www.ebook3000.com doa73699_fm_i-xxvii.qxd 11/26/09 12:31 PM Page i Applied Statistics in Business and Economics Third Edition David P Doane Oakland University Lori E Seward University of Colorado www.ebook3000.com doa73699_fm_i-xxvii.qxd 12/4/09 11:01 PM Page ii APPLIED STATISTICS IN BUSINESS AND ECONOMICS Published by McGraw-Hill/Irwin, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY, 10020 Copyright © 2011, 2009, 2007 by The McGraw-Hill Companies, Inc All rights reserved No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper WDQ/WDQ ISBN 978-0-07-337369-0 MHID 0-07-337369-9 Vice president and editor-in-chief: Brent Gordon Editorial director: Stewart Mattson Publisher: Tim Vertovec Executive editor: Steve Schuetz Director of development: Ann Torbert Senior developmental editor: Wanda J Zeman Vice president and director of marketing: Robin J Zwettler Marketing director: Sankha Basu Marketing manager: Michelle Heaster Vice president of editing, design and production: Sesha Bolisetty Lead project manager: Pat Frederickson Full service project manager: Manjot Singh Dhodi Production supervisor: Michael McCormick Designer: Matt Diamond Senior photo research coordinator: Lori Kramer Photo researcher: Allison Grimes Senior media project manager: Kerry Bowler Typeface: 10/12 Times New Roman Compositor: MPS Limited, A Macmillan Company Printer: Worldcolor Library of Congress Cataloging-in-Publication Data Doane, David P Applied statistics in business and economics / David P Doane, Lori E Seward — 3rd ed p cm — (The McGraw-Hill/Irwin series, operations and decision sciences) Includes index ISBN-13: 978-0-07-337369-0 (alk paper) ISBN-10: 0-07-337369-9 (alk paper) Commercial statistics Management—Statistical methods Economics—Statistical methods I Seward, Lori Welte, 1962- II Title HF1017.D55 2011 519.5—dc22 2009045547 www.mhhe.com www.ebook3000.com doa73699_fm_i-xxvii.qxd 11/26/09 12:31 PM Page iii ABOUT THE AUTHORS David P Doane David P Doane is a professor of quantitative methods in Oakland University’s Department of Decision and Information Sciences He earned his Bachelor of Arts degree in mathematics and economics at the University of Kansas and his PhD from Purdue University’s Krannert Graduate School His research and teaching interests include applied statistics, forecasting, and statistical education He is corecipient of three National Science Foundation grants to develop software to teach statistics and to create a computer classroom He is a longtime member of the American Statistical Association and INFORMS, serving in 2002 as president of the Detroit ASA chapter, where he remains on the board He has consulted with government, health care organizations, and local firms He has published articles in many academic journals and is the author of LearningStats (McGraw-Hill, 2003, 2007) and co-author of Visual Statistics (McGraw-Hill, 1997, 2001) Lori E Seward Lori E Seward is an instructor in the Decisions Sciences Department in the College of Business at the University of Colorado at Denver and Health Sciences Center She earned her Bachelor of Science and Master of Science degrees in Industrial Engineering at Virginia Tech After several years working as a reliability and quality engineer in the paper and automotive industries, she earned her PhD from Virginia Tech She served as the chair of the INFORMS Teachers’ Workshop for the annual 2004 meeting Prior to joining UCDHSC in 2008, Dr Seward served on the faculty at the Leeds School of Business at the University of Colorado–Boulder for 10 years Her teaching interests focus on developing pedagogy that uses technology to create a collaborative learning environment in both large undergraduate and MBA statistics courses Her most recent article was published in The International Journal of Flexible Manufacturing Systems (Kluwer Academic Publishers, 2004) DEDICATION To Robert Hamilton Doane-Solomon David To all my students who challenged me to make statistics relevant to their lives Lori iii www.ebook3000.com doa73699_fm_i-xxvii.qxd 11/26/09 12:31 PM Page iv FROM THE “How often have you heard people/students say about a particular subject, ‘I’ll never use this in the real world?’ I thought statistics was a bit on the ‘math-geeky’ side at first Imagine my horror when I saw ␣, R2, and correlations on several financial reports at my current job (an intern position at a financial services company) I realized then that I had better try to understand some of this stuff.” —Jill Odette (an introductory statistics student) As recently as a decade ago our students used to ask us, “How I use statistics?” Today we more often hear, “Why should I use statistics?” Applied Statistics in Business and Economics has attempted to provide real meaning to the use of statistics in our world by using real business situations and real data and appealing to your need to know why rather than just how With over 50 years of teaching statistics between the two of us, we feel we have something to offer Seeing how students have changed as the new century unfolds has required us to adapt and seek out better ways of instruction So we wrote Applied Statistics in Business and Economics to meet four distinct objectives Objective 1: Communicate the Meaning of Variation in a Business Context Variation exists everywhere in the world around us Successful businesses know how to measure variation They also know how to tell when variation should be responded to and when it should be left alone We’ll show how businesses this Objective 2: Use Real Data and Real Business Applications Examples, case studies, and problems are taken from published research or real applications whenever possible Hypothetical data are used when it seems the best way to illustrate a concept You can usually tell the difference by examining the footnotes citing the source Objective 3: Incorporate Current Statistical Practices and Offer Practical Advice With the increased reliance on computers, statistics practitioners have changed the way they use statistical tools We’ll show the current practices and explain why they are used the way they are We will also tell you when each technique should not be used Objective 4: Provide More In-Depth Explanation of the Why and Let the Software Take Care of the How It is critical to understand the importance of communicating with data Today’s computer capabilities make it much easier to summarize and display data than ever before We demonstrate easily mastered software techniques using the common software available We also spend a great deal of time on the idea that there are risks in decision making and those risks should be quantified and directly considered in every business decision Our experience tells us that students want to be given credit for the experience they bring to the college classroom We have tried to honor this by choosing examples and exercises set in situations that will draw on students’ already vast knowledge of the world and knowledge gained from other classes Emphasis is on thinking about data, choosing appropriate analytic tools, using computers effectively, and recognizing limitations of statistics What’s New in This Third Edition? In this third edition we have listened to you and have made many changes that you asked for We sought advice from students and faculty who are currently using the textbook, objective reviewers at a variety of colleges and universities, and participants in focus groups on teaching statistics with technology At the end of this preface is a detailed list of chapter-bychapter improvements, but here are just a few of them: • Revised learning objectives mapped to topics within chapter sections • Step-by-step instructions on using Excel 2007 for descriptive statistics, histograms, scatter plots, line charts, fitting trends, and editing charts • More “practice” exercises and more worked examples in the textbook • Sixteen large, real data sets that can be downloaded for class projects • Many updated exercises and new skill-focused “business context” exercises • Appendix on writing technical business reports and presenting them orally • Expanded treatment of business ethics and critical thinking skills • Closer compatibility between textbook exercises and Connect online grading • Rewritten instructor’s manual with step-by-step solutions iv www.ebook3000.com doa73699_fm_i-xxvii.qxd 11/26/09 12:31 PM Page v AUTHORS • New Mini Cases featuring Vail Resorts, Inc., a mountain resort company • Consistent notation for random variables and event probabilities • Improved flow of normal distribution concepts and matching exercises • Restructured material on sampling distributions, estimation, and hypothesis testing • Intuitive explanations and illustrations of p-values and steps in hypothesis testing • New format for hypotheses in tests of two means or two proportions • Moved two-sample confidence intervals to chapter on two-sample hypothesis tests • More coverage of covariance and its role in financial analysis • More emphasis on interpretation of regression results • End of each chapter guides to downloads from the Online Learning Center (simulations, demonstrations, tips, and ScreenCam video tutorials for Excel, MegaStat, and MINITAB) Software Excel is used throughout this book because it is available everywhere But calculations are illustrated using MegaStat, an Excel add-in whose Excel-based menus and spreadsheet format offer more capability than Excel’s Data Analysis Tools MINITAB menus and examples are also included to point out similarities and differences of these tools To assist students who need extra help or “catch up” work, the text Web site contains tutorials or demonstrations on using Excel, MINITAB, or MegaStat for the tasks of each chapter At the end of each chapter is a list of LearningStats demonstrations that illustrate the concepts from the chapter These demonstrations can be downloaded from the text Web site (www.mhhe.com/doane3e) Math Level The assumed level of mathematics is pre-calculus, though there are rare references to calculus where it might help the better-trained reader All but the simplest proofs and derivations are omitted, though key assumptions are stated clearly The learner is advised what to when these assumptions are not fulfilled Worked examples are included for basic calculations, but the textbook does assume that computers will all calculations after the statistics class is over Thus, interpretation is paramount End-of-chapter references and suggested Web sites are given so that interested readers can deepen their understanding Exercises Simple practice exercises are placed within each section End-of-chapter exercises tend to be more integrative or to be embedded in more realistic contexts The end-of-chapter exercises encourage the learner to try alternative approaches and discuss ambiguities or underlying issues when the statistical tools not quite “fit” the situation Some exercises invite miniessays (at least a sentence or two) rather than just quoting a formula Answers to most odd-numbered exercises are in the back of the book (all answers are in the instructor’s manual) LearningStats LearningStats is intended to let students explore data and concepts at their own pace, ignoring material they already know and focusing on things that interest them LearningStats includes explanations on topics that are not covered in other software packages, such as how to write effective reports, how to perform calculations, how to make effective charts, or how the bootstrap method works It also includes some topics that did not appear prominently in the textbook (e.g., stem-and-leaf plots, finite population correction factor, and bootstrap simulation techniques) Instructors can use LearningStats PowerPoint presentations in the classroom, but students can also use them for self-instruction No instructor can “cover everything,” but students can be encouraged to explore LearningStats data sets and/or demonstrations perhaps with an instructor’s guidance, or even as an assigned project David P Doane Lori E Seward v www.ebook3000.com doa73699_fm_i-xxvii.qxd 11/26/09 12:31 PM Page vi HOW ARE CHAPTERS ORGANIZED Chapter Contents Chapter Contents Each chapter begins with a short list of section topics that are covered in the chapter 1.1 What Is Statistics? 1.2 Why Study Statistics? 1.3 Uses of Statistics 1.4 Statistical Challenges 1.5 Critical Thinking Chapter Learning Objectives Each chapter includes a list of learning objectives students should be able to attain upon reading and studying the chapter material Learning objectives give students an overview of what is expected and identify the goals for learning Learning objectives also appear next to chapter topics in the margins Chapter Learning Objectives When you finish this chapter you should be able to LO1 Define statistics and explain some of its uses in business LO2 List reasons for a business student to study statistics LO3 State the common challenges facing business professionals using statistics LO4 List and explain common statistical pitfalls Section Exercises SECTION EXERCISES Multiple section exercises are found throughout the chapter so that students can focus on material just learned Instructions for Exercises 12.21 and 12.22: (a) Perform a regression using MegaStat or Excel (b) State the null and alternative hypotheses for a two-tailed test for a zero slope (c) Report the p-value and the 95 percent confidence interval for the slope shown in the regression results (d) Is the slope significantly different from zero? Explain your conclusion Mini Cases Every chapter includes two or three mini cases, which are solved applications They show and illlustrate the analytical application of specific statistical concepts at a deeper level than the examples 12.21 College Student Weekly Earnings in Dollars (n = 5) WeekPay 12.22 Phone Hold Time for Concert Tickets in Seconds (n = 5) CallWait Hours Worked (X) Weekly Pay (Y) Operators (X) Wait Time (Y) 10 15 20 20 35 93 171 204 156 261 385 335 383 344 288 Mini Case 4.7 Vail Resorts Customer Satisfaction Figure 4.37 is a matrix showing correlations between several satisfaction variables from a sample of respondents to a Vail Resorts’ satisfaction survey The correlations are all positive, suggesting that greater satisfaction with any one of these criteria tends to be associated with greater satisfaction with the others (positive covariance) The highest correlation (r = 0.488) is between SkiSafe (attention to skier safety) and SkiPatV (Ski Patrol visibility) This makes intuitive sense When a skier sees a ski patroller, you would expect increased perception that the organization is concerned with skier safety While many of the correlations seem small, they are all statistically significant (as you will learn in Chapter 12) FIGURE 4.37 Correlation Matrix Skier Satisfaction Variables (n = 502) VailGuestSat LiftOps LiftWait TrailVar SnoAmt GroomT SkiSafe LiftOps 1.000 LiftWait 0.180 1.000 TrailVar 0.206 0.128 1.000 SnoAmt 0.242 0.227 0.373 1.000 GroomT 0.271 0.251 0.221 0.299 1.000 SkiSafe 0.306 0.196 0.172 0.200 0.274 1.000 SkiPatV 0.190 0.207 0.172 0.184 0.149 0.488 vi www.ebook3000.com SkiPatV 1.000 doa73699_fm_i-xxvii.qxd 11/26/09 12:31 PM Page vii TO PROMOTE STUDENT LEARNING? Figures and Tables Throughout the text, there are hundreds of charts, graphs, tables, and spreadsheets to illustrate statistical concepts being applied These visuals help stimulate student interest and clarify the text explanations FIGURE 4.21 Central Tendency versus Dispersion Machine A Machine B Too Much Process Variation Incorrectly Centered Process 12 10 15 Percent 10 0 4.996 5.000 5.004 Diameter of Hole 5.008 5.012 30 50 90 130 170 260 450 1,020 Examples 10 30 50 96 131 176 268 450 1,050 15 15 35 35 50 53 100 100 139 140 185 198 270 279 474 484 1,200 1,341 EXAMPLE U.S Trade USTrade 20 36 55 100 145 200 295 495 20 39 60 100 150 200 309 553 20 40 60 100 150 200 345 600 22 40 60 103 153 220 350 720 23 40 67 105 153 232 366 777 25 40 75 118 156 237 375 855 26 47 78 125 160 252 431 960 26 50 86 125 163 259 433 987 Figure 3.18 shows the U.S balance of trade The arithmetic scale shows that growth has been exponential Yet, although exports and imports are increasing in absolute terms, the log graph suggests that the growth rate in both series may be slowing, because the log graph is slightly concave On the log graph, the recently increasing trade deficit is not relatively as large Regardless how it is displayed, the trade deficit remains a concern for policymakers, for fear that foreigners may no longer wish to purchase U.S debt instruments to finance the trade deficit (see The Wall Street Journal, July 24, 2005, p Cl) FIGURE 3.18 Comparison of Arithmetic and Log Scales USTrade U.S Balance of Trade, 1960–2005 U.S Balance of Trade, 1960–2005 2,500 10,000 Exports 2,000 Imports 1,500 1,000 500 Imports 1,000 100 (a) Arithmetic scale 05 00 20 95 20 90 19 85 19 80 19 75 70 19 19 65 19 05 00 20 95 20 90 19 85 19 19 80 75 19 70 19 19 65 10 19 19 60 Exports 60 Billions of Current Dollars Examples of interest to students are taken from published research or real applications to illustrate the statistics concept For the most part, examples are focused on business but there are also some that are more general and don’t require any prerequisite knowledge And there are some that are based on student projects 4.992 4.994 4.996 4.998 5.000 5.002 5.004 Diameter of Hole 100 ATM Deposits (dollars) ATMDeposits TABLE 4.7 19 4.992 19 4.988 Billions of Current Dollars Percent 20 (b) Log scale Data Set Icon A data set icon is used throughout the text to identify data sets used in the figures, examples, and exercises that are included on the Online Learning Center (OLC) for the text USTrade vii www.ebook3000.com doa73699_fm_i-xxvii.qxd 11/26/09 12:31 PM Page viii HOW DOES THIS TEXT REINFORCE Chapter Summary CHAPTER SUMMARY For a set of observations on a single numerical variable, a dot plot displays the individual data values, while a frequency distribution classifies the data into classes called bins for a histogram of frequencies for each bin The number of bins and their limits are matters left to your judgment, though Sturges’ Rule offers advice on the number of bins The line chart shows values of one or more time series variables plotted against time A log scale is sometimes used in time series charts when data vary by orders of magnitude The bar chart or column chart shows a numerical data value for each category of an attribute However, a bar chart can also be used for a time series A scatter plot can reveal the association (or lack of association) between two variables X and Y The pie chart (showing a numerical data value for each category of an attribute if the data values are parts of a whole) is common but should be used with caution Sometimes a simple table is the best visual display Creating effective visual displays is an acquired skill Excel offers a wide range of charts from which to choose Deceptive graphs are found frequently in both media and business presentations, and the consumer should be aware of common errors Chapter summaries provide an overview of the material covered in the chapter Key Terms KEY TERMS arithmetic scale, 79 bar chart, 82 column chart, 82 central tendency, 59 dispersion, 59 dot plot, 61 frequency distribution, 64 frequency polygon, 72 histogram, 66 Key terms are highlighted and defined within the text They are also listed at the ends of chapters, along with chapter page references, to aid in reviewing Commonly Used Formulas Some chapters provide a listing of commonly used formulas for the topic under discussion left-skewed, 71 line chart, 77 logarithmic scale, 79 modal class, 71 ogive, 72 outlier, 71 Pareto chart, 82 pie chart, 95 pivot table, 92 right-skewed, 71 scatter plot, 86 shape, 59 stacked bar chart, 83 stacked dot plot, 62 Sturges’ Rule, 65 symmetric, 71 trend line, 89 Commonly Used Formulas in Descriptive Statistics Sample mean: x¯ = n n xi i=1 Geometric mean: G = √ n x1 x2 · · · xn Range: R = xmax − xmin Midrange: Midrange = xmin + xmax n (xi − x) ¯ Sample standard deviation: s = Chapter Review Each chapter has a list of questions for student selfreview or for discussion CHAPTER REVIEW i=1 n−1 (a) What is a dot plot? (b) Why are dot plots attractive? (c) What are their limitations? (a) What is a frequency distribution? (b) What are the steps in creating one? (a) What is a histogram? (b) What does it show? (a) What is a bimodal histogram? (b) Explain the difference between left-skewed, symmetric, and right-skewed histograms (c) What is an outlier? (a) What is a scatter plot? (b) What scatter plots reveal? (c) Sketch a scatter plot with a moderate positive correlation (d) Sketch a scatter plot with a strong negative correlation viii www.ebook3000.com doa73699_fm_i-xxvii.qxd 11/26/09 12:31 PM Page ix STUDENT LEARNING? Chapter Exercises DATA SET A Advertising Dollars as Percent of Sales in Selected Industries (n = 30) Ads Industry Percent Accident and health insurance Apparel and other finished products Beverages 0.9 5.5 7.4 … … Exercises give students an opportunity to test their understanding of the chapter material Exercises are included at the ends of sections and at the ends of chapters Some exercises contain data sets, identified by data set icons Data sets can be accessed on the Online Learning Center and used to solve problems in the text 4.75 (a) Choose a data set and prepare a brief, descriptive report.You may use any computer software you wish (e.g., Excel, MegaStat, MINITAB) Include relevant worksheets or graphs in your report If some questions not apply to your data set, explain why not (b) Sort the data (c) Make a histogram Describe its shape (d) Calculate the mean and median Are the data skewed? (e) Calculate the standard deviation (f) Standardize the data and check for outliers (g) Compare the data with the Empirical Rule Discuss (h) Calculate the quartiles and interpret them (i) Make a box plot Describe its appearance Steel works and blast furnaces Tires and inner tubes Wine, brandy, and spirits 1.9 1.8 11.3 Source: George E Belch and Michael A Belch, Advertising and Promotion, pp 219–220 Copyright © 2004 Richard D Irwin Used with permission of McGraw-Hill Companies, Inc Online Learning Resources LearningStats, included on the Online Learning Center (OLC; www.mhhe.com/doane3e), provides a means for students to explore data and concepts at their own pace Applications that relate to the material in the chapter are identified by topic at the ends of chapters under Online Learning Resources Exam Review Questions At the end of a group of chapters, students can review the material they covered in those chapters This provides them with an opportunity to test themselves on their grasp of the material CHAPTER Online Learning Resources The Online Learning Center (OLC) at www.mhhe.com/doane3e has several LearningStats demonstrations to help you understand continuous probability distributions Your instructor may assign one or more of them, or you may decide to download the ones that sound interesting Topic LearningStats demonstrations Calculations Normal Areas Probability Calculator Normal approximations Evaluating Rules of Thumb Random data Random Continuous Data Visualizing Random Normal Data Tables Table C—Normal Probabilities Key: = Excel EXAM REVIEW QUESTIONS FOR CHAPTERS 5–7 Which type of probability (empirical, classical, subjective) is each of the following? a On a given Friday, the probability that Flight 277 to Chicago is on time is 23.7% b Your chance of going to Disney World next year is 10% c The chance of rolling a on two dice is 1/18 For the following contingency table, find (a) P(H ʝ T ); (b) P(S | G); (c) P(S) R S T G 10 50 30 Row Total 90 H 20 50 40 110 Col Total 30 100 70 200 If P(A) = 30, P(B) = 70, and P(A ʝ B) = 25 are A and B independent events? Explain ix www.ebook3000.com doa73699_index.qxd 826 11/20/09 5:56 PM Page 826 Index Redundancy, 186–188 applications of, 187–188 Space Shuttle case study, 186–187 Refrigerators, prices of, 567–568 Regional binary predictors, 566 Regional voting patterns, and predictors, 566–567, 574–575 Regions Financial, 167 Regression, 488–533, 545 best subsets, 576 bivariate, 494–496, 545 cockpit noise, applied to, 514–515 criteria for assessment, 549 and data, 529–530 estimated, 495 exam scores, applied to, 506–508 fitted, 548 fuel consumption, applied to, 498 and intercepts, 496–497 multiple, 545–549 and outliers, 528 prediction using, 496 problems with, 528–531 and residual tests, 518–524 retail sales, applied to, 508–509 on scatter plots, 497 simple regression defined, 495 and slopes, 496–497 standard error of the, 504, 558 stepwise, 576 variation explained by the, 511 Regression equation, 497 Regression lines, 497 Regression modeling, 549, 577–578 Rejection region, 350 Relative frequencies, 64 Relative frequency approach (see Empirical approach) Relative index, 628 Replacement, sampling with and without, 35–36 Replication, 464 two-factor ANOVA with, 464–470 two-factor ANOVA without, 456–464 Reports: appearance of, 804 presenting, 805–806 writing, 803–805 Research Industry Coalition, Inc., 47 Residual plots, heteroscedastic, 576–577 Residual tests, 518–524, 575 Residuals: body fat, applied to, 527–528 defined, 497 histogram of, 518, 519 standardized, 518, 519 in two-predictor regression, 551 unusual, 524–525, 579 Response error, 42, 43 Response variable, 439, 495, 547 Restaurant quality, Mann-Whitney test applied to, 692–694 Retail sales: interval estimate, applied to, 516–517 regression example for, 508–509 Return Exchange, 366–367 Reuters/Zogby poll, 324–325 Rex Stores, 690 RFID (radio frequency identification) tags, 366 Right-sided tests, 348–350 Right-skewed (positively skewed), 121 Right-skewed histograms, 71 Right-tail chi-square test, 645–647 Right-tailed test, 348, 350, 353–356, 370, 371, 689 Risk, 744 Robust design, 745 Rockwell, 167 Rockwell Collins, 39 Roderick, J A., 169 Roell, Stephen A., 39 Rohm & Haas Co., 60, 691 Romig, Harry G., 719 Roosevelt, Franklin D., 140 Roosevelt, Theodore, 140 Rose Bowl, 101 Rotated graphs, 98 Rounding, 25–26 Row/column data arrays, 37 Rubin, Donald B., 169 Rudestam, Kjell Erik, 21 Rule of Three, 325 Runs test, one-sample, 686–688 Russell 3000 Index, 36 Ryan, Thomas M., 39 Ryan, Thomas P., 591 S S (see Sample space; Seasonal) s (sample standard deviation), 131 S charts, 741 s2 (sample variance), 131 Saab, 63 Safety applications, two-sample tests for, 391 Sahei, Hardeo, 21 Sales data: deseasonalization of, 622–625 exponential smoothing of, 618–619 seasonal binaries in, 625–626 Sally Beauty Co., Inc., 690 Salmon, wild vs farm-bred, 53 Sample accuracy, 325 Sample correlation coefficient, 149–150, 490 Sample covariance, 150, 151 Sample mean(s), 299, 300, 303–304 Sample proportions, 299 Sample size, 375 determination for mean, 327–328 determination for proportions, 329–331 and difference of two means, 403 effect of, 376–377 and effectively infinite populations, 36 and standard error, 304–305 Sample space (S), 173–174 Sample standard deviation (s), 131, 299 Sample statistics, and test statistics, 393 Sample variance (s2), and dispersion, 131 Samples: census vs., 32–33 defined, 31–32 effect of larger, 371 errors in reading, 15 good, 314–315 large, 15, 362–363 non-normality, 373–374 nonrandom, 15 preliminary, 327, 330 doa73699_index.qxd 11/20/09 5:56 PM Page 827 Index prior, 330 size of, 42 small, 15, 373–374, 688 Sampling, 32–33 acceptance, 744 cluster, 40–41 computer methods for, 36–37 convenience, 41 focus groups, 42 judgment, 41 quota, 41 random vs nonrandom, 35 randomizing in Excel, 37 with replacement, 35 without replacement, 35, 238–242 scanner accuracy case example, 42 simple random, 35 single vs double, 744 size needed for, 42 stratified, 39 systematic, 37–38 using data arrays, 37 Sampling distributions, of an estimator, 297 Sampling error, 42–43, 60, 297–299 Sampling frames, 33 Sampling variation, 295–296 Sara Lee, 60, 691, 705, 711 Sarbanes-Oxley Act, 337 SARS, 53 SAT scores, 160, 385 Saturn, 63 Scales: arithmetic, 79 log, 79–80 logarithmic, 79 Scallops, 54 Scanner accuracy, sampling example about, 42 Scatter plots, 86–89, 489 in Excel, 88–89 MBA applicants, applied to, 491–492 and regressions in Excel, 497 regressions on, 497 Schaeffer, Richard L., 57 Scheffé, H., 434 Schenker, Nathan, 434 Schindler, Pamela S., 57 Schlumberger, 144, 145 Schmidt, Eric E., 39 Scholarships, for athletes, 209 Scientific inquiry, 343–344 Scion (automobile brand), 63 Scott, H Lee, Jr., 39 SE (standard error of the regression), 558 Sears, Roebuck, 42 Seasonal (S), 598–599 Seasonal binaries, 625, 627 Seasonal forecasts, using binary predictors, 625 Seasonality, 622–626 deseasonalization of data, 622–626 exponential smoothing with, 620 Seglin, Jeffrey L., 21 Selection bias, 42, 43 Sensitivity, to α, 360 Sensitivity of test, 346 Sentencing, of convicts, 568 “Serial exchangers,” 366 Shape, 59, 71, 113, 226, 227 Shenton, L R., 682 Sherringham, Philip R., 39 Sherwin-Williams Co., 690 Shewhart, Walter A., 719, 721 Shift variable, 561 Shingo, Shigeo, 721 Shivery, Charles W., 39 Shoemaker, Lewis F., 434 Sigma (σ): confidence intervals for, 333 confidence intervals for a mean with known σ, 307–310 confidence intervals for a mean with unknown σ, 310–317 estimating, 140, 327 σ2 (see Population variance) Significance: F test for, 551–552 level of, 345, 346, 375 practical importance vs., 16, 351, 357, 361 statistical, 392, 575 testing for, 561 Simple events, 174 Simple line charts, 77 Simple random sampling, 35 Simple regression (see Regression) Singh, Ravindra, 57 Single exponential smoothing, in MINITAB, 618–620 Single sampling, 744 Six Sigma, 723, 745–746 Size effect, 530 Skewed left, 121 Skewed population, 303–304 Skewed right, 121 Skewness, 154–155 and central tendency, 121 of histogram, 71 Skewness coefficient, 154–155 Sky (aircraft manufacturer), 167, 536 Slopes, 499–500 confidence intervals for, 504–505 in exponential trend calculations, 604 in linear trend calculations, 601 and regressions, 496–497 Small samples, 15, 688 Smartphone devices, 364 Smith, Gerald M., 756 Smith, Neil, 39 Smith International, 144, 145 Smithfield Foods, 711 Smoking: and gender (case study), 193–194, 202–203 vaccine for, 433 Smoothing constant (␣), 617 Smoothing models, 599 Socata (aircraft manufacturer), 167, 536 Software packages: binary dependent variables in, 576 contingency tables in, 651 for visual description, 61 Solutions, search for, 718 Som, R K., 57 Sony, 540 Sources of variation, 439, 465–466 SouthTrust Corp., 167 Southwest Airlines, 107, 342 Sovereign Bancorp, 39 827 doa73699_index.qxd 828 11/20/09 5:56 PM Page 828 Index Space launch, patterns in, 599 Space Shuttle, 14, 186–187 Spam e-mail, 52 SPC (statistical process control), 723 Spearman’s rank correlation test, 702–704 Spearman’s rho, 702 Special cause variation, 717, 723 Special law of addition, 181 Specification limits, upper and lower, 736 Spelling, 804 Spirit Airlines, 107 Sports drinks, potassium content of, 358 Sports drinks, sodium content of, 386 Spurious correlation, 530–531 SQC (see Statistical quality control) Squares, sum of, 490 SSE (unexplained variation errors, random error), 439, 502, 511–512 SSR (variation explained by the regression), 502, 511–512 SST (total variation around the mean), 502, 511–512 Stacked bar charts, 83, 84 Stacked data, 448n Stacked dot plots, 62–63 Standard & Poor’s 500 Index, 60 Standard deviation, 130–133 calculating, 132 characteristics of, 133 and discrete distributions, 219–220 in grouped data, 153–154 true, 375 two-sum formula for calculating, 132 Standard error: as a measure of fit, 612–614 of a regression, 504 and sample size, 304–305 true, 525 Standard error of the mean, 300, 304–305, 728 Standard error of the proportion, 321, 366 Standard error of the regression (SE), 558 Standard normal distributions, 262–274 Standard normal distributions, cumulative, 764–765 Standardized data, 137–141 Chebyshev’s Theorem, 137 defining variables with, 139 Empirical Rule, 137, 138 estimating sigma using, 140 and outliers, 138–140 unusual observations in, 138 Standardized residuals, 518, 519 in Excel, 524, 525 in MegaStat, 525 in MINITAB, 525 Standardized variables, 139, 268 Staples Inc., 60 Starwood Hotels, 60 State and Metropolitan Area Data Book, 45 State and Metropolitan Area Data Book, 44 State Farm, 186 State St Corp., 167 Statistical Abstract of the United States, 44, 45 Statistical estimation, 296 Statistical generalization, 16 Statistical hypothesis tests, 348 Statistical process control (SPC), 723, 746 Statistical quality control (SQC), 722, 723 Statisticians: and quality control, 717 traits of, 10 Statistics (data), 3, 33, 113 leverage, 526–527 t statistic, 491 Statistics (discipline), challenges in, 10 descriptive, inferential, pitfalls of, 14–16 Statistics Canada (Web site), 45 Stefanski, Leonard A., 177n Stephens, Michael A., 682 Stepwise regression, 576 Stochastic process, 208 Stock prices, MINITAB sample of, 156–157 Stocks, 596 Stratified sampling, 39 Stryker Corporation, 60 Student work, 652–653 Student’s t, for unknown population variance, 359–361 Student’s t distribution, 310–314, 316–317 Sturges’ Rule, 65, 667 Subaru, 63, 364 Subgroup size, 724 Subjective approach (probability), 178–179 Sum of squares, 490 Sum of squares, partitioned, 444 Sum of squares error, 502 Sums of random variables: and discrete distributions, 245 gasoline application of, 245 project scheduling application of, 245–246 SunTrust Banks, 167 Supply-chain management, 744–745 Surveys, 45–50 coding, 48 on colleges, role of, 50 of customer satisfaction, 325 and data file formats, 49 data screening, 48 Likert scales, 50 poor quality methods, 15 questionnaire design for, 47 response rates for, 46 sources of error in, 42–43 types of, 46 wording of, 47–48 Sutter Home, 536, 703 Suzuki, 63 Swordfish, 54 Symantec Corp., 60 Symmetric data, 120 Symmetric histograms, 71 Symmetric triangular distributions, 284 Syms Corporation, 690 Synovus Financial Corp., 167 Systematic error, 298 Systematic population, 302–303 Systematic sampling, 37–38 T T (see Trend) t distribution (see Student’s t distribution) t statistic, 491 t test, paired, 404, 408–409 doa73699_index.qxd 11/20/09 5:56 PM Page 829 Index Tables, 92–94 contingency, 192–193 presenting, 13 in technical reports, 804, 805 3-way, 651 Tabulated data, for Poisson goodness-of-fit-tests, 661–662 Taft, William, 140 Taguchi, Genichi, 719, 721, 745 Taguchi method, 723 Talbots, Inc., 690 Target (store), 42, 689 Target population, 33 Tax returns, estimated time for preparing, 53 Taylor, Zachary, 140 TD Ameritrade Holding, 39 Technical literacy, studying statistics and, Technical reports, 804–805 Telefund calling, 243 Telemarketers, predictive dialing by, 385 Telephone surveys, 45, 46 Television ratings, 34 Test preparation companies, 385 Test statistic, calculating, 350, 357 Tests for one variance, 381–382 Thode, Henry C., Jr., 693 Thompson, Steven K., 57 Three P’s, 806 3-D bar charts, 82–83 3-D graphs, 98 3-D pie charts, 95, 96 3-way tables, 651 Tiffany & Co., 690 Tiger (aircraft manufacturer), 167, 536 Time, 100 Time series: for growth rates, 127 in software packages, 633 Time-series analysis, 594–631 Time-series data, 30–31, 595–596 bar charts for, 84 in line charts, 80 Time-series decomposition, 597 Time-series graphs, 595–596, 598–599 Time-series variable, 595 TINV (Excel function), 316 TMA (trailing moving average), 614–616 TMR Inc., 50 Tootsie Roll, 270, 335 Total cost, 245 Total quality control, 721 Total quality management (TQM), 721, 744 Total variation around the mean (see SST) Totals, problem of, 530 Touchstar, 385 Toyota, 6, 63, 90, 329, 401, 536, 585 TQM (see Total quality management) Trade, arithmetic scale example in, 80 Traffic fatalities, uniform GOF test of, 656–657 Trailing moving average (TMA), 614–616 Transformations, linear, 244 Transformations of random variables, 244–246 Transocean, 144, 145 Transplants, graph of trends for, 599 Transportation, U.S Department of, 342 TransUnion, 186 Treatment, 439, 456 Tree diagrams, 196, 197 Trend (T), 597–599, 620 Trend fitting: criteria for, 608–609 in Excel, 607, 609 and U.S trade deficit, 611–612 Trend forecasting, 599–609 Trend line, 88, 89 Trend models, 600 Trend pattern (control charts), 734–736 Trendless data, 614 Trendline, in Excel, 497n Triangular distributions, 282–285 characteristics of, 282–284 symmetric, 284 uses of, 285 Tribune Company, 60 Tri-Cities Tobacco Coalition, 372 Trimmed mean, and central tendency, 116, 127–128 Troppo Malo, 290 True mean, 375 True standard deviation, 375 True standard error, 525 Truman, Harry, 140 Tufte, Edward R., 109 Tukey, John Wilder, 450 Tukey t tests, 451–452 Tukey’s studentized range tests, 450–451 decision rule for, 450–451 of pairs and means, 470 Turbine data, stepwise regression of, 576 Two events: intersection of, 180 union of, 179–180 Two means, comparing, 393–401 hypotheses for, 393 and independent samples, 393–401 issues of, 399 with known variances, 394 and paired samples, 404–409 with unknown variances, 394 Two means, difference of, 401–403 2k models, 476 Two-factor ANOVA with replication, 464–470 hypothesis for, 464–465 sources of variation in, 465–466 and turbine engine thrust example, 472–473 using MegaStat for, 468 Two-factor ANOVA without replication, 456–464 calculation of nonreplicated, 458–460 limitations of, 461–462 using MegaStat, 460–461 Two-factor model, 457 Two-predictor regression, 551 Two-sample hypothesis tests (see Hypothesis tests, two-sample) Two-scale line charts, 77 Two-sided tests, 348–350 Two-sum formula, for standard deviations, 132 Two-tailed p-values, and software packages, 554 Two-tailed tests, 348, 350, 356–357 in bivariate regression, 512 for means with known population variance, 356–357 for power curves and OC curves, 380 p-value method in, 369, 414 in Wilcoxan signed-rank test, 689 for a zero difference, 406–407 829 doa73699_index.qxd 830 11/20/09 5:56 PM Page 830 Index 2004 election, 324–325 Type I error(s) (α), 344, 346, 392–393, 734 choosing, 347 effect of varying, 369–370 in one-sample hypothesis tests, 344, 345–353 probability of, 345, 346 sensitivity to, 360 and Type II error, 352–353 Type II error(s) (β), 344, 347, 392–393, 734 consequences of, 347 in one-sample hypothesis tests, 344 and power of curve, 374–379 and Type I error, 352–353 U UCL (upper control limit), 725 Unbiased estimator, 298 Unconditional probability (prior probability), 198 Unconscious bias, 16 Unexplained variation errors (see SSE) Uniform continuous distributions, 257–259 Uniform discrete distributions, 221–224 copier codes application of, 224 gas pumping application of, 223 lotteries application of, 223 Uniform distributions, 302–303, 656 Uniform goodness-of-fit tests, 654–659 for grouped data, 656–657 for raw data, 657–659 Uniform model, 259–260 Uniform population: all possible samples from, 305–306 assuming, 327 Uniform random integers, 223 Unimodal classes, 71 Uninsured patients, 225 Union, of two events, 179–180 Union Pacific, 39, 60 Union Planters Corp., 167 Unit rectangular distribution, 259 United Airlines, 107 customer service, 342 dot plot example using, 148–149 United Auto Group, 680 United Nations Department of Economic and Social Affairs, 45 U.S Bureau of the Census, 45 U.S Centers for Disease Control and Prevention, 17, 54 U.S Congress, 32 U.S Customs Service, 338 U.S Energy Information Administration, 34 U.S Federal Statistics (Web site), 45 U.S Fisheries and Wildlife Service, 54 U.S Food and Drug Administration (FDA), 18, 45, 53, 54 U.S News & World Report, 100 U.S presidents, standardized data example using, 140–141 U.S trade deficit, and trend fitting, 611–612 Units of measure, 328, 331 Univariate data sets, 24, 59 Unknown population variances, 359–363 hypothesis tests of mean with, 359–363 p-value method for, 361 Student’s t method for, 359–361 using MegaStat to find, 362 Unstacked data, 448n Unusual data, 138, 146–147 Unusual leverage, 580 Unusual observations, 579–580 Unusual residuals, 524–525, 579 Upper control limit (UCL), 725 Upper specification limit (USL), 736–739 US Airways, 107 US Airways Flight 1549, 609 USA Today, 100 USL (upper specification limit), 736–739 Utts, Jessica, 16, 21 V Vail Resorts, Inc.: correlation matrix for, 152–153 employee age, 433 employee pay, 272, 351, 364 employee rehiring, 427 employee seniority, 401 guest age, 310 ISO certification, 747 Likert scales for, 28, 30 market research surveys, 85 regression modeling, 550, 557 satisfaction surveys, 338, 401, 550, 557, 575 use of statistics by, 4–5 Vallee, Roy, 39 Van Belle, Gerald, 339 Van Buren, Martin, 140 Vanguard, Varco International, 144, 145 Vardeman, Stephen B., 21 Variable(s): binary, 25 continuous, 255 continuous random, 256–257 defined, 23 dependent, 495 discrete, 255 dummy, 561 independent, 495, 644 indicator, 561 random, 208, 215–216, 261–262 response, 439, 495, 547 shift, 561 standardized, 139 time-series, 595 Variable control charts, 724 Variable data, 724 Variable selection, 548 Variable transform, 531 Variance(s): analysis of, 511–512 Bay Street Inn example of, 219–220 comparing two, 417–424 comparing two means with known variances, 394 comparing two means with unknown variances, 394 comparison of, in one-tailed test, 421–422 comparison of, in two-tailed test, 419–421 of continuous random variable, 256–257 critical values for two, 418 decomposition of, 511 and discrete distributions, 219–220 F test for two, 417–418 folded F test for two, 421 homogeneity of, 452–454 hypothesis tests for one, 381–382 doa73699_index.qxd 11/20/09 5:56 PM Page 831 Index hypothesis tests for two, 417 nonconstant, 519 population, 130–131, 331–332 sample, 131 tests for one, 381–382 two, applied to collision damage, 418–419 Variance inflation, 571 Variance inflation factor (VIF), 572–573 Variation around the mean, total (SST), 502, 511–512 Variation explained by the regression (SSR), 502, 511–512 Variations: among observations, 31 coefficient of, 133 common cause, 717, 723 reduced, 716–717 special cause, 717, 723 unexplained error (SSE), 439, 502, 511–512 zero, 716–717 Vellemen, Paul F., 57 Venn diagram, 174 Verbal anchors, for interval data, 28 Verify-1 (software), 366 Verizon, 372–373 Video preference, viewer age and, 194 VIF (see Variance inflation factor) Visual data, 59–100 bar charts, 82–84 dot plots, 61–64 frequency distributions, 67–69 histograms, 66–71 line charts, 77–80 measuring, 60 pie charts, 95–96 pivot tables, 92–94 scatter plots, 86–89 sorting, 60–61 stacked dot plots, 62–63 tables, 92–94 Visual displays, 61, 489, 490 Visual Statistics (software program): Anderson-Darling test in, 672 equal expected frequencies in, 667, 668 Poisson GOF test in, 664 Volkswagen, 63, 536, 585 Volunteers, effect on sampling, 34 Volvo, 63, 90, 536, 585 Voting patterns, regional, 566–567 W The W Edwards Deming Institute, 720 Wachovia Corp., 167 Wainscott, James L., 39 Wald, Abraham, 686 Wald-Wolfowitz test (one-sample runs test), 686–688 The Wall Street Journal, 5, 6, 100, 119, 148, 179 The Wall Street Journal Index, 45 Wallis, W Allen, 695 Walmart, 6, 39, 401, 563, 564 Wang, Y., 434 Warehousing, 246 Washington, George, 140 Web surveys, 46 Weight Watchers, 712 Weighted index, 629 Welch’s adjusted degrees of freedom, 394 Welch’s formula, 401–402 Welch-Satterthwaite test, 394 Well-conditioned data, 529–530 Wells Fargo, 167 Wendy’s International Incorporated, 60, 691 Wheelwright, Steven C., 641 Whirlpool Corporation, 60, 691 Whitaker, D., 20 Whitney, D R., 692 Wichern, Dean W., 641 Wilcoxan, Frank, 689 Wilcoxan rank-sum test (Mann-Whitney test), 692–694 Wilcoxan signed-rank test, 689–691 Wilkinson, Leland, 57 William of Occam, 608 William Wrigley, Jr., 705 Williamson, Bruce A., 39 Williams-Sonoma, 680 Willoughby, Floyd G., 695 Wilson, J Holton, 641 Wilson, Woodrow, 140 Wireless routers, encryption for, 428 Wirthlin Worldwide, 53 Wm Wrigley, Jr (company), 711 Wolfowitz, Jacob, 686 World Bank, 45 World Demographics (Web site), 45 World Health Organization, 45 Written communication skills, 803–805 executive summaries, 805 technical reports, 804–805 Wynn, Stephen A., 39 Wynn Resorts, 39 X x charts, 724, 725, 733 Y Y: confidence intervals for, 558–559 interval estimate for, 515–516 Yahoo, 108 Young, James R., 39 Yum! Brands Inc., 60, 691 Z z: in sample size, 328 Student’s t distribution vs., 316 Zale, 680, 690 Zelazny, Gene, 109 Zero, meaningful, 29 Zero correlation, zero slope and, 506 Zero difference: testing for, 409 two-tailed test for a, 406–407 Zero slope: in bivariate regression, 512 test for, 506 Zero variation, 716–717 Zions Bancorp, 167 Zocor, 395–398 Zone charts, 742 z-scores, 139 z-values, 367 831 doa73699_index.qxd 11/20/09 5:56 PM Page 832 doa73699_index.qxd 11/20/09 5:56 PM Page 833 doa73699_index.qxd 11/20/09 5:56 PM Page 834 doa73699_index.qxd 11/20/09 5:56 PM Page 835 doa73699_index.qxd 11/20/09 5:56 PM Page 836 doa73699_index.qxd 11/20/09 5:56 PM Page 837 doa73699_Endpaper_spread 11/14/09 8:53 PM Page STANDARD NORMAL AREAS This table shows the normal area between and z Example: P(0 < z < 1.96) = 4750 z z 00 01 02 03 04 05 06 07 08 09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 0000 0398 0793 1179 1554 1915 2257 2580 2881 3159 3413 3643 3849 4032 4192 4332 4452 4554 4641 4713 4772 4821 4861 4893 4918 4938 4953 4965 4974 4981 49865 49903 49931 49952 49966 49977 49984 49989 0040 0438 0832 1217 1591 1950 2291 2611 2910 3186 3438 3665 3869 4049 4207 4345 4463 4564 4649 4719 4778 4826 4864 4896 4920 4940 4955 4966 4975 4982 49869 49906 49934 49953 49968 49978 49985 49990 0080 0478 0871 1255 1628 1985 2324 2642 2939 3212 3461 3686 3888 4066 4222 4357 4474 4573 4656 4726 4783 4830 4868 4898 4922 4941 4956 4967 4976 4982 49874 49910 49936 49955 49969 49978 49985 49990 0120 0517 0910 1293 1664 2019 2357 2673 2967 3238 3485 3708 3907 4082 4236 4370 4484 4582 4664 4732 4788 4834 4871 4901 4925 4943 4957 4968 4977 4983 49878 49913 49938 49957 49970 49979 49986 49990 0160 0557 0948 1331 1700 2054 2389 2704 2995 3264 3508 3729 3925 4099 4251 4382 4495 4591 4671 4738 4793 4838 4875 4904 4927 4945 4959 4969 4977 4984 49882 49916 49940 49958 49971 49980 49986 49991 0199 0596 0987 1368 1736 2088 2422 2734 3023 3289 3531 3749 3944 4115 4265 4394 4505 4599 4678 4744 4798 4842 4878 4906 4929 4946 4960 4970 4978 4984 49886 49918 49942 49960 49972 49981 49987 49991 0239 0636 1026 1406 1772 2123 2454 2764 3051 3315 3554 3770 3962 4131 4279 4406 4515 4608 4686 4750 4803 4846 4881 4909 4931 4948 4961 4971 4979 4985 49889 49921 49944 49961 49973 49981 49987 49992 0279 0675 1064 1443 1808 2157 2486 2794 3078 3340 3577 3790 3980 4147 4292 4418 4525 4616 4693 4756 4808 4850 4884 4911 4932 4949 4962 4972 4979 4985 49893 49924 49946 49962 49974 49982 49988 49992 0319 0714 1103 1480 1844 2190 2517 2823 3106 3365 3599 3810 3997 4162 4306 4429 4535 4625 4699 4761 4812 4854 4887 4913 4934 4951 4963 4973 4980 4986 49896 49926 49948 49964 49975 49983 49988 49992 0359 0753 1141 1517 1879 2224 2549 2852 3133 3389 3621 3830 4015 4177 4319 4441 4545 4633 4706 4767 4817 4857 4890 4916 4936 4952 4964 4974 4981 4986 49900 49929 49950 49965 49976 49983 49989 49992 doa73699_Endpaper_spread 11/14/09 8:53 PM Page CUMULATIVE STANDARD NORMAL DISTRIBUTION This table shows the normal area less than z Example: P(z < −1.96) = 0250 This table shows the normal area less than z Example: P(z < 1.96) = 9750 z z z 00 01 02 03 04 05 06 07 08 09 z 00 01 02 03 04 05 06 07 08 09 −3.7 −3.6 −3.5 −3.4 −3.3 −3.2 −3.1 −3.0 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2.0 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3 −1.2 −1.1 −1.0 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 −0.0 00011 00016 00023 00034 00048 00069 00097 00135 0019 0026 0035 0047 0062 0082 0107 0139 0179 0228 0287 0359 0446 0548 0668 0808 0968 1151 1357 1587 1841 2119 2420 2743 3085 3446 3821 4207 4602 5000 00010 00015 00022 00032 00047 00066 00094 00131 0018 0025 0034 0045 0060 0080 0104 0136 0174 0222 0281 0351 0436 0537 0655 0793 0951 1131 1335 1562 1814 2090 2389 2709 3050 3409 3783 4168 4562 4960 00010 00015 00022 00031 00045 00064 00090 00126 0018 0024 0033 0044 0059 0078 0102 0132 0170 0217 0274 0344 0427 0526 0643 0778 0934 1112 1314 1539 1788 2061 2358 2676 3015 3372 3745 4129 4522 4920 00010 00014 00021 00030 00043 00062 00087 00122 0017 0023 0032 0043 0057 0075 0099 0129 0166 0212 0268 0336 0418 0516 0630 0764 0918 1093 1292 1515 1762 2033 2327 2643 2981 3336 3707 4090 4483 4880 00009 00014 00020 00029 00042 00060 00084 00118 0016 0023 0031 0041 0055 0073 0096 0125 0162 0207 0262 0329 0409 0505 0618 0749 0901 1075 1271 1492 1736 2005 2296 2611 2946 3300 3669 4052 4443 4841 00009 00013 00019 00028 00040 00058 00082 00114 0016 0022 0030 0040 0054 0071 0094 0122 0158 0202 0256 0322 0401 0495 0606 0735 0885 1056 1251 1469 1711 1977 2266 2578 2912 3264 3632 4013 4404 4801 00008 00013 00019 00027 00039 00056 00079 00111 0015 0021 0029 0039 0052 0069 0091 0119 0154 0197 0250 0314 0392 0485 0594 0721 0869 1038 1230 1446 1685 1949 2236 2546 2877 3228 3594 3974 4364 4761 00008 00012 00018 00026 00038 00054 00076 00107 0015 0021 0028 0038 0051 0068 0089 0116 0150 0192 0244 0307 0384 0475 0582 0708 0853 1020 1210 1423 1660 1922 2206 2514 2843 3192 3557 3936 4325 4721 00008 00012 00017 00025 00036 00052 00074 00104 0014 0020 0027 0037 0049 0066 0087 0113 0146 0188 0239 0301 0375 0465 0571 0694 0838 1003 1190 1401 1635 1894 2177 2483 2810 3156 3520 3897 4286 4681 00008 00011 00017 00024 00035 00050 00071 00100 0014 0019 0026 0036 0048 0064 0084 0110 0143 0183 0233 0294 0367 0455 0559 0681 0823 0985 1170 1379 1611 1867 2148 2451 2776 3121 3483 3859 4247 4641 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 5000 5398 5793 6179 6554 6915 7257 7580 7881 8159 8413 8643 8849 9032 9192 9332 9452 9554 9641 9713 9772 9821 9861 9893 9918 9938 9953 9965 9974 9981 99865 99903 99931 99952 99966 99977 99984 99989 5040 5438 5832 6217 6591 6950 7291 7611 7910 8186 8438 8665 8869 9049 9207 9345 9463 9564 9649 9719 9778 9826 9864 9896 9920 9940 9955 9966 9975 9982 99869 99906 99934 99953 99968 99978 99985 99990 5080 5478 5871 6255 6628 6985 7324 7642 7939 8212 8461 8686 8888 9066 9222 9357 9474 9573 9656 9726 9783 9830 9868 9898 9922 9941 9956 9967 9976 9982 99874 99910 99936 99955 99969 99978 99985 99990 5120 5517 5910 6293 6664 7019 7357 7673 7967 8238 8485 8708 8907 9082 9236 9370 9484 9582 9664 9732 9788 9834 9871 9901 9925 9943 9957 9968 9977 9983 99878 99913 99938 99957 99970 99979 99986 99990 5160 5557 5948 6331 6700 7054 7389 7704 7995 8264 8508 8729 8925 9099 9251 9382 9495 9591 9671 9738 9793 9838 9875 9904 9927 9945 9959 9969 9977 9984 99882 99916 99940 99958 99971 99980 99986 99991 5199 5596 5987 6368 6736 7088 7422 7734 8023 8289 8531 8749 8944 9115 9265 9394 9505 9599 9678 9744 9798 9842 9878 9906 9929 9946 9960 9970 9978 9984 99886 99918 99942 99960 99972 99981 99987 99991 5239 5636 6026 6406 6772 7123 7454 7764 8051 8315 8554 8770 8962 9131 9279 9406 9515 9608 9686 9750 9803 9846 9881 9909 9931 9948 9961 9971 9979 9985 99889 99921 99944 99961 99973 99981 99987 99992 5279 5675 6064 6443 6808 7157 7486 7794 8078 8340 8577 8790 8980 9147 9292 9418 9525 9616 9693 9756 9808 9850 9884 9911 9932 9949 9962 9972 9979 9985 99893 99924 99946 99962 99974 99982 99988 99992 5319 5714 6103 6480 6844 7190 7517 7823 8106 8365 8599 8810 8997 9162 9306 9429 9535 9625 9699 9761 9812 9854 9887 9913 9934 9951 9963 9973 9980 9986 99896 99926 99948 99964 99975 99983 99988 99992 5359 5753 6141 6517 6879 7224 7549 7852 8133 8389 8621 8830 9015 9177 9319 9441 9545 9633 9706 9767 9817 9857 9890 9916 9936 9952 9964 9974 9981 9986 99900 99929 99950 99965 99976 99983 99989 99992 doa73699_Endpaper_1.qxd 11/20/09 6:38 PM Page STUDENT’S t CRITICAL VALUES This table shows the t-value that defines the area for the stated degrees of freedom (d.f.) .80 d.f Confidence Level 90 95 98 99 80 20 Significance Level for Two-Tailed Test 10 05 02 01 10 Significance Level for One-Tailed Test 05 025 01 005 Confidence Level 90 95 t 98 99 20 Significance Level for Two-Tailed Test 10 05 02 01 d.f .10 Significance Level for One-Tailed Test 05 025 01 005 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 63.656 9.925 5.841 4.604 4.032 36 37 38 39 40 1.306 1.305 1.304 1.304 1.303 1.688 1.687 1.686 1.685 1.684 2.028 2.026 2.024 2.023 2.021 2.434 2.431 2.429 2.426 2.423 2.719 2.715 2.712 2.708 2.704 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 41 42 43 44 45 1.303 1.302 1.302 1.301 1.301 1.683 1.682 1.681 1.680 1.679 2.020 2.018 2.017 2.015 2.014 2.421 2.418 2.416 2.414 2.412 2.701 2.698 2.695 2.692 2.690 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 46 47 48 49 50 1.300 1.300 1.299 1.299 1.299 1.679 1.678 1.677 1.677 1.676 2.013 2.012 2.011 2.010 2.009 2.410 2.408 2.407 2.405 2.403 2.687 2.685 2.682 2.680 2.678 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 55 60 65 70 75 1.297 1.296 1.295 1.294 1.293 1.673 1.671 1.669 1.667 1.665 2.004 2.000 1.997 1.994 1.992 2.396 2.390 2.385 2.381 2.377 2.668 2.660 2.654 2.648 2.643 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 80 85 90 95 100 1.292 1.292 1.291 1.291 1.290 1.664 1.663 1.662 1.661 1.660 1.990 1.988 1.987 1.985 1.984 2.374 2.371 2.368 2.366 2.364 2.639 2.635 2.632 2.629 2.626 26 27 28 29 30 1.315 1.314 1.313 1.311 1.310 1.706 1.703 1.701 1.699 1.697 2.056 2.052 2.048 2.045 2.042 2.479 2.473 2.467 2.462 2.457 2.779 2.771 2.763 2.756 2.750 110 120 130 140 150 1.289 1.289 1.288 1.288 1.287 1.659 1.658 1.657 1.656 1.655 1.982 1.980 1.978 1.977 1.976 2.361 2.358 2.355 2.353 2.351 2.621 2.617 2.614 2.611 2.609 31 32 33 34 35 1.309 1.309 1.308 1.307 1.306 1.696 1.694 1.692 1.691 1.690 2.040 2.037 2.035 2.032 2.030 2.453 2.449 2.445 2.441 2.438 2.744 2.738 2.733 2.728 2.724 ∞ 1.282 1.645 1.960 2.326 2.576 Note: As n increases, critical values of Student’s t approach the z-values in the last line of this table A common rule of thumb is to use z when n > 30, but that is not conservative