities for Means applet 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 permissionrequest@cengage.com 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 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 when statistical assumptions have been violated INDEX ❍ Notation factorial, 139 for measures of variability, 62 Null hypothesis analysis of variance and, 454, 470 explanation of, 344–345 rejection of, 347, 353, 357, 645 use of, 349, 350, 373, 393 Number of degrees of freedom (df ) associated with s2, 388 Numbers, random, 256 Numerical measures calculation of s and, 70–71 of center, 53–57 explanation of, 53 five-number summary and box plot and, 80–83 MINITAB and, 88–89 for quantitative bivariate data, 105–108 of relative standing, 75–80 standard deviation and, 66–70 of variability, 60–65 Observational studies, 256, 448 1-in-k systematic random samples, 258 One-sided confidence bounds, 328–329 One-tailed test of hypothesis, 345, 347, 349, 401 One-way classification, 450 See also Completely randomized designs Orderings, 139 Outliers examination of, 22, 24 isolation of, 80–82 median and, 56 z-score and, 76 Output, Paired comparisons computer programs performing, 465 Tukey’s method for, 463, 464, 484 Paired-difference experiment, 640 Paired-difference test, 410–414, 639 Paired t, 414, 641 Parameters explanation of, 53, 255 inferences about, 298–299 testing hypotheses about population, 344 values of, 255 Pareto charts, 13 Partial regression coefficients explanation of, 553, 554, 556 testing significance of, 557 Partial slopes, 553, 556 p chart, 283–285 Pearson, Karl, 595 Pearson product moment sample coefficient of correlation, 533–534 Pearson’s chi-square statistic See Chi-square statistic Percentage, 11 Percentiles, 76, 77 Permutations counting rule for, 140, 141 explanation of, 139 Pie charts explanation of, 12, 13 for quantitative data, 17–19 side-by-side, 98–100 using MINITAB, 38–40 Plane, 553 Plot of residual versus fit, 523, 558 Point estimate, 299 Point estimation explanation of, 299 large-sample, 325–326 of population parameter, 302–303 use of, 300–305 Point estimator explanation of, 299, 315 sampling distribution of, 301 variability of, 303 Poisson approximation, to binomial distribution, 201–202, 237 Poisson probabilities, 237 Poisson probability distribution explanation of, 197–202 formula for, 198 graphs of, 200 MINITAB, 202, 209–211 Poisson random variable, 197–198 Polls, 1–3 Polynomial regression models, 560–562 Pooled method, 403, 405, 406 Pooled t test, 410 Population correlation coefficients, 536 Population means estimating difference between two, 318–321 estimation of, 303, 304, 318–321 explanation of, 54, 166 F test for comparing, 455 large-sample confidence interval for, 310–311 large-sample test about, 347–360 large-sample test for difference between, 363–366 ranking, 462–465 small-sample inferences concerning, 391–392 small-sample inferences for differences between two, 399–406 use of, 55 Population model, 503–506 Population rank correlation coefficient, 664 Populations comparing multinomial, 610–612 explanation of, 3, hypothetical, 257 identification of, normal, 266 skewed, 266 symmetric, 266 Population standard deviation, 167, 223 Population variances comparing two, 424–430 estimation of, 64 explanation of, 62 formula for, 167 hypothesis testing of, 420–421 inference concerning, 417–423 inferences concerning, 417–423 Posterior probabilities, 160, 161 Power of statistical test, 357–360, 643 of z-test, 360 Power curve, 357 741 Prediction use of fitted line for, 527–531 use of regression model for, 559 value of, 520 Prediction equation, 503 Prediction intervals, 529 Predictor variables explanation of, 504, 552 in regression models, 566–571 Principle of least squares, 506–507 Prior probabilities, 160 Probabilistic model, simple linear, 503–506 Probabilities Bayes’ Rule and, 158–161 binomial, 201 conditional, 149–151, 159–160 counting rules and, 137–142 cumulative binomial, 188–189 event relations and, 144–148 events and sample space and, 128–131 for general normal random variables, 229–232 independence and, 149–154 laws of total, 159 Multiplication Rule and, 149–152, 154, 159 Poisson, 198–202 posterior, 160, 161 prior, 160 relationship between statistics and, 128 of sample mean, 268 simple-event, 131–134 unconditional, 159, 603 for unions and complements, 146–148 Probability density function, 192, 221, 222 Probability distributions binomial, 184–193, 237–243 chi-square, 418–420 continuous, 220–222 for continuous random variables, 220–223 for discrete random variables, 164–170 explanation of, 163–164, 221 graphs of, 168 hypergeometric, 205–207 MINITAB and, 173–175 normal, 68, 223–232 Poisson, 197–202 requirements for discrete, 164–165 Probability histograms, 165 Probability tables, 131, 148, 159 Process mean, 281–283 Proportions of defectives, 283–285 estimating difference between two binomial, 324–326 estimation of, 303 sample, 275–279 pth percentile, 76 p-value calculation of, 351–355, 395 explanation of, 345, 346 hypothesis tests and, 365, 378, 394, 422 Quadratic model, 559–562 Qualitative variables explanation of, 10–11, 163 graphs for, 98–100 predictor, 566–571 statistical tables for, 11–12 742 ❍ INDEX Quantitative data graphs for, 17–24 numerical measures for bivariate, 105–107 Quantitative variables explanation of, 10–11, 17, 163 graphs of, 19 predictor, 566–571 scatterplots for two, 102–104 Quartiles calculation of sample, 78–80 explanation of, 78 lower, 77, 78 upper, 77, 78 Quota samples, 258 R2 adjusted value of, 557–558 explanation of, 556–557 Random error, 504, 505 Randomized assignment, 451 Randomized block designs analysis of variance for, 467–473 cautions regarding, 473 explanation of, 413–414, 466–467 Friedman Fr-test for, 656–659 tests for, 471 Random numbers, 256 Random numbers table, 706–707 Random samples explanation of, 256 independent, 399–406, 630–637 1-in-k systematic, 258 simple, 255–256 stratified, 257–258 Random variables binomial, 184, 186–188, 237, 275 continuous, 163, 170, 220–223 discrete, 163–170, 221 explanation of, 163 exponential, 222 hypergeometric, 205–206 normal, 225–232, 266 Poisson, 197–198 probability density function for, 221 uniform, 222 Range approximation of, 70–71 explanation of, 60–61 interquartile, 78–79 Rank correlation coefficient, 660–664 Rank sum, 631, 644 Regression, 109 See also Linear regression; Multiple regression analysis Regression analysis computer software for, 517 misinterpretation of, 580–581 predicting value of, 522 stepwise, 579–580 Regression coefficients partial, 575–577 testing sets of, 575–577 Regression line See also Least-squares regression line calculation of, 111–112 explanation of, 109, 506 Rejection region, 346, 347, 349, 350, 352, 355, 393 Relative efficiency, 644 Relative frequencies explanation of, 11, 12, 100 sum of, 221 Relative frequency distributions for increasingly large sample sizes, 220 probability distributions and, 166 showing extreme values on mean and median, 56 Relative frequency histograms construction of, 26–28, 35–37 explanation of, 24–25 uses for, 28–29 using MINITAB, 27 Relative standing measures explanation of, 75 MINITAB and, 80 sample quartiles and, 78 types of, 75–78 Replications, of experiment, 480 Residual, 488 Residual error, 511, 523 Residual plots explanation of, 488–490, 523–524, 558 interpretation of, 578–579 Response, in experiments, 448 Response variables, 504, 552 Right-tailed test explanation of, 346, 454 use of, 596 Robust, 391, 433, 449 s, calculation of, 70–71 s2 calculation of, 63, 64, 401, 418 explanation of, 62 number of degrees of freedom (df ) associated with, 388, 400 Sample mean calculating probabilities for, 268 formula for, 54 sampling distribution of, 266–272 use of, 55 Sample proportion calculating probabilities for, 277–279 sampling distribution for, 275–279 Samples cluster, 258 convenience, 258 elements of, explanation of, 3, 8, 55 judgment, 258 quote, 258 selection of, 255, 256 variance of, 62 Sample size See also Large-sample confidence interval; Large-sample estimation; Large-sample tests of hypotheses; Small-sample inference; Small-sample techniques binomial experiments and, 186 Central Limit Theorem and, 266 choice of, 329–333 formulas to determine, 333 margin of error and, 305 Sample space, 129, 130 Sample surveys objectives of, 314 problems related to, 256–257, 315 Sample variance calculation of, 64 explanation of, 62–63 Sample z-scores, 75 Sampling, Sampling design See Experimental design Sampling distributions Central Limit Theorem and, 263–266 MINITAB, 264 of point estimator, 301 of sample mean, 266–272, 318–321 of sample proportion, 275–279 sampling plans and experimental designs and, 255–258 statistical process control method and, 281–285 statistics and, 260–262 Sampling error, 305 Sampling plans, 255–258, 329, 448 See also Experimental design Sampling procedure, 4–5 Satterthwaite’s approximation, 406 Scales, examination of, 22, 24 Scatterplots applets, 104, 504, 505 explanation of, 102 to show correlation, 535–536 for two quantitative variables, 102–104 Screening tests, 159–160 Second-order models, 560, 567 Sequential sums of squares, 556 Shape, of data distribution, 22, 24 Shared information, 581 Shortcut method for calculating s2, 63 Side-by-side pie charts, 98–100 Sigma (⌺), 54 Significance level explanation of, 347, 348, 352, 356 practical importance and, 370–371 Sign test normal approximation for, 640–641 for paired experiment, 639–640 Simple events applet, 138 explanation of, 129–131 probabilities of, 131–134, 164 Simple linear probabilistic model, 503–506 Simple random samples, 255–256 See also Random samples Simulation to approximate discrete probability distributions, 165 to approximate sampling distributions, 260–261, 265 Monte Carlo procedure and, 295 Skewed distributions, 22–23, 56 Slope confidence interval for, 517 explanation of, 108 of line of means, 514–516 partial, 553 test for, 516 Small-sample inference See also Inference concerning population mean, 391–396 concerning population variance, 417–423 independent random samples and, 399–406 paired-difference test and, 410–414 INDEX ❍ Small-sample techniques assumptions of, 432–433 comparing two population variances, 424–430 explanation of, 387 MINITAB, 434–436 Student’s t distribution, 387–391 use of, 630 Sources of variation, 468, 481 Spearman rs, 660 Spearman’s rank correlation coefficient critical values of, 705 explanation of, 660–664 Spearman’s rank correlation test, 663–664 Stacked bar charts explanation of, 98 use of, 99, 100 Standard deviation for binomial random variables, 186–188 calculation of, 65 for discrete random variables, 166–170 explanation of, 62–63 for Poisson probability distribution, 198 population, 167, 223 practical significance of, 66–70 in research results, 304 Standard error of estimator, 267, 528 explanation of, 313 of mean, 267, 392 in research results, 304 Standardized normal distribution, 225, 229–230 Standardized test statistic, 348 Standard normal random variable, 225–229 Standard normal z distribution, 388 States of nature, 160 Statistical inference See Inference Statistical process control (SPC) control chart for process mean and, 281–283 control chart for proportion defective and, 283–285 explanation of, 281 Statistical significance, 352, 370–371 Statistical software, Statistical tables, 11, 12, 14 Statistical tests See also Hypothesis testing comparison of, 643–644 equivalence of, 614–615 essentials of, 348–350 explanation of, 344–347, 378 large-sample, 350–351, 363–366, 369–370, 373–376 left-tailed, 346 power curve for, 357 power of, 356–360 right-tailed, 346, 454, 596 Statistical theorems, 261, 262 Statistics descriptive, explanation of, 53, 260 inferential, 4–5 relationship between probability and, 128 sampling distributions and, 260–262 training your brain for, 5–6 Stem and leaf plots, 20–22 Stepwise regression analysis, 579–580 Strata, 257 Stratified random samples, 257–258 Studentized range explanation of, 463 percentage points of, 708–711 Student’s t distribution applet, 389 assumptions behind, 391 explanation of, 388–389 statistical computing packages and, 395 Student’s t probabilities, 389, 392, 412 Student’s t table, 389 Sum of sample measurements ⌺xi, 265 Sum of squares for error (SSE), 452, 480, 506 Sum of squares for treatments (SST), 451–452 Sums of squares calculation of, 481 main effect, 480 sequential, 556 use of, 507, 509 Symbols, for process of summing, 54 Symmetric distributions, 22, 56, 223 Tchebysheff’s Theorem calculation of s and, 70 explanation of, 66–69 use of, 66, 68–70, 193 z-scores and, 76 Tests of homogeneity, 611 Test statistic analysis of variance, 454, 455 explanation of, 345, 346, 405 modification of, 375–376 standardized, 348 use of, 349, 393 for Wilcoxon signed-rank test, 644 Tied observations, 639 Time-dependent multinomials, 615–616 Time series, 19, 523 Total sum of squares (TSS), 451, 452 t-probabilities, 537 Treatment means estimating differences in, 456–458, 464 testing equality of, 454–456 Treatments in experiments, 448 identifying differences in, 472–473 randomized block design and, 467 testing equality of, 470–471 Tree diagrams, 130–131 Trend, 19 t statistic degree of freedom for, 400 as robust, 391 use of, 432 t-test paired, 414, 641 pooled, 410 two-sample, 404, 406 use of, 641 Tukey’s method for paired comparisons, 463, 464, 484 Two-sample t test, 404, 406 Two-sided confidence intervals, 328 See also Confidence intervals Two-tailed test of hypothesis, 345, 349, 350, 401, 645 743 ϫ factorial experiments, 480 Type I error, 347 Type II error, 356, 357 Unbiased estimator, 301–303 Unconditional probabilities, 159, 603 Undercoverage, in sample surveys, 257 Uniform random variables, 222 Unimodal distributions, 23 Unions calculating probabilities for, 146–148 of events, 144–146 Univariate data, Upper confidence limit (UCL), 309, 328 Upper quartiles, 77, 78 Variability estimator, 301–303 measures of, 60–65 rules for describing, 67 Variables continuous, 10, 11, 17 continuous random, 163, 170, 220–223 dependent, 108 discrete, 10, 11, 17 dummy, 567 explanation of, 8–9, 163 independent, 108 predictor, 552 qualitative, 10, 11, 98–100, 163, 566–571 quantitative, 10, 11, 17, 102–104, 566–571 random, 163–170 response, 552 types of, 10–11 Variance See also Analysis of variance (ANOVA) calculation of, 63 common, 449 explanation of, 62 for grouped data, 74 MINITAB, 534 notation for, 62 population, 62, 64, 167, 417–430 sample, 62, 64 Venn diagrams events in, 146, 147 explanation of, 130 Weighted average, 400 Wilcoxon, Frank, 631, 644 Wilcoxon rank sum test explanation of, 630–634 normal approximation for, 634–636 use of, 637 Wilcoxon signed-rank test critical values of T for, 704 normal approximation for, 647–648 for paired experiment, 644–647 Wording bias, in sample surveys, 257 x– chart, 282–283 y-intercept, 108, 109, 530 z-scores applet, 232–233 explanation of, 75–76 z-test, 358–360 Credits This page constitutes an extension of the copyright page We have made every effort to trace the ownership of all copyrighted material and to secure permission from copyright holders In the event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in future printings Thanks are due to the following authors, publishers, and agents for permission to use the material indicated Introduction 1: © Mark Karrass/CORBIS; 2: “Hot News: 98.6 Not Normal,” © McClatchy-Tribune Information Services All Rights Reserved Reprinted with permission Chapter 7: © Jupiterimages/Brand X/CORBIS; 9: Portions of the input and output contained in this publication/book are printed with permission of Minitab® Inc All material remains the exclusive property and copyright of Minitab®, Inc All rights reserved www.minitab.com; 31, Exercise 1.29: Adapted from “Top Ten Organized Religions of the World,” www.infoplease.com/ipa/A0904108.html, as it appeared on November 15, 2007 Info Please Database, © Pearson Education, Inc Reproduced by permission of Pearson Education, Inc publishing as Info Please All rights reserved; 47, 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Consumer Reports® for educational purposes only No commercial use or reproduction permitted www.ConsumerReports.orgđ Chapter 12 502: â Justin Sullivan/Getty Images; 549, Exercise 12.80: From “Ratings: Walking Shoes,” Copyright 2006 by Consumers Union of U.S., Inc., Yonkers, NY 10703-1057, a nonprofit organization Reprinted with permission from the October 2006 issue of Consumer Reports® for educational purposes only No commercial use or reproduction permitted www.ConsumerReports.orgđ Chapter 13 551: â Will & Deni McIntyre/CORBIS; 590, Exercise 13.33: From “Tuna Goes Upscale,” 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.orgđ Chapter 14 594: â Dave Bartruff/CORBIS; 601, Exercises 14.13, 14.14: M&M’s® and M® are registered trademarks owned by Mars, Incorporated and its affiliates These trademarks are used with permission Mars, Incorporated is not associated with Cengage Learning Market Group worldwide © Mars, Inc 2008 Chapter 15 629: © Don Carstens/Brand X/CORBIS; 677: From “Eggs Substitutes Range in Quality” by K Sakekel in The San Francisco Chronicle, Febuary 10, 1993, p Copyright © 1993 San Francisco Chronicle Appendix 691: From “Table of Percentage Points of the t-Distribution,” Biometrika 32 (1941):300 Reproduced by permission of the Biometrika Trustees; 692: From “Tables of the Percentage Points of the x2-Distribution,” Biometrika Tables for Statisticians, Vol 1, 3rd ed (1966) Reproduced by permission of the Biometrika Trustees; 694: A portion of “Tables of Percentage Points of the Inverted Beta (F) Distribution,” Biometrika, Vol 33 (1943) by M Merrington and C.M Thompson and from Table 18 of Biometrika Tables for Statisticians, Vol 1, Cambridge University Press, 1954, edited by E.S Pearson and 746 ❍ CREDITS H.O Hartley Reproduced with permission of the authors, editors, and Biometrika Trustees; 702, Tables 7(a) and 7(b): Data from “An Extended Table of Critical Values for the Mann-Whitney (Wilcoxon) Two-Sample Statistic” by Roy C Milton, pp 925–934, in the Journal of the American Statistical Association, Vol 59, No 307, Sept 1964 Reprinted with permission from the Journal of the American Statistical Association Copyright 1964 by the American Statistical Association All rights reserved; 704: From “Some Rapid Approximate Statistical Procedures” (1964) 28, by F Wilcoxon and R.A Wilcox Reproduced with the kind permission of Lederle Laboratories, a division of American Cyanamid Company; 705: From “Distribution of Sums of Squares of Rank Differences for Small Samples” by E.G Olds, Annals of Mathematical Statistics (1938) Reproduced with the permission of the editor, Annals of Mathematical Statistics; 706: From Handbook of Tables for Probability and Statistics, 2nd ed., edited by William H Beyer (CRC Press) Used by permission of William H Beyer Answers to MyPersonal Trainer Exercises Chapter A 90 12.86 15 5.9 98 1.0 200 25 25 B 0 to < 15 0 to < 1.0 15 to < 30 1.0 to < 2.0 500 500 to < 525 525 to < 550 Chapter A Data Set Sorted n 2, 5, 7, 1, 1, 2, 1, 1, 2, 2, 5, 7, Position of Q1 Position of Q3 Lower Quartile, Q1 Upper Quartile, Q3 2nd 6th 5, 0, 1, 3, 1, 5, 5, 2, 4, 4, 0, 1, 1, 1, 2, 3, 4, 4, 5, 5, 11 3rd 9th B Sorted Data Set Position of Q1 Adjacent Values 0, 1, 4, 4, 5, 1.75 and ϩ 75(1) ϭ 75 5.25 and ϩ 25(4) ϭ 0, 1, 3, 3, 4, 7, 7, 2.25 and ϩ 25(2) ϭ 1.5 6.75 and 7 ϩ 75(0) ϭ and ϩ 5(1) ϭ 1.5 7.5 and ϩ 5(2) ϭ 1, 1, 2, 5, 6, 6, 7, 9, 2.5 Chapter Q1 Position of Q3 Adjacent Values A x y xy Calculate: Covariance nϭ3 10 sx ϭ (Sx)(Sy) Sxy Ϫ ᎏᎏ n sxy ϭ ᎏᎏ ϭ nϪ1 sy ϭ 2.082 Sx ϭ Sy ϭ Sxy ϭ 18 Correlation Coefficient sy r ϭ ᎏxᎏ ϭ 240 sx sy B From Part A From Part A Calculate: Sx ϭ sx ϭ xෆ ϭ Sy ϭ sy ϭ 2.082 yෆ ϭ 2.667 r ϭ 240 Chapter P(A) P(B) Slope ΂ ΃ s b ϭ r ᎏᎏy ϭ 25 sx Conditions for Events A and B Mutually exclusive Independent Mutually exclusive and dependent Independent A .010, 087, 317, 663, 922, 1.000 a ϭ yෆ Ϫ bxෆ ϭ 2.167 Regression Line: y ϭ 2.167 ϩ 25x Chapter Section 5.2 y-Intercept P(A ʝ B) P(A ʜ B) P(A͉B) 12 58 10 B 0, 1, 2, 3, P(x Յ 4) n/a 922 4, P(x Ն 4) Ϫ P (x Յ 3) 337 P(x Ͼ 4) Ϫ P (x Յ 4) 078 0, 1, 2, P(x Ͻ 4) P (x Յ 3) 663 2, 3, P(2 Յ x Յ 4) P(x Յ 4) Ϫ P (x Յ 1) 835 P (x ϭ 4) P(x Յ 4) Ϫ P (x Յ 3) 259 Q3 Chapter Section 5.3 A B Ϫ1.5 223, 558, 809, 934, 981, 996, 999, 1.000 1.5 e ᎏᎏ, 223 0! 1.51eϪ1.5 ᎏᎏ, 335 1! 0, 1, 558 Chapter Section 6.3 C 0, 1, 2, P(x Յ 3) n/a 934 3, 4, 5, P(x Ն 3) Ϫ P(x Յ 2) 191 4, 5, 6, P(x Ͼ 3) Ϫ P(x Յ 3) 066 0, 1, P(x Ͻ 3) P(x Յ 2) 809 2, 3, P(2 Յ x Յ 4) P(x Յ 4) Ϫ P (x Յ 1) 423 P(x ϭ 3) P(x Յ 3) Ϫ P (x Յ 2) 125 1.5 n/a 9332 Ϫ P (z Յ 2) Ϫ 9772 ϭ 0228 2.33 Ϫ P (z Յ 2.33) Ϫ 9901 ϭ 0099 Ϫ1.96, 1.96 P(z Ͻ 1.96) Ϫ P(z Ͻ Ϫ1.96) 9750 Ϫ 0250 ϭ 9500 Ϫ1.24, 2.37 P(z Ͻ 2.37) Ϫ P(z Ͻ Ϫ1.24) 9911 Ϫ 1075 ϭ 8836 Ϫ1 n/a 1587 Chapter Section 6.4 A 12; 18 B yes 20, 21, , 30 12; 2.683 20; 19.5 2.80 2.80; 9974; 0026 Chapter Section 7.5 A B C P(xෆ Ͼ 80); 2.5; 80; 2.5; 9938; 0062 normal; 75; P(70 Ͻ xෆ Ͻ 72); Ϫ2.5; Ϫ1.5; 70; 72; Ϫ2.5; Ϫ1.5; 0668; 0062; 0606 Chapter Section 7.6 A B normal; 4; 08165 Chapter C P(pˆ Ͼ 5); 1.22; 5; 1.22; 8888; 1112 Type or MOE Ί๶ pq 1.96 ᎏᎏ n Quantitative One Ί๶๶ s2 s2 ᎏᎏ1 ϩ ᎏᎏ2 n n Binomial Chapter Two P (.5 Ͻ pˆ Ͻ 6); 1.22; 2.45; 5; 6; 1.22; 2.45; 9929; 8888; 1041 Solve Sample Size Ί๶ 4(.6) 1.96 ᎏᎏ Յ n n Ն 93 1.96ᎏᎏ Յ ͙nෆ n Ն 139 Ί๶๶ n1 Ն 70 n2 Ն 70 Ί๶๶ n1 Ն 738 n2 Ն 738 36 36 1.96 ᎏᎏ ϩ ᎏᎏ Յ n n 4(.6) 4(.6) 1.96 ᎏᎏ ϩ ᎏᎏ Յ 05 n n A B Critical Value Rejection Region Conclusion p-value p-value Ͻ a? Conclusion 1.645 z Ͼ 1.645 Do not reject H0 0808 No Do not reject H0 2.33 z Ͼ 2.33 Reject H0 0069 Yes Reject H0 1.96 z Ͼ 1.96 or z Ͻ Ϫ1.96 Do not reject H0 4592 No Do not reject H0 2.58 z Ͼ 2.58 or z Ͻ Ϫ2.58 Reject H0 Ϸ0 Yes Reject H0 ... 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... 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. .. 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