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Utama The Basic Practice of Statistics: w/Student CD
The Basic Practice of Statistics: w/Student CDDavid S. Moore, William I. Notz, Michael A. Fligner
The Basic Practice of Statistics: w/Student CD
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FM.indd Page xxv 11/9/11 3:58:39 PM user-s163 user-F452 FM.indd Page i 11/10/11 3:45:17 PM user-s163 user-F452 The Basic Practice of Statistics SIXTH EDITION D AV I D S . M O O R E Purdue University WILLIAM I. NOTZ The Ohio State University MICHAEL A. FLIGNER The Ohio State University W. H. Freeman and Company New York FM.indd Page ii 11/9/11 3:58:32 PM user-s163 Publisher: Ruth Baruth Acquisitions Editor: Karen Carson Executive Marketing Manager: Jennifer Somerville Developmental Editors: Andrew Sylvester and Leslie Lahr Senior Media Acquisitions Editor: Roland Cheyney Senior Media Editor: Laura Capuano Associate Editor: Katrina Wilhelm Assistant Media Editor: Catriona Kaplan Editorial Assistant: Tyler Holzer Photo Editor: Cecilia Varas Photo Researcher: Elyse Rieder Cover and Text Designer: Blake Logan Senior Project Editor: Mary Louise Byrd Illustrations: Macmillan Solutions Production Coordinator: Susan Wein Composition: Aptara®, Inc. Printing and Binding: Quad Graphics Library of Congress Control Number: 2011934674 Student Edition (Hardcover w/cd) Student Edition (Paperback w/cd) Student Edition (Looseleaf w/cd) ISBN-13: 978-1-4641-0254-7 ISBN-13: 978-1-4641-0434-3 ISBN-13: 978-1-4641-0433-6 ISBN-10: 1-4641-0254-6 ISBN-10: 1-4641-0434-4 ISBN-10: 1-4641-0433-6 © 2013, 2010, 2007, 2004 by W. H. Freeman and Company All rights reserved Printed in the United States of America First printing W. H. Freeman and Company 41 Madison Avenue New York, NY 10010 Houndmills, Basingstoke RG21 6XS, England www.whfreeman.com user-F452 FM.indd Page iii 11/18/11 11:54:13 PM user-s163 user-F452 Brief Contents Pa r t I 1 Exploring Data Exploring Data: Variables and Distributions CHAPTER 1 Picturing Distributions with Graphs 3 CHAPTER 2 Pa r t I I I Describing Distributions with Numbers 39 Quantitative Response Variable Inference about a Population Mean 437 Two-Sample Problems 465 Categorical Response Variable CHAPTER 20 Inference about a Population Proportion 493 The Normal Distributions 69 Exploring Data: Relationships CHAPTER 4 Scatterplots and Correlation 97 CHAPTER 19 CHAPTER 5 Regression 125 CHAPTER 6 Two-Way Tables* CHAPTER 7 Exploring Data: Part I Review Pa r t I I From Exploration to Inference 197 159 175 CHAPTER 21 Comparing Two Proportions CHAPTER 22 Inference about Variables: Part III Review 533 Pa r t I V Inference about Relationships Producing Data Producing Data: Sampling 199 Producing Data: Experiments Commentary: Data Ethics* Probability and Sampling Distributions 246 CHAPTER 10 Introducing Probability 259 CHAPTER 9 435 CHAPTER 18 CHAPTER 3 CHAPTER 8 Inference about Variables Sampling Distributions 285 CHAPTER 12 General Rules of Probability* 307 Binomial Distributions* 331 Foundations of Inference CHAPTER 14 Confidence Intervals: The Basics 351 CHAPTER 15 Tests of Significance: The Basics 369 CHAPTER 16 Inference in Practice CHAPTER 17 From Exploration to Inference: Part II Review 417 551 CHAPTER 23 Two Categorical Variables: The Chi-Square Test 553 CHAPTER 24 Inference for Regression CHAPTER 25 One-Way Analysis of Variance: Comparing Several Means 623 Pa r t V Optional Companion Chapters 223 CHAPTER 11 515 CHAPTER 13 587 (available on the BPS CD and online) 391 CHAPTER 26 Nonparametrics Tests 26-3 CHAPTER 27 Statistical Process Control CHAPTER 28 Multiple Regression* CHAPTER 29 More about Analysis of Variance 29-3 27-3 28-3 *Starred material is not required for later parts of the text. iii FM.indd Page iv 11/9/11 3:58:32 PM user-s163 user-F452 Detailed Table of Contents To the Instructor viii Media and Supplements xix About the Authors xxiv To the Student xxvi Pa r t I CHAPTER 4 Scatterplots and Correlation 1 Exploring Data CHAPTER 1 Picturing Distributions with Graphs 3 Individuals and variables 3 Categorical variables: pie charts and bar graphs Quantitative variables: histograms 11 Interpreting histograms 15 Quantitative variables: stemplots 20 Time plots 23 6 Measuring center: the mean 40 Measuring center: the median 41 Comparing the mean and the median 42 Measuring spread: the quartiles 43 The five-number summary and boxplots 45 Spotting suspected outliers* 48 Measuring spread: the standard deviation 49 Choosing measures of center and spread 51 Using technology 53 Organizing a statistical problem 55 Regression lines 125 The least-squares regression line 128 Using technology 130 Facts about least-squares regression 132 Residuals 135 Influential observations 139 Cautions about correlation and regression 142 Association does not imply causation 144 CHAPTER 6 Two-Way Tables* 159 Marginal distributions 160 Conditional distributions 162 Simpson's paradox 166 CHAPTER 7 Exploring Data: Part I Review Part I summary 177 Test yourself 180 Supplementary exercises 175 191 69 Density curves 69 Describing density curves 73 Normal distributions 75 The 68–95–99.7 rule 77 The standard Normal distribution 80 Finding Normal proportions 81 Using the standard Normal table 83 Finding a value given a proportion 86 *Starred material is not required for later parts of the text. iv Explanatory and response variables 97 Displaying relationships: scatterplots 99 Interpreting scatterplots 101 Adding categorical variables to scatterplots 104 Measuring linear association: correlation 106 Facts about correlation 108 CHAPTER 5 Regression 125 CHAPTER 2 Describing Distributions with Numbers 39 CHAPTER 3 The Normal Distributions 97 Pa r t I I From Exploration to Inference CHAPTER 8 Producing Data: Sampling 199 Population versus sample 199 How to sample badly 202 Simple random samples 203 197 FM.indd Page v 11/9/11 3:58:32 PM user-s163 user-F452 • Inference about the population 208 Other sampling designs 209 Cautions about sample surveys 210 The impact of technology 213 CHAPTER 9 Producing Data: Experiments CHAPTER 13 Binomial Distributions* 232 351 The reasoning of tests of significance 370 Stating hypotheses 372 P-value and statistical significance 374 Tests for a population mean 378 Significance from a table* 382 253 CHAPTER 16 Inference in Practice 391 Conditions for inference in practice 392 Cautions about confidence intervals 395 Cautions about significance tests 397 Planning studies: sample size for confidence intervals 401 Planning studies: the power of a statistical test* 402 268 CHAPTER 11 Sampling Distributions 285 Parameters and statistics 285 Statistical estimation and the law of large numbers Sampling distributions 290 _ The sampling distribution of x 293 The central limit theorem 295 CHAPTER 12 General Rules of Probability* CHAPTER 14 Confidence Intervals: The Basics CHAPTER 15 Tests of Significance: The Basics 369 259 The idea of probability 260 The search for randomness* 262 Probability models 264 Probability rules 266 Finite and discrete probability models Continuous probability models 271 Random variables 275 Personal probability* 276 331 The reasoning of statistical estimation 352 Margin of error and confidence level 354 Confidence intervals for a population mean 357 How confidence intervals behave 361 Commentary: Data Ethics* 246 Institutional review boards 248 Informed consent 248 Confidentiality 250 Clinical trials 252 Behavioral and social science experiments v The binomial setting and binomial distributions 331 Binomial distributions in statistical sampling 333 Binomial probabilities 334 Using technology 336 Binomial mean and standard deviation 338 The Normal approximation to binomial distributions 340 223 Observation versus experiment 223 Subjects, factors, treatments 225 How to experiment badly 228 Randomized comparative experiments 229 The logic of randomized comparative experiments Cautions about experimentation 234 Matched pairs and other block designs 236 CHAPTER 10 Introducing Probability D E T A I L E D T A BL E O F CON TE N TS 287 Part II summary 419 Test yourself 423 Supplementary exercises Pa r t I I I 307 Independence and the multiplication rule The general addition rule 312 Conditional probability 314 The general multiplication rule 316 Independence again 318 Tree diagrams 318 CHAPTER 17 From Exploration to Inference: Part II Review 431 Inference about 435 Variables 308 417 CHAPTER 18 Inference about a Population Mean 437 Conditions for inference about a mean 437 The t distributions 438 The one-sample t confidence interval 440 FM.indd Page vi 11/18/11 11:53:50 PM user-s163 vi user-F452 DETA ILED TA B LE O F CO N T E N T S The one-sample t test 443 Using technology 446 Matched pairs t procedures 449 Robustness of t procedures 452 The chi-square test statistic 560 Cell counts required for the chi-square test 561 Using technology 562 Uses of the chi-square test 567 The chi-square distributions 570 The chi-square test for goodness of fit* 572 CHAPTER 19 Two-Sample Problems 465 CHAPTER 24 Inference for Regression Two-sample problems 465 Comparing two population means 466 Two-sample t procedures 469 Using technology 474 Robustness again 477 Details of the t approximation* 480 Avoid the pooled two-sample t procedures* 481 Avoid inference about standard deviations* 482 CHAPTER 25 One-Way Analysis of Variance: Comparing Several Means 623 The sample proportion p̂ 494 Large-sample confidence intervals for a proportion 496 Accurate confidence intervals for a proportion 499 Choosing the sample size 502 Significance tests for a proportion 504 Comparing several means 625 The analysis of variance F test 625 Using technology 628 The idea of analysis of variance 631 Conditions for ANOVA 633 F distributions and degrees of freedom Some details of ANOVA* 640 515 Two-sample problems: proportions 515 The sampling distribution of a difference between proportions 516 Large-sample confidence intervals for comparing proportions 517 Using technology 518 Accurate confidence intervals for comparing proportions Significance tests for comparing proportions 522 Notes and Data Sources Tables 520 CHAPTER 22 Inference about Variables: Part III Review 533 Part III summary 536 Test yourself 538 Supplementary exercises 587 Conditions for regression inference 589 Estimating the parameters 590 Using technology 593 Testing the hypothesis of no linear relationship 597 Testing lack of correlation 598 Confidence intervals for the regression slope 600 Inference about prediction 602 Checking the conditions for inference 607 CHAPTER 20 Inference about a Population Proportion 493 CHAPTER 21 Comparing Two Proportions • 655 675 TABLE A TABLE B TABLE C TABLE D TABLE E Standard Normal probabilities 676 Random digits 678 t distribution critical values 679 Chi-square distribution critical values 680 Critical values of the correlation r 681 Answers to Selected Exercises Index 545 Inference about Relationships CHAPTER 23 Two Categorical Variables: The Chi-Square Test 553 Two-way tables 553 The problem of multiple comparisons 556 Expected counts in two-way tables 558 551 682 733 Pa r t V Pa r t I V 637 Optional Companion Chapters (available on the BPS CD and online) CHAPTER 26 Nonparametric Tests 26-3 Comparing two samples: the Wilcoxon rank sum test The Normal approximation for W 26-8 26-4 FM.indd Page vii 11/9/11 3:58:32 PM user-s163 user-F452 • Using technology 26-10 What hypotheses does Wilcoxon test? 26-13 Dealing with ties in rank tests 26-14 Matched pairs: the Wilcoxon signed rank test 26-19 The Normal approximation for W ⫹ 26-22 Dealing with ties in the signed rank test 26-24 Comparing several samples: the Kruskal-Wallis test 26-27 Hypotheses and conditions for the Kruskal-Wallis test 26-29 The Kruskal-Wallis test statistic 26-29 CHAPTER 27 Statistical Process Control 27-3 Processes 27-4 Describing processes 27-4 The _ idea of statistical process control 27-9 x charts for process monitoring 27-10 s charts for process monitoring 27-16 Using control charts 27-23 Setting up control charts 27-25 Comments on statistical control 27-32 Don't confuse control with capability! 27-34 Control charts for sample proportions 27-36 Control limits for p charts 27-37 D E T A I L E D T A BL E O F CON TE N TS CHAPTER 28 Multiple Regression* 28-3 Parallel regression lines 28-4 Estimating parameters 28-8 Using technology 28-13 Inference for multiple regression 28-16 Interaction 28-26 The multiple linear regression model 28-32 The woes of regression coefficients 28-39 A case study for multiple regression 28-41 Inference for regression parameters 28-53 Checking the conditions for inference 28-58 CHAPTER 29 More about Analysis of Variance 29-3 Beyond one-way ANOVA 29-3 Follow-up analysis: Tukey pairwise multiple comparisons 29-8 Follow-up analysis: contrasts* 29-12 Two-way ANOVA: conditions, main effects, and interaction 29-16 Inference for two-way ANOVA 29-23 Some details of two-way ANOVA* 29-32 vii FM.indd Page viii 11/9/11 3:58:33 PM user-s163 user-F452 To the Instructor: About this Book elcome to the sixth edition of The Basic Practice of Statistics. This book is the cumulation of 40 years of teaching undergraduates and 20 years of writing texts. Previous editions have been very successful, and we think that this new edition is the best yet. In this preface we describe for instructors the nature and features of the book and the changes in this sixth edition. BPS is designed to be accessible to college and university students with limited quantitative background—"just algebra" in the sense of being able to read and use simple equations. It is usable with almost any level of technology for calculating and graphing—from a $15 "two-variable statistics" calculator through a graphing calculator or spreadsheet program through full statistical software. Of course, graphs and calculations are less tedious with good technology, so we recommend making available to students the most effective technology that circumstances permit. Despite its rather low mathematical level, BPS is a "serious" text in the sense that it wants students to do more than master the mechanics of statistical calculations and graphs. Even quite basic statistics is very useful in many fields of study and in everyday life, but only if the student has learned to move from a real-world setting to choose and carry out statistical methods and then carry conclusions back to the original setting. These translations require some conceptual understanding of such issues as the distinction between data analysis and inference, the critical role of where the data come from, the reasoning of inference, and the conditions under which we can trust the conclusions of inference. BPS tries to teach both the mechanics and the concepts needed for practical statistical work, at a level appropriate for beginners. BPS is designed to reflect the actual practice of statistics, where data analysis and design of data production join with probability-based inference to form a coherent science of data. There are good pedagogical reasons for beginning with data analysis (Chapters 1 to 7), then moving to data production (Chapters 8 and 9), and then to probability (Chapters 10 to 13) and inference (Chapters 14 to 29). In studying data analysis, students learn useful skills immediately and get over some of their fear of statistics. Data analysis is a necessary preliminary to inference in practice, because inference requires clean data. Designed data production is the surest foundation for inference, and the deliberate use of chance in random sampling and randomized comparative experiments motivates the study of probability in a course that emphasizes data-oriented statistics. BPS gives a full presentation of basic probability and inference (20 of the 29 chapters) but places it in the context of statistics as a whole. W GUIDING PRINCIPLES AND THE GAISE GUIDELINES David Moore has based BPS on three principles: balanced content, experience with data, and the importance of ideas. These principles are widely accepted by statisticians concerned about teaching and are directly connected to and reflected by the viii FM.indd Page ix 11/9/11 3:58:33 PM user-s163 user-F452 • T O T H E I N STRUC TOR themes of the College Report of the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Project. The GAISE Guidelines includes six recommendations for the introductory statistics course. The content, coverage, and features of BPS are closely aligned to these recommendations: 1. Emphasize statistical literacy and develop statistical thinking. The intent of BPS is to be modern and accessible. The exposition is straightforward and concentrates on major ideas and skills. One principle of writing for beginners is not to try to tell students everything you know. Another principle is to offer frequent stopping points, marking off digestible bites of material. Statistical literacy is promoted throughout BPS in the many examples and exercises drawn from the popular press and from many fields of study. Statistical thinking is promoted in examples and exercises that give enough background to allow students to consider the meaning of their calculations. Exercises often ask for conclusions that are more than a number (or "reject H0"). Some exercises require judgment in addition to right-or-wrong calculations and conclusions. Statistics, more than mathematics, depends on judgment for effective use. BPS begins to develop students' judgment about statistical studies. 2. Use real data. The study of statistics is supposed to help students work with data in their varied academic disciplines and in their unpredictable later employment. Students learn to work with data by working with data. BPS is full of data from many fields of study and from everyday life. Data are more than mere numbers—they are numbers with a context that should play a role in making sense of the numbers and in stating conclusions. Examples and exercises in BPS, though intended for beginners, use real data and give enough background to allow students to consider the meaning of their calculations. 3. Stress conceptual understanding rather than mere knowledge of procedures. A first course in statistics introduces many skills, from making a stemplot and calculating a correlation to choosing and carrying out a significance test. In practice (even if not always in the course), calculations and graphs are automated. Moreover, anyone who makes serious use of statistics will need some specific procedures not taught in her college stat course. BPS therefore tries to make clear the larger patterns and big ideas of statistics, not in the abstract, but in the context of learning specific skills and working with specific data. Many of the big ideas are summarized in graphical outlines. Three of the most useful appear inside the front cover. Formulas without guiding principles do students little good once the final exam is past, so it is worth the time to slow down a bit and explain the ideas. 4. Foster active learning in the classroom. Fostering active learning is the business of the teacher, though an emphasis on working with data helps. To this end, we have created interactive applets to our specifications and made them available online and on the text CD. The applets are designed primarily to help in learning statistics rather than in doing statistics. An icon calls ix FM.indd Page x 11/9/11 3:58:33 PM user-s163 x user-F452 TO THE INS TR U CTO R • attention to comments and exercises based on the applets. We suggest using selected applets for classroom demonstrations even if you do not ask students to work with them. The Correlation and Regression, Confidence Interval, and P-value applets, for example, convey core ideas more clearly than any amount of chalk and talk. We also provide Web exercises at the end of each chapter. Our intent is to take advantage of the fact that most undergraduates are "Web savvy." These exercises require students to search the Web for either data or statistical examples and then evaluate what they find. Teachers can use these as classroom activities or assign them as homework projects. 5. Use technology for developing conceptual understanding and analyzing data. Automating calculations increases students' ability to complete problems, reduces their frustration, and helps them concentrate on ideas and problem recognition rather than mechanics. At a minimum, students should have a "two-variable statistics" calculator with functions for correlation and the least-squares regression line as well as for the mean and standard deviation. Many instructors will take advantage of more elaborate technology, as ASA/MAA and GAISE recommend. And many students who don't use technology in their college statistics course will find themselves using (for example) Excel on the job. BPS does not assume or require use of software except in Parts IV and V, where the work is otherwise too tedious. It does accommodate software use and tries to convince students that they are gaining knowledge that will enable them to read and use output from almost any source. There are regular "Using Technology" sections throughout the text. Each one displays and comments on output from the same three technologies, representing graphing calculators (the Texas Instruments TI-83 or TI-84), spreadsheets (Microsoft Excel), and statistical software (Minitab and CrunchIt!). The output always concerns one of the main teaching examples so that students can compare text and output. 6. Use assessments to improve and evaluate student learning. Within chapters, a few "Apply Your Knowledge" exercises follow each new idea or skill for a quick check of basic mastery and also to mark off digestible bites of material. Each of the first three parts of the book ends with a review chapter that includes a point-by-point outline of skills learned, problems students can use to test themselves, and several supplementary exercises. (Instructors can choose to cover any or none of the chapters in Parts IV and V, so each of these chapters includes a skills outline.) The review chapters present supplemental exercises without the "I just studied that" context, thus asking for another level of learning. We think it is helpful to assign some supplemental exercises. Many instructors will find that the review chapters appear at the right points for preexamination review. The "Test Yourself" questions can be used by students to review, self-assess, and prepare for such an examination. In addition, assessment materials in the form of a test bank and quizzes are available online. FM.indd Page xi 11/17/11 6:11:46 PM user-s163 user-F452 • T O T H E I N STRUC TOR WHAT'S NEW? As always, a new edition of BPS brings many new examples and exercises. There are new data sets from a variety of sources, including marketing (methods for waitpersons to improve the size of tips), exercise physiology (improving swimming performance), psychology (do colors affect performance on cognitive tasks), and the environment (water quality in Ohio State Parks). The old favorite Florida manatee regression example returns to Chapters 4, 5, and 24 now that current data are available. These are just a few of a large number of new data settings in this edition. A new edition is also an opportunity to introduce new features and polish the exposition in ways intended to help students learn. Here are some of the changes. ■ Chapter summaries consist of two sections. One, titled "Chapter Specifics," summarizes the material presented in the chapter. The second section titled "Link It," relates the chapter content to material in previous and upcoming chapters. Together, Chapter Specifics and Link it help students understand how individual chapters relate to each other and to the overall practice of statistics. ■ Each chapter now contains a few exercises that have students investigate data or statistical issues that can be found on the Web. Problems include locating data on the Web and exploring the data to answer questions about it, discussing statistical statements found online (much like we do in exercises asking students to evaluate statistical statements found in newspapers and magazines), or learning about special applications of statistics (for example, the Six Sigma movement in industry). These exercises appear in a section called "Exploring the Web" at the end of each chapter. ■ The Part Reviews have been revised to include a section titled "Test Yourself." This section provides multiple-choice, calculations, and short-answer questions to help students review the basic ideas and skills presented in that Part's chapters. These questions replace the review exercises in previous editions. ■ Data icons are located next to examples and exercises involving data sets. Data sets are now given descriptive names and are available in a variety of formats online. ■ In response to reviewer requests, we have returned to the three-chapter coverage of inference used in the fourth edition of BPS. The sixth edition introduces this topic with separate chapters for confidence intervals (Chapter 14), tests of significance (Chapter 15), and inference in practice (Chapter 16). The revised organization allows students to focus on the basics of each procedure before exploring their uses in practice. The approach adopted in the fifth edition was intended to emphasize the relation between confidence intervals and tests of significance for means. Although confidence intervals and hypothesis tests have a natural connection, we realize it is simpler for students to first learn them as separate ideas and then explore the connection. We do not discuss one-sided intervals, so it may be prudent to avoid overemphasizing the connection and raising questions about how one-sided tests connect to confidence intervals. xi c02DescribingDistributionsWithNu39 Page 39 8/17/11 5:40:37 PM user-s163 user-F452 FM.indd Page xii 11/9/11 3:58:33 PM user-s163 TO THE INS TR U CTO R • FEATURES OF THE BASIC PRACTICE OF STATISTICS, Sixth Edition Chapter 2 IN THIS CHAPTER WE COVER... In this chapter we cover . . . Each chapter opener gives a brief look at where the chapter is heading, often with references to previous chapters, and includes this bulleted list of the major topics covered. 4-Step Examples In Chapter 2, students learn how to use the four-step process for working through statistical problems: State, Plan, Solve, and Conclude. By observing this process at work in selected examples c02DescribingDistributionsWithNu56 Page 56 8/17/11 5:41:20 PM user-s163 throughout the text and practicing it in selected exercises, students develop the ability to solve and write reports on real statistical problems encountered outside of the classroom setting. ■ Measuring center: the mean ■ Measuring center: the median ■ Comparing the mean and the median ■ Measuring spread: the quartiles ■ The five-number summary and boxplots ■ Spotting suspected outliers* ■ Measuring spread: the standard deviation ■ Choosing measures of center and spread ■ Using technology ■ Organizing a statistical problem user-F452 E X A M P L E 2 . 9 Comparing tropical flowers STATE: Ethan Temeles of Amherst College, with his colleague W. John Kress, studied the relationship between varieties of the tropical flower Heliconia on the island of Dominica and the different species of hummingbirds that fertilize the flowers.9 Over time, the researchers believe, the lengths of the flowers and the forms of the hummingbirds' beaks have evolved to match each other. If that is true, flower varieties fertilized by different hummingbird species should have distinct distributions of length. Table 2.1 gives length measurements (in millimeters) for samples of three varieties of Heliconia, each fertilized by a different species of hummingbird. Do the three varieties display distinct distributions of length? How do the mean lengths compare? PLAN: Use graphs and numerical descriptions to describe and compare these three distributions of flower length. SOLVE: We might use boxplots to compare the distributions, but stemplots preserve more detail and work well for data sets of these sizes. Figure 2.4 displays stemplots with the stems lined up for easy comparison. The lengths have been rounded to the nearest tenth of a millimeter. The bihai and red varieties have somewhat skewed distributions, so we might choose to compare the five-number summaries. But because the researchers plan to use x and s for further analysis, we instead calculate these measures: Art Wolfe/Getty Images xii user-F452 TROPICALFLOWER Variety Mean length Standard deviation bihai red yellow 47.60 39.71 36.18 1.213 1.799 0.975 CONCLUDE: The three varieties differ so much in flower length that there is little overlap among them. In particular, the flowers of bihai are longer than either red or yellow. The mean lengths are 47.6 mm for H. bihai, 39.7 mm for H. caribaea red, and 36.2 mm for H. caribaea yellow. ■ FM.indd Page xiii 11/9/11 3:58:36 PM user-s163 user-F452 c02DescribingDistributionsWithNu53 Page 53 8/17/11 5:41:19 PM user-s163 04ScatterplotsandCorrelation.in108 Page 108 8/17/11 7:24:09 PM user-s163 user-F452 • A P P LY Y O U R K N O W L E D G E 4.10 Coral reefs. Exercise 4.2 discusses a study in which scientists examined data on mean sea surface temperatures (in degrees Celsius) and mean coral growth (in millimeters per year) over a several-year period at locations in the Red Sea. Here are c01PicturingDistributionsWithGra20 Page 20 8/17/11 9:41:20 PM user-s163 CORAL the data:7 Sea surface temperature Growth 29.68 29.87 30.16 30.22 30.48 30.65 30.90 2.63 2.58 2.60 2.48 2.26 2.38 2.26 T O T H E I N STRUC TOR xi i i Apply Your Knowledge Major concepts are immediately reinforced with problems that appear throughout the chapter (often following examples), allowing students to user-F452 practice their skills as they work through the text. (a) Make a scatterplot. Which is the explanatory variable? The plot shows a negative linear pattern. (b) Find the correlation r step-by-step. You may wish to round off to two decimal places in each step. First find the mean and standard deviation of each variable. Then find the seven standardized values for each variable. Finally, use the formula for r. Explain how your value for r matches your graph in (a). (c) Enter these data into your calculator or software and use the correlation function to find r. Check that you get the same result as in (b), up to roundoff error. Using Technology Located throughout the text, these special sections display and comment on the output from graphing calculators, spreadsheets, and statistical software in the context of examples from the text. c08ProducingDataSampling.indd Page 216 9/6/11 USING TECHNOLOGY Although a calculator with "two-variable statistics'' functions will do the basic calculations we need, more elaborate tools are helpful. Graphing calculators and computer software will do calculations and make graphs as you command, freeing you to concentrate on choosing the right methods and interpreting your results. Figure 2.3 displays output describing the travel times to work of 20 people in New York State (Example 2.3). Can you find x, s, and the five-number summary in each output? The big message of this section is: once you know what to look for, you can output from any technological tool. 4:28:22 PMread user-s163 The vital few Skewed distributions can show us where to concentrate our efforts. Ten percent of the cars on the road account for half of all carbon dioxide emissions. A histogram of CO2 emissions would show many cars with small or moderate values and a few with very high values. Cleaning up or replacing these cars would reduce pollution at a cost much lower than that of programs aimed at all cars. Statisticians who work at improving quality in industry make a principle of this: distinguish "the vital few'' from "the trivial many.'' Statistics in Your World These brief asides, found in each chapter, illustrate major concepts or present cautionary tales through entertaining and relevant stories, allowing students to take a break from the exposition while staying engaged. Chapter Summary and Link It Each chapter concludes with a summary of the chapter specifics, including major terms and processes, followed by a brief discussion of how the chapter "links" to material in the previous and upcoming chapters. LINK IT The methods of Chapters 1 to 6 can be used to describe data regardless of how the data were obtained. However, if we want to reason from data to give answers to specific questions or to draw conclusions about the larger population, then the method that was used to collect the data is important. Sampling is one way to collect data, but it does not guarantee that we can draw meaningful conclusions. Biased sampling methods, such as convenience sampling and voluntary response samples, produce data that can be misleading, resulting in incorrect conclusions. Simple random sampling avoids bias and produces data that can lead to valid conclusions regarding the population. Even with perfect sampling methods, there is still sample-to-sample variation; we will begin our study of the connection between sampling variation and drawing conclusions in Chapter 11. user-F452 FM.indd Page xiv 11/9/11 3:58:37 PM user-s163 user-F452 c21ComparingTwoProportions.indd Page 531 9/23/11 4:34:28 PM user-s163 xiv user-F452 TO THE INS TR U CTO R • CHAPTER 21 EXERCISES 21.34 Does preschool help? To study the long-term effects of preschool programs for poor children, the High/Scope Educational Research Foundation has followed two groups of Michigan children since early childhood.27 One group of 62 attended preschool as three- and four-year-olds. A control group of 61 children from the same area and similar backgrounds did not attend preschool. Over a 10-year period as adults, 38 of the preschool sample and 49 of the control sample needed social services (mainly welfare). Does the study provide significant evidence that children who attend preschool have less need for social services as adults? How large is the difference between the proportions of the preschool and no-preschool populations that require social services? Do inference to answer both questions. Be sure to explain exactly what inference you choose to do. 21.35 Hand sanitizers. Hand disinfection is frequently recommended for prevention of transmission of the rhinovirus that causes the common cold. In particular, hand lotion containing 2% citric acid and 2% malic acid in 70% ethanol (HL⫹) has been found to have both immediate Check Your Skills and Chapter Exercises Each chapter ends with a series of multiple-choice problems that test students' understanding of basic concepts and students' ability to apply the concepts to real-world statistical situations. The multiple-choice problems are followed by a set of more in-depth exercises that allow students to make judgments and draw conclusions based on c02DescribingDistributionsWithNu67 Page 67 9/28/11 8:02:34 PM user-s163 real data and real scenarios. Web Exercises A final set of exercises asks students to investigate data or statistical issues by researching online. These exercises tend to be more involved and provide an opportunity for students to dig deep into contemporary issues and special applications of statistics. EXPLORING THE WEB 2.55 Crime rates and outliers. The Statistical Abstract of the United States is a comprehensive summary of statistics on the social, political, and economic organization of the United States. It can be found at the Web site www.census.gov/compendia/ statab/. Go to the section Law Enforcement, Courts and Prisons, and then to the subsection Crimes and Crime Rates. Several tables of data will be available. (a) Open the Table on Crime Rates by State and Type for the latest year given. Why do you think they use rates per 100,000 population rather than the number of crimes committed? The District of Columbia is a high outlier in almost every crime category. (b) Open the Table on Crime Rates by Type for Selected Large Cities. This table includes the District of Columbia, which is listed as Washington, DC. Without doing any formal calculations, does the District of Columbia look like a high outlier in the table for large cities? Whether or not the District of Columbia is an outlier depends on more than its crime rate. It also depends on the other observations included in the data set. Which data set do you feel is more appropriate for the District of Columbia? WHY DID YOU DO THAT? There is no single best way to organize our presentation of statistics to beginners. That said, our choices reflect thinking about both content and pedagogy. Here are comments on several "frequently asked questions" about the order and selection of material in BPS. Why does the distinction between population and sample not appear in Part I? There is more to statistics than inference. In fact, statistical inference is appropriate only in rather special circumstances. The chapters in Part I present tools and tactics for describing data—any data. These tools and tactics do not depend on the idea of inference from sample to population. Many data sets in these chapters (for example, the several sets of data about the 50 states) do not lend themselves to inference because they represent an entire population. John Tukey of Bell Labs and Princeton, the philosopher of modern data analysis, insisted that the population-sample distinction be avoided when it is not relevant. He used the word "batch" for data sets in general. We see no need for a special word, but we think Tukey was right. user-F452 FM.indd Page xv 11/9/11 3:58:38 PM user-s163 user-F452 • T O T H E I N STRUC TOR Why not begin with data production? It is certainly reasonable to do so— the natural flow of a planned study is from design to data analysis to inference. But in their future employment most students will use statistics mainly in settings other than planned research studies. We place the design of data production (Chapters 8 and 9) after data analysis to emphasize that data-analytic techniques apply to any data. One of the primary purposes of statistical designs for producing data is to make inference possible, so the discussion in Chapters 8 and 9 opens Part II and motivates the study of probability. Why do Normal distributions appear in Part I? Density curves such as the Normal curves are just another tool to describe the distribution of a quantitative variable, along with stemplots, histograms, and boxplots. Professional statistical software offers to make density curves from data just as it offers histograms. We prefer not to suggest that this material is essentially tied to probability, as the traditional order does. And we find it very helpful to break up the indigestible lump of probability that troubles students so much. Meeting Normal distributions early does this and strengthens the "probability distributions are like data distributions" way of approaching probability. Why not delay correlation and regression until late in the course, as was traditional? BPS begins by offering experience working with data and gives a conceptual structure for this nonmathematical but essential part of statistics. Students profit from more experience with data and from seeing the conceptual structure worked out in relations among variables as well as in describing single-variable data. Correlation and least-squares regression are very important descriptive tools and are often used in settings where there is no population-sample distinction, such as studies of all a firm's employees. Perhaps most important, the BPS approach asks students to think about what kind of relationship lies behind the data (confounding, lurking variables, association doesn't imply causation, and so on), without overwhelming them with the demands of formal inference methods. Inference in the correlation and regression setting is a bit complex, demands software, and often comes right at the end of the course. We find that delaying all mention of correlation and regression to that point means that students often don't master the basic uses and properties of these methods. We consider Chapters 4 and 5 (correlation and regression) essential and Chapter 24 (regression inference) optional. What about probability? Much of the usual formal probability appears in the optional Chapters 12 and 13. Chapters 10 and 11 present in a less formal way the ideas of probability and sampling distributions that are needed to understand inference. These two chapters follow a straight line from the idea of probability as long-term regularity, through concrete ways of assigning probabilities, to the central idea of the sampling distribution of a statistic. The law of large numbers and the central limit theorem appear in the context of discussing the sampling distribution of a sample mean. What is left to Chapters 12 and 13 is mostly "general probability rules" (including conditional probability) and the binomial distributions. xv FM.indd Page xvi 11/9/11 3:58:38 PM user-s163 xvi user-F452 TO THE INS TR U CTO R • We suggest that you omit Chapters 12 and 13 unless you are constrained by external forces. Experienced teachers recognize that students find probability difficult. Research on learning confirms our experience. Even students who can do formally posed probability problems often have a very fragile conceptual grasp of probability ideas. Attempting to present a substantial introduction to probability in a data-oriented statistics course for students who are not mathematically trained is in our opinion unwise. Formal probability does not help these students master the ideas of inference (at least not as much as we teachers often imagine), and it depletes reserves of mental energy that might better be applied to essentially statistical ideas. Why use the z procedures for a population mean to introduce the reasoning of inference? This is a pedagogical issue, not a question of statistics in practice. Sometime in the golden future we will start with resampling methods. We think that permutation tests make the reasoning of tests clearer than any traditional approach. For now the main choices are z for a mean and z for a proportion. We find z for means quite a bit more accessible to students. Positively, we can say up front that we are going to explore the reasoning of inference in the overly simple setting described in the box on page 352 titled "Simple Conditions for Inference about a Mean." As this box suggests, exactly Normal population and true simple random sample are as unrealistic as known . All the issues of practice—robustness against lack of Normality and application when the data aren't an SRS as well as the need to estimate —are put off until, with the reasoning in hand, we discuss the practically useful t procedures. This separation of initial reasoning from messier practice works well. And it allows us to discuss optional topics such as Type I and II errors and the power of a statistical test. Negatively, starting with inference for p introduces many side issues: no exact Normal sampling distribution, but a Normal approximation to a discrete distribution; use of p̂ in both the numerator and denominator of the test statistic to estimate both the parameter p and p̂'s own standard deviation; loss of the direct link between test and confidence interval. Once upon a time we had at least the compensation of developing practically useful procedures. Now the often gross inaccuracy of the traditional z confidence interval for p is better understood. See the following explanation. Why does the presentation of inference for proportions go beyond the traditional methods? Computational and theoretical work has demonstrated convincingly that the standard confidence intervals for proportions can be trusted only for very large sample sizes. It is hard to abandon old friends, but we think that a look at the graphs in Section 2 of the paper by Brown, Cai, and DasGupta in the May 2001 issue of Statistical Science is both distressing and persuasive.1 The standard intervals often have a true confidence level much less than what was requested, and requiring larger samples encounters a maze of "lucky" and "unlucky" sample sizes until very large samples are reached. Fortunately, there is a simple cure: just add two successes and two failures to your data. We present these "plus four intervals" in Chapters 20 and 21, along with guidelines for use. FM.indd Page xvii 11/9/11 3:58:38 PM user-s163 user-F452 • T O T H E I N STRUC TOR xv i i Why didn't you cover Topic X? Introductory texts ought not to be encyclopedic. Including each reader's favorite special topic results in a text that is formidable in size and intimidating to students. We chose topics on two grounds: they are the most commonly used in practice, and they are suitable vehicles for learning broader statistical ideas. Students who have completed the core of BPS, Chapters 1–11 and 14–22, will have little difficulty moving on to more elaborate methods. There are, of course, seven chapters that discuss more advanced topics, three in this volume and four available on CD and online, to begin the next stages of learning. ACKNOWLEDGMENTS We are grateful to colleagues from two-year and four-year colleges and universities who commented on successive drafts of the manuscript. Thanks to Professor Patricia Humphrey of Georgia Southern University who read the manuscript in detail and offered advice. Thanks to Professor Jackie Miller of The Ohio State University for carefully reading all chapters, spotting many errors, and making numerous useful suggestions. Others who provided comments: James Adcock, University of Western Ontario Brad Bailey, North Georgia College and State University E. N. Barron, Loyola University, Chicago Jennifer Beineke, Western New England College Diane Benner, Harrisburg Area Community College Zoubir Benzaid, University of Wisconsin, Oshkosh Jennifer Borrello, Baylor University Smiley Cheng, University of Manitoba, Winnipeg Patti Collings, Brigham Young University Tadd Colver, Purdue University Patti Costello, Eastern Kentucky University James Curl, Modesto Junior College DeWayne Derryberry, Idaho State University Jonathan Duggins, Virginia Tech Lacy Echols, Butler University Chris Edwards, University of Wisconsin, Oshkosh Margaret Elrich, Georgia Perimeter College Karen Estes, St. Petersburg College Eugene Galperin, East Stroudsburg University Mark Gebert, Eastern Kentucky University Kim Gilbert, Clayton State University Aaron Gladish, Austin Community College Ellen Gundlach, Purdue University Arjun Gupta, Bowling Green State University Brenda Gunderson, University of Michigan Leslie Hendrix, University of South Carolina Jeanne Hill, Baylor University Dawn Holmes, University of California, Santa Barbara Patricia Humphrey, Georgia Southern University Thomas Ilvento, University of Delaware Mark Jacobson, University of Northern Iowa Marc Kirschenbaum, John Carroll University Greg Knofczynski, Armstrong Atlantic State University Mike Kowalski, University of Alberta Zhongshan Li, Georgia State University Michael Lichter, University of Buffalo Tom Linton, Central College William Liu, Bowling Green—Firelands College Amy Maddox, Grand Rapids Community College Steve Marsden, Glendale Community College Catherine Matos, Clayton State University Darcy Mays, Virginia Commonwealth University Andrew McDougall, Montclair State University Bill Meisel, Florida Community College—Jacksonville Nancy Mendell, State University of New York, Stony Brook Scott McClintock, West Chester University Lynne Nielsen, Brigham Young University Helen Noble, San Diego State University Richard Numrich, College of Southern Nevada Melvin Nyman, Alma College FM.indd Page xviii 11/9/11 3:58:38 PM user-s163 xviii user-F452 TO THE INS TR U CTO R • Darlene Olsen, Norwich University Eric Packard, Mesa State University Mary Parker, Austin Community College, Rio Grande Campus Don Porter, Beloit College Bob Price, East Tennessee State University Asoka Ramanayake, University of Wisconsin, Oshkosh Eric Rudrud, St. Cloud State University Christopher Richter, Queens University Scott Richter, University of North Carolina, Greensboro Corlis Robe, East Tennessee State University Deborah Rumsey, The Ohio State University Therese Shelton, Southwestern University Rob Sinn, North Georgia College Eugenia Skirta, East Stroudsburg University Jim Smart, Tallahassee Community College Dianna Spence, North Georgia College and State University Tim Swartz, Simon Fraser University Suzhong Tian, Husson College Suzanne Tourville, Columbia College Christopher Tripler, Endicott College Gail Tudor, Husson College Ramin Vakilian, California State University, Northridge David Vlieger, Northwest Missouri State University Joseph Walker, Georgia State University Steve Waters, Pacific Union College Yuanhui Xiao, Georgia State University Yichuan Zhao, Georgia State University We are also grateful to Craig Bleyer, Ruth Baruth, Karen Carson, Bruce Kaplan, Shona Burke, Mary Louise Byrd, Andrew Sylvester, Leslie Lahr, R. Scott Linder, Blake Logan, Pamela Bruton, and the other editorial and design professionals who have contributed greatly to the attractiveness of this book. Special thanks to Denise Showers at Aptara for her patience and careful attention. Finally, we are indebted to the many statistics teachers with whom we have discussed the teaching of our subject over many years; to people from diverse fields with whom we have worked to understand data; and especially to students whose compliments and complaints have changed and improved our teaching. Working with teachers, colleagues in other disciplines, and students constantly reminds us of the importance of hands-on experience with data and of statistical thinking in an era when computer routines quickly handle statistical details. David S. Moore, Michael Fligner, and William I. Notz FM.indd Page xix 11/9/11 3:58:38 PM user-s163 user-F452 Media and Supplements FOR STUDENTS www.yourstatsportal.com (Access code or online purchase required.) StatsPortal is the digital gateway to The Basic Practice of Statistics, Sixth Edition, designed to enrich the course and enhance students' study skills through a collection of Web-based tools. StatsPortal integrates a rich suite of diagnostic, assessment, tutorial, and enrichment features, enabling students to master statistics at their own pace. StatsPortal is organized around three main teaching and learning components: 1. Interactive eBook offers a complete and customizable online version of the text, fully integrated with all the media resources available with the sixth edition of BPS. The eBook allows students to quickly search the text, highlight key areas, and add notes about what they're reading. Instructors can customize the eBook to add, hide, and reorder content, add their own material, and highlight key text for students. 2. Resources organizes all the resources for BPS into one location for ease of use. Student Resources ■ ■ ■ ■ ■ ■ ■ ■ NEW! Statistical Video Series consisting of StatClips, StatClips Examples, and Statistically Speaking "Snapshots." View animated lecture videos, whiteboard lessons, and documentary-style footage that illustrate key statistical concepts and help students visualize statistics in real world scenarios. StatTutor Tutorials offer audio-multimedia tutorials tied directly to the textbook, containing videos, applets, and animations. NEW! is a formative quizzing system that offers immediate feedback at the question level to help students master course material. Statistical Applets offer a series of interactive applets to help students master key statistical concepts and work exercises from the text. CrunchIt!® Statistical Software allows users to analyze data from any Internet location. Designed with the novice user in mind, the software is not only easily accessible but also easy to use. CrunchIt!® offers all the basic statistical routines covered in introductory statistics courses and more. Stats@Work Simulations put students in the role of statistical consultant, helping them better understand statistics interactively within the context of real-life scenarios. EESEE Case Studies, developed by The Ohio State University Statistics Department, teach students to apply their statistical skills by exploring actual case studies using real data. Data sets are available in ASCII, Excel, TI, Minitab, SPSS, an IBM Company*, S-PLUS, and JMP formats. *SPSS was acquired by IBM in October 2009. xix FM.indd Page xx 11/9/11 3:58:39 PM user-s163 xx user-F452 M EDIA AND S U P P LEM EN T S • ■ Student Solutions Manual provides solutions to the odd-numbered exercises, with stepped out solutions to select problems. ■ Statistical Software Manuals for TI-83/84, Minitab, Excel, JMP, and SPSS provide instruction, examples, and exercises using specific statistical software packages. ■ Interactive Table Reader allows students to use statistical tables interactively to seek the information they need. ■ Tables Resources for instructors only ■ Instructor's Guide with Full Solutions includes teaching suggestions, chapter comments, and detailed solutions to all exercises. ■ Test Bank offers hundreds of multiple choice questions. ■ Lecture PowerPoint slides offer a detailed lecture presentation of statistical concepts covered in each chapter of BPS. ■ NEW! SolutionMaster is a Web-based version of the solutions in the Instructor's Guide with Full Solutions. This easy-to-use tool allows instructors to generate a solution file for any set of homework exercises. Solutions can be downloaded in PDF format for convenient printing and posting. For more information or a demonstration, contact your local W. H. Freeman sales representative. 3. Assignments organizes assignments and guides instructors through an easyto-create assignment process providing access to questions from the Test Bank and exercises from the text, including many algorithmic problems. The Assignment Center enables instructors to create their own assignments from a variety of question types for machine-gradable assignments. This powerful assignment manager allows instructors to select their preferred policies for scheduling, maximum attempts, time limitations, feedback, and more! Online Study Center www.whfreeman.com/osc/bps6e (Access code or online purchase required.) The Online Study Center offers all the resources available in StatsPortal except the eBook and Assignment Center. Companion Web site www.whfreeman.com/bps6e This open-access Web site includes statistical applets, data sets, supplementary exercises, statistical profiles, and self-quizzes. The Web site also offers four optional companion chapters covering bootstrap methods and permutation tests and statistics for quality control and capability. FM.indd Page xxi 11/9/11 3:58:39 PM user-s163 user-F452 • M E D I A A N D SU PPLE ME N TS Interactive Student CD-ROM Included with every new copy of the sixth edition of BPS, the CD contains access to all the content available on the Companion Web site. CrunchIt!® statistical software and EESEE case studies are available via an access-code-protected Web site. (An access code is included with every new text.) Special Software Packages Student versions of JMP, Minitab, S-PLUS, and SPSS are available on a CDROM packaged with the textbook. This software is not sold separately; it must be packaged with a text or a manual. Contact your W. H. Freeman representative for information or visit www.whfreeman.com. NEW! Video Tool Kit (Access code or online purchase required.) This new Statistical Video Series consists of three types of videos aimed to illustrate key statistical concepts and help students visualize statistics in real-world scenarios: ■ StatClips lecture videos, created and presented by Alan Dabney, PhD, Texas A&M University, are innovative visual tutorials that illustrate key statistical concepts. In 3 to 5 minutes, each StatClips video combines dynamic animation, data sets, and interesting scenarios to help students understand the concepts in an introductory statistics course. ■ In StatClips Examples, Alan Dabney walks students through step-by-step examples related to the StatClips lecture videos to reinforce the concepts through problem solving. ■ SnapShots videos are abbreviated, student friendly versions of the Statistically Speaking video series, and they bring the world of statistics into the classroom. In the same vein as the successful PBS series Against All Odds Statistics, Statistically Speaking uses new and updated documentary footage and interviews that show real people using data analysis to make important decisions in their careers and in their daily lives. From business to medicine, from the environment to understanding the census, SnapShots focus on why statistics is important for students' careers, and how statistics can be a powerful tool to understand their world. Printed Student Solutions Manual This printed manual provides step-by-step solutions for all odd-numbered exercises in the text. ISBN: 1-4292-8000-X Software Manuals Software manuals covering Minitab, Excel, SPSS, TI-83/84, and JMP are offered within StatsPortal and the Online Study Center. These manuals are also available xxi FM.indd Page xxii 11/9/11 3:58:39 PM user-s163 xxii user-F452 M EDIA AND S U P P LEM EN T S • in printed versions through custom publishing. They serve as basic introductions to popular statistical software options and guides to their use with the sixth edition of BPS. FOR INSTRUCTORS Instructor's Web site www.whfreeman.com/bps6e Requires user registration as an instructor and features all the student Web materials plus: ■ Instructor version of EESEE (Electronic Encyclopedia of Statistical Examples and Exercises), with solutions to the exercises in the student version. ■ PowerPoint slides containing all textbook figures and tables. ■ Lecture PowerPoint slides offering a detailed lecture presentation of statistical concepts covered in each chapter of BPS 6e. ■ Full answers to the Supplementary Exercises supplement on the student Web site. Printed Instructor's Guide with Full Solutions This printed guide includes full solutions to all exercises and provides additional examples and data sets for class use, Internet resources, and sample examinations. It also contains brief discussions of the BPS approach for each chapter. ISBN: 1-4641-1406-4 Test Bank The test bank contains hundreds of multiple-choice questions to use in generating quizzes and tests for each chapter of the text. Available in print as well as electronically on CD-ROM (for Windows and Mac), where questions can be downloaded, edited, and resequenced to suit each instructor's needs. Printed Version, ISBN: 1-4641-1492-7 Computerized (CD) Version, ISBN: 1-4641-0891-9 Enhanced Instructor's Resource CD-ROM The CD allows instructors to search and export (by key term or chapter) all the material from the student CD, plus: ■ All text images and tables ■ Instructor's Guide with full solutions ■ PowerPoint files and lecture slides ■ Test bank files ISBN: 1-4641-0893-5 FM.indd Page xxiii 11/9/11 3:58:39 PM user-s163 user-F452 • M E D I A A N D SU PPLE ME N TS Course Management Systems W. H. Freeman and Company provides courses for Blackboard, WebCT (Campus Edition and Vista), Angel, Desire2Learn, Moodle, and Sakai course management systems. These solutions are completely integrated, and you can easily customize and adapt them to meet your teaching goals and course objectives. Visit www. bfwpub.com/lms for more information. i-clicker i-clicker is a two-way radio frequency classroom response solution developed by educators for educators. University of Illinois physicists Tim Stelzer, Gary Gladding, Mats Selen, and Benny Brown created the i-clicker system after using competing classroom response solutions and discovering that they were neither classroomappropriate nor student-friendly. Each step of i-clicker's development has been informed by teaching and learning. i-clicker is superior to other systems from both a pedagogical and technical standpoint. To learn more about packaging i-clicker with this textbook, please contact your local sales rep or visit www.iclicker.com. xxi i i FM.indd Page xxiv 11/9/11 3:58:39 PM user-s163 user-F452 About the Authors David S. Moore is Shanti S. Gupta Distinguished Professor of Statistics, Emeritus, at Purdue University and was the 1998 president of the American Statistical Association. He received his A.B. from Princeton and his Ph.D. from Cornell, both in mathematics. He has written many research papers in statistical theory and served on the editorial boards of several major journals. Professor Moore is an elected fellow of the American Statistical Association and of the Institute of Mathematical Statistics and an elected member of the International Statistical Institute. He has served as program director for statistics and probability at the National Science Foundation. In recent years, Professor Moore has devoted his attention to the teaching of statistics. He was the content developer for the Annenberg/Corporation for Public Broadcasting college-level telecourse Against All Odds: Inside Statistics and for the series of video modules Statistics: Decisions through Data, intended to aid the teaching of statistics in schools. He is the author of influential articles on statistics education and of several leading texts. Professor Moore has served as president of the International Association for Statistical Education and has received the Mathematical Association of America's national award for distinguished college or university teaching of mathematics. William I. Notz is Professor of Statistics at The Ohio State University. He received his B.S. in physics from the Johns Hopkins University and his Ph.D. in mathematics from Cornell University. His first academic job was as an assistant professor in the Department of Statistics at Purdue University. While there, he taught the introductory concepts course with Professor Moore and as a result of this experience he developed an interest in statistical education. Professor Notz is a coauthor of EESEE (the Electronic Encyclopedia of Statistical Examples and Exercises) and coauthor of Statistics: Concepts and Controversies. Professor Notz's research interests have focused on experimental design and computer experiments. He is the author of several research papers and of a book on the design and analysis of computer experiments. He is an elected fellow of the American Statistical Association. He has served as the editor of the journal Technometrics and as editor of the Journal of Statistics Education. He has served as the director of the Statistical Consulting Service, as acting chair of the Department of Statistics for a year, and as an associate dean in the College of Mathematical and Physical Sciences at The Ohio State University. He is a winner of The Ohio State University's Alumni Distinguished Teaching Award. Michael A. Fligner is Professor Emeritus at The Ohio State University. He received his B.S. in mathematics from the State University of New York at Stony Brook and his Ph.D. from the University of Connecticut. He has spent his entire professional career at The Ohio State University where he was vice chair of the xxiv FM.indd Page xxv 11/9/11 3:58:39 PM user-s163 user-F452 • A BO U T T H E AUTH ORS Department for over 10 years and also served as Director of the Statistical Consulting Service. He has done consulting work with several large corporations in Central Ohio. Professor Fligner's research interests are in nonparametric statistical methods, and he received the Statistics in Chemistry award from the American Statistical Association for work on detecting biologically active compounds. He is a coauthor of the book Statistical Methods for Behavioral Ecology and received a Fulbright scholarship under the American Republics Research program to work at the Charles Darwin Research Station in the Galápagos Islands. He has been an associate editor of the Journal of Statistical Education. xxv FM.indd Page xxvi 11/9/11 3:58:39 PM user-s163 user-F452 To the Student STATISTICAL THINKING What genes are active in a tissue? Answering this question can unravel basic questions in biology, distinguish cancer cells from normal cells, and distinguish between closely related types of cancer. To learn the answer, apply the tissue to a "microarray" that contains thousands of snippets of DNA arranged in a grid on a chip about the size of your thumb. As DNA in the tissue binds to the snippets in the array, special recorders pick up spots of light of varying color and intensity across the grid and store what they see as numbers. What's hot in popular music this week? SoundScan knows. SoundScan collects data electronically from the cash registers in more than 14,000 retail outlets and also collects data on download sales from Web sites. When you buy a CD or download a digital track, the checkout scanner or Web site is probably telling SoundScan what you bought. SoundScan provides this information to Billboard magazine, MTV, and VH1, as well as to record companies and artists' agents. Should women take hormones such as estrogen after menopause, when natural production of these hormones ends? In 1992, several major medical organizations said "Yes." In particular, women who took hormones seemed to reduce their risk of a heart attack by 35% to 50%. The risks of taking hormones appeared small compared with the benefits. But in 2002, the National Institutes of Health declared these findings wrong. Use of hormones after menopause immediately plummeted. Both recommendations were based on extensive studies. What happened? DNA microarrays, SoundScan, and medical studies all produce data (numerical facts), and lots of them. Using data effectively is a large and growing part of most professions. Reacting to data is part of everyday life. That's why statistics is important: STATISTICS IS THE SCIENCE OF LEARNING FROM DATA Data are numbers, but they are not "just numbers." Data are numbers with a context. The number 10.5, for example, carries no information by itself. But if we hear that a friend's new baby weighed 10.5 pounds at birth, we congratulate her on the healthy size of the child. The context engages our background knowledge and allows us to make judgments. We know that a baby weighing 10.5 pounds is quite large, and that a human baby is unlikely to weigh 10.5 ounces or 10.5 kilograms. The context makes the number informative. xxvi FM.indd Page xxvii 11/9/11 3:58:40 PM user-s163 user-F452 • T O T H E S TUD E N T To gain insight from data, we make graphs and do calculations. But graphs and calculations are guided by ways of thinking that amount to educated commonsense. Let's begin our study of statistics with an informal look at some principles of statistical thinking.1 WHERE THE DATA COME FROM MATTERS What's behind the flip-flop in the advice offered to women about hormone replacement? The evidence in favor of hormone replacement came from a number of observational studies that compared women who were taking hormones with others who were not. But women who choose to take hormones are very different from women who do not: they are richer and better educated and see doctors more often. These women do many things to maintain their health. It isn't surprising that they have fewer heart attacks. Large and careful observational studies are expensive, but they are easier to arrange than careful experiments. Experiments don't let women decide what to do. They assign women to either hormone replacement or to dummy pills that look and taste the same as the hormone pills. The assignment is done by a coin toss, so that all kinds of women are equally likely to get either treatment. Part of the difficulty of a good experiment is persuading women to accept the result—invisible to them–of the coin toss. By 2002, several experiments agreed that hormone replacement does not reduce the risk of heart attacks, at least for older women. Faced with this better evidence, medical authorities changed their recommendations.2 Of course, observational studies are often useful. We can learn from observational studies how chimpanzees behave in the wild or which popular songs sold best last week or what percent of workers were unemployed last month. SoundScan's data on popular music and the government's data on employment and unemployment come from sample surveys, an important kind of observational study that chooses a part(the sample) to represent a larger whole. Opinion polls interview perhaps 1000 of the 235 million adults in the United States to report the public's views on current issues. Can we trust the results? We'll see that this isn't a simple yes-or-no question. Let's just say that the government's published unemployment rate is much more trustworthy than opinion poll results, and not just because the Bureau of Labor Statistics interviews 60,000 people rather than 1000. We can, however, say right away that some samples can't be trusted. The advice columnist Ann Landers once asked her readers, "If you had it to do over again, would you have children?" A few weeks later, her column was headlined "70% OF PARENTS SAY KIDS NOT WORTH IT." Indeed, 70% of the nearly 10,000 parents who wrote in said they would not have children if they could make the choice again. Those 10,000 parents were upset enough with their children to write Ann Landers. Most parents are happy with their kids and don't bother to write. Statistically designed samples, even opinion polls, don't let people choose themselves for the sample. They interview people selected by impersonal chance so that everyone has an equal opportunity to be in the sample. Such a poll showed xxv i i FM.indd Page xxviii 11/9/11 3:58:40 PM user-s163 xxviii user-F452 TO THE S TU DENT • that 91% of parents would have children again. Where data come from matters a lot. If you are careless about how you get your data, you may announce 70% "No" when the truth is close to 90% "Yes." ALWAYS LOOK AT THE DATA 0 500 Votes for Buchanan 1000 1500 2000 2500 3000 3500 Yogi Berra said it: "You can observe a lot by just watching." That's a motto for learning • Palm Beach County from data. A few carefully chosen graphs are often more instructive than great piles of numWhat happened bers. Consider the outcome of the 2000 presiin Palm Beach County? dential election in Florida. Elections don't come much closer: after much recounting, state officials declared that George W. Bush had carried Florida by 537 votes out of almost 6 million votes cast. Florida's vote decided the election and made George W. Bush, rather than Al Gore, president. Let's look at some data. Figure 1 • displays a graph that plots votes for the • • third-party candidate Pat Buchanan against • • •• •• •• • votes for the Democratic candidate Al Gore •••••• •• • • in Florida's 67 counties. ••••••••••••• • • •• What happened in Palm Beach County? 0 50,000 100,000 150,000 200,000 250,000 300,000 350,000 400,000 The question leaps from the graph. In this Votes for Gore large and heavily Democratic county, a conservative third-party candidate did far better FIGURE 1 relative to the Democratic candidate than Votes in the 2000 presidential election for Al Gore and Patrick Buchanan in Florida's in any other county. The points for the other 67 counties. What happened in Palm Beach County? 66 counties show votes for both candidates increasing together in a roughly straight-line pattern. Both counts go up as county population goes up. Based on this pattern, we would expect Buchanan to receive around 800 votes in Palm Beach County. He actually received more than 3400 votes. That difference determined the election result in Florida and in the nation. The graph demands an explanation. It turns out that Palm Beach County used a confusing "butterfly" ballot, in which candidate names on both left and right pages led to a voting column in the center. It would be easy for a voter who intended to vote for Gore to in fact cast a vote for Buchanan. The FM.indd Page xxix 11/9/11 3:58:41 PM user-s163 user-F452 • T O T H E S TUD E N T graph is convincing evidence that this in fact happened, more convincing than the complaints of voters who (later)were unsure where their votes ended up. BEWARE THE LURKING VARIABLE Women who chose hormone replacement after menopause were on the average richer and better educated than those who didn't. No wonder they had fewer heart attacks. Children who play soccer tend to have prosperous and well-educated parents. No wonder they do better in school (on the average) than children who don't play soccer. We can't conclude that hormone replacement reduces heart attacks or that playing soccer increases school grades just because we see these relationships in data. In both examples, education and affluence are lurking variables, background factors that help explain the relationships between hormone replacement and good health and between soccer and good grades. Almost all relationships between two variables are influenced by other variables lurking in the background. To understand the relationship between two variables, you must often look at other variables. Careful statistical studies try to think of and measure possible lurking variables in order to correct for their influence. As the hormone saga illustrates, this doesn't always work well. News reports often just ignore possible lurking variables that might ruin a good headline like "Playing soccer can improve your grades." The habit of asking, "What might lie behind this relationship?" is part of thinking statistically. VARIATION IS EVERYWHERE The company's sales reps file into their monthly meeting. The sales manager rises. "Congratulations! Our sales were up 2% last month, so we're all drinking champagne this morning. You remember that when sales were down 1% last month I fired half of our reps." This picture is only slightly exaggerated. Many managers overreact to small short-term variations in key figures. Here is Arthur Nielsen, head of the country's largest market research firm, describing his experience: Too many business people assign equal validity to all numbers printed on paper. They accept numbers as representing Truth and find it difficult to work with the concept of probability. They do not see a number as a kind of shorthand for a range that describes our actual knowledge of the underlying condition.3 Business data such as sales and prices vary from month to month for reasons ranging from the weather to a customer's financial difficulties to the inevitable errors in gathering the data. The manager's challenge is to say when there is a real pattern behind the variation. We'll see that statistics provides tools for understanding variation and for seeking patterns behind the screen of variation. Let's look at some more data. Figure 2 plots the average price of a gallon of regular unleaded gasoline each week from September 1990 to June 2011.4 There certainly is variation! But a close look shows a yearly pattern: gas prices go up during the summer driving season, then down as demand drops in the fall. On top of this regular pattern we see the effects of international events. For xxi x FM.indd Page xxx 11/9/11 3:58:41 PM user-s163 xxx user-F452 TO THE S TU DENT • 19 90 19 9 19 1 92 19 93 19 94 19 9 19 5 96 19 9 19 7 98 19 9 20 9 00 20 0 20 1 02 20 0 20 3 04 20 0 20 5 06 20 0 20 7 08 20 0 20 9 10 20 11 20 12 Gasoline price (dollars per gallon) example, prices rose when the 1990 Gulf War threatened oil supplies 4.00 High demand, and dropped when the world econMiddle East unrest, omy turned down after the Septem3.50 dollar loses value ber 11, 2001, terrorist attacks in the United States. The years 2007 3.00 and 2008 brought the perfect storm: 2.50 the ability to produce oil and refine gasoline was overwhelmed by high Gulf oil 2.00 demand from China and the United spill Gulf War States and continued turmoil in the 1.50 oil-producing areas of the Middle September 11 East and Nigeria. Add a rapid fall in attacks, world 1.00 economy slumps the value of the dollar, and prices at the pump skyrocketed to more than $4 per gallon. In 2010 the Gulf oil spill also affected supply and hence Year prices. The data carry an important message: because the United States FIGURE 2 Variation is everywhere: the average retail price of regular unleaded gasoline, 1990 to mid 2011. imports most of its oil, we can't control the price we pay for gasoline. Variation is everywhere. Individuals vary; repeated measurements on the same individual vary; almost everything varies over time. One reason we need to know some statistics is that statistics helps us deal with variation. CONCLUSIONS ARE NOT CERTAIN Cervical cancer is second only to breast cancer as a cause of cancer deaths in women. Almost all cervical cancers are caused by human papillomavirus (HPV). The first vaccine to protect against the most common varieties of HPV became available in 2006. The Centers for Disease Control and Prevention recommend that all girls be vaccinated at age 11 or 12. How well does the vaccine work? Doctors rely on experiments (called "clinical Trials" in medicine) that give some women the new vaccine and others a dummy vaccine. (This is ethical when it is not yet known whether or not the vaccine is safe and effective.) The conclusion of the most important trial was that an estimated 98% of women up to age 26 who are vaccinated before they are infected with HPV will avoid cervical cancers over a 3-year period. On the average, women who get the vaccine are much less likely to get cervical cancer. But because variation is everywhere, the results are different for different women. Some vaccinated women will get cancer, and many who are not vaccinated will escape. Statistical conclusions are "on the average" statements only. Well then, can we at least be certain that the vaccine reduces risk on the average? No. We can be very confident, but we can't be certain. FM.indd Page xxxi 11/9/11 3:58:41 PM user-s163 user-F452 • T O T H E S TUD E N T Because variation is everywhere, conclusions are uncertain. Statistics gives us a language for talking about uncertainty that is used and understood by statistically literate people everywhere. In the case of HPV vaccine, the medical journal used that language to tell us: "Vaccine efficiency . . . was 98% (95 percent confidence interval 86% to 100%)."5 That "98% effective" is, in Arthur Nielsen's words, "shorthand for a range that describes our actual knowledge of the underlying condition." The range is 86% to 100%, and we are 95 percent confident that the truth lies in that range. We will soon learn to understand this language. We can't escape variation and uncertainty. Learning statistics enables us to live more comfortably with these realities. STATISTICAL THINKING AND YOU What Lies Ahead in This Book The purpose of The Basic Practice of Statistics (BPS) is to give you a working knowledge of the ideas and tools of practical statistics. We will divide practical statistics into three main areas: 1. Data analysis concerns methods and strategies for exploring, organizing, and describing data using graphs and numerical summaries. Only organized data can illuminate reality. Only thoughtful exploration of data can defeat the lurking variable. Part I of BPS (Chapters 1–7) discusses data analysis. 2. Data production provides methods for producing data that can give clear answers to specific questions. Where the data come from really is important. Basic concepts about how to select samples and design experiments are the most influential ideas in statistics. These concepts are the subject of Chapters 8 and 9. 3. Statistical inference moves beyond the data in hand to draw conclusions about some wider universe, taking into account that variation is everywhere and that conclusions are uncertain. To describe variation and uncertainty, inference uses the language of probability, introduced in Chapters 10 and 11. Because we are concerned with practice rather than theory, we need only a limited knowledge of probability. Chapters 12 and 13 offer more probability for students who want it. Chapters 14, 15, and 16 discuss the reasoning of statistical inference. These chapters are the key to the rest of the book. Chapters 18–21 present inference as used in practice in the most common settings. Chapters 23–25 and the Optional Companion Chapters 26–29 on the text CD and online concern more advanced or specialized kinds of inference. Because data are numbers with a context, doing statistics means more than manipulating numbers. You must state a problem in its real-world context, plan your specific statistical work in detail, solve the problem by making the necessary graphs and calculations, and conclude by explaining what your findings say about the real-world setting. We'll make regular use of this four-step process to encourage good habits that go beyond graphs and calculations to ask, "What do the data tell me?" xxxi FM.indd Page xxxii 11/9/11 3:58:41 PM user-s163 xxxii user-F452 TO THE S TU DENT • Statistics does involve lots of calculating and graphing. The text presents the techniques you need, but you should use technology to automate calculations and graphs as much as possible. Because the big ideas of statistics don't depend on any particular level of access to technology, BPS does not require software or a graphing calculator until we reach the more advanced methods in Part IV of the text. Even if you make little use of technology, you should look at the "Using Technology" sections throughout the book. You will see at once that you can read and apply the output from almost any technology used for statistical calculations. The ideas really are more important than the details of how to do the calculations. Unless you have constant access to software or a graphing calculator, you will need a basic calculator with some built-in statistical functions. Specifically, your calculator should find means and standard deviations and calculate correlations and regression lines. Look for a calculator that claims to do "two-variable statistics" or "regression." Because graphing and calculating are automated in statistical practice, the most important assets you can gain from the study of statistics are an understanding of the big ideas and the beginnings of good judgment in working with data. BPS tries to explain the most important ideas of statistics, not just teach methods. Some examples of big ideas that you will meet (one from each of the three areas of statistics) are "always plot your data," "randomized comparative experiments," and "statistical significance." You learn statistics by doing statistical problems. As you read, you will see several levels of exercises, arranged to help you learn. Short "Apply Your Knowledge" problem sets appear after each major idea. These straightforward exercises help you solidify the main points as you read. Be sure you can do these exercises before going on. The end-of-chapter exercises begin with multiple-choice "Check Your Skills" exercises (all answers are in the back of the book). Use them to check your grasp of the basics. The regular "Chapter Exercises" help you combine all the ideas of a chapter. Finally, the three part review chapters (Chapters 7, 17, and 22) look back over major blocks of learning. They begin with a section of multiplechoice, basic calculations and short-answer questions to help you "Test Yourself" on this set of chapters. The supplemental review exercises in the review chapters often require combining ideas learned in several chapters. For each type of exercise, you are given different information regarding the specific statistical ideas and skills the problem will require, so each type of exercise requires a different level of understanding. The part review chapters (and the individual chapters in Part IV) include point-by-point lists of specific things you should be able to do. Go through the lists, and be sure you can say "I can do that" to each item. Then try some of the "Test Yourself" problems. The key to learning is persistence. The main ideas of statistics, like the main ideas of any important subject, took a long time to discover and take some time to master. The gain will be worth the pain. FrontEndpapers.indd Page 2 11/15/11 3:01:53 PM user-s163 user-F452 ST A TIS TIC S IN SUM MA RY Analyzing Data for One Variable Plot your data: Stemplot, histogram Interpret what you see: Shape, center, spread, outliers Numerical summary? x and s, five-number summary? Density curve? Normal distribution? ST A TIS TIC S IN SUM MA RY Analyzing Data for Two Variables Plot your data: Scatterplot Interpret what you see: Direction, form, strength. Linear? Numerical summary? x, y, sx, sy, and r? Regression line? ST A TIS TIC S IN SUM MA RY Mean (quantitative data) Chap. 18 t= Proportion (categorical data) Chap. 20 z= Compare means (quantitative data) Chap. 19 x̄ − µ √ s/ n One sample Test a claim Significance test t= p̂ − p0 p0(1−p0) n (x̄1 − x̄2 ) − (µ1 − µ2 ) s2 1 n1 + s2 2 n2 Two samples Compare proportions (categorical data) p̂1 − p̂2 Chap. 21 z= Mean (quantitative data) Chap. 18 s x̄ ± t∗ √ n Proportion (categorical data) Chap. 20 p̂ ± z ∗ Chap. 19 (x̄1 − x̄2 ) ± t∗ s2 s21 + 2 n1 n2 Chap. 21 (p̂1 − p̂2 ) ± z ∗ p̂1 (1 − p̂1 ) p̂2 (1 − p̂2 ) + n1 n2 p̂(1 − p̂)( n11 + 1 n2 ) p̂ = pooled proportion State problem One sample Estimate a parameter Confidence interval Difference of means (quantitative data) p̂(1 − p̂) n (use plus four) Two samples Difference of proportions (categorical data) (use plus four) FrontEndpapers.indd Page 3 11/15/11 3:01:53 PM user-s163 ORGANIZING A STATISTICAL PROBLEM: A Four-Step Process STATE: What is the practical question, in the context of the real-world setting? PLAN: What specific statistical operations does this problem call for? SOLVE: Make the graphs and carry out the calculations needed for this problem. CONCLUDE: Give your practical conclusion in the setting of the real-world problem. CONFIDENCE INTERVALS: The Four-Step Process STATE: What is the practical question that requires estimating a parameter? PLAN: Identify the parameter, choose a level of confidence, and select the type of confidence interval that fits your situation. SOLVE: Carry out the work in two phases: 1. Check the conditions for the interval you plan to use. 2. Calculate the confidence interval. CONCLUDE: Return to the practical question to describe your results in this setting. TESTS OF SIGNIFICANCE: A Four-Step Process STATE: What is the practical question that requires a statistical test? PLAN: Identify the parameter, state null and alternative hypotheses, and choose the type of test that fits your situation. SOLVE: Carry out the test in three phases: 1. Check the conditions for the test you plan to use. 2. Calculate the test statistic. 3. Find the P-value. CONCLUDE: Return to the practical question to describe your results in this setting. user-F452 c01PicturingDistributionsWithGra1 Page 1 8/17/11 9:41:10 PM user-s163 Exploring Data "W hat do the data say?'' is the first question we ask in any statistical study. Data analysis answers this question by open-ended exploration of the data. The tools of data analysis are graphs such as histograms and scatterplots and numerical measures such as means and correlations. At least as important as the tools are principles that organize our thinking as we examine data. The seven chapters in Part I present the principles and tools of statistical data analysis. They equip you with skills that are immediately useful whenever you deal with numbers. These chapters reflect the strong emphasis on exploring data that characterizes modern statistics. Sometimes we hope to draw conclusions that apply to a setting that goes beyond the data in hand. This is statistical inference, the topic of much of the rest of the book. Data analysis is essential if we are to trust the results of inference, but data analysis isn't just preparation for inference. Roughly speaking, you can always do data analysis but inference requires rather special conditions. One of the organizing principles of data analysis is to first look at one thing at a time and then at relationships. Our presentation follows this principle. In Chapters 1, 2, and 3 you will study variables and their distributions. Chapters 4, 5, and 6 concern relationships among variables. Chapter 7 reviews this part of the text. user-F452 Part I EXPLORING DATA: Variables and Distributions Picturing Distributions with Graphs CHAPTER 1 Describing Distributions with Numbers CHAPTER 2 CHAPTER 3 The Normal Distributions EXPLORING DATA: Relationships CHAPTER 4 Scatterplots and Correlation CHAPTER 5 Regression CHAPTER 6 Two-Way Tables* CHAPTER 7 Exploring Data: Part I Review 1 c01PicturingDistributionsWithGra2 Page 2 8/17/11 9:41:10 PM user-s163 user-F452 c01PicturingDistributionsWithGra3 Page 3 8/17/11 9:41:12 PM user-s163 user-F452 Picturing Distributions with Graphs tatistics is the science of data. The volume of data available to us is overwhelming. For example, the U.S. Census Bureau's American Community Survey collects data from about 3,000,000 housing units each year. Astronomers work with data on tens of millions of galaxies. The checkout scanners at Walmart's 8000 stores in 15 countries record hundreds of millions of transactions every week, all saved to inform both Walmart and its suppliers. The first step in dealing with such a flood of data is to organize our thinking about data. Fortunately, we can do this without looking at millions of data points. S INDIVIDUALS AND VARIABLES Chapter 1 IN THIS CHAPTER WE COVER... ■ Individuals and variables ■ Categorical variables: pie charts and bar graphs ■ Quantitative variables: histograms ■ Interpreting histograms ■ Quantitative variables: stemplots ■ Time plots Any set of data contains information about some group of individuals. The information is organized in variables. INDIVIDUALS AND VARIABLES Individuals are the objects described by a set of data. Individuals may be people, but they may also be animals or things. A variable is any characteristic of an individual. A variable can take different values for different individuals. A college's student data base, for example, includes data about every currently enrolled student. The students are the individuals described by the data set. For each individual, the data contain the values of variables such as date of birth, choice of major, and grade point average (GPA). In practice, 3 Courtesy U.S. Census c01PicturingDistributionsWithGra4 Page 4 8/17/11 9:41:15 PM user-s163 4 CHAP TER 1 • Picturing Distributions with Graphs any set of data is accompanied by background information that helps us understand the data. When you plan a statistical study or explore data from someone else's work, ask yourself the following questions: What's that number? You might think that numbers, unlike words, are universal. Think again. A "billion'' in the United States means 1,000,000,000 (nine zeros). In Europe, a "billion'' is 1,000,000,000,000 (twelve zeros). OK, those are words that describe numbers. But those commas in big numbers are periods in many other languages. This is so confusing that international standards call for spaces instead, so that an American billion is written 1 000 000 000. And the decimal point of the English-speaking world is the decimal comma in many other languages, so that 3.1416 in the United States becomes 3,1416 in Europe. So what is the number 10,642.389? Depends on where you are. 1. Who? What individuals do the data describe? How many individuals appear in the data? 2. What? How many variables do the data contain? What are the exact definitions of these variables? In what unit of measurement is each variable recorded? Weights, for example, might be recorded in pounds, in thousands of pounds, or in kilograms. 3. Where? Student GPAs and SAT scores (or lack of them) will vary from college to college depending on many variables, including admissions "selectivity'' for the college. 4. When? Students change from year to year, as do prices, salaries, etc. 5. Why? What purpose do the data have? Do we hope to answer some specific questions? Do we want answers for just these individuals or for some larger group that these individuals are supposed to represent? Are the individuals and variables suitable for the intended purpose? Some variables, like a person's sex or college major, simply place individuals into categories. Others, like height and grade point average, take numerical values for which we can do arithmetic. It makes sense to give an average income for a company's employees, but it does not make sense to give an "average'' sex. We can, however, count the numbers of female and male employees and do arithmetic with these counts. CATEGORICAL AND QUANTITATIVE VARIABLES A categorical variable places an individual into one of several groups or categories. A quantitative variable takes numerical values for which arithmetic operations such as adding and averaging make sense. The values of a quantitative variable are usually recorded in a unit of measurement such as seconds or kilograms. E X A M P L E 1 . 1 The American Community Survey At the U.S. Census Bureau Web site, you can view the detailed data collected by the American Community Survey, though of course the identities of people and housing units are protected. If you choose the file of data on people, the individuals are the people living in the housing units contacted by the survey. Over 100 variables are recorded for each individual. Figure 1.1 displays a very small part of the data. Each row records data on one individual. Each column contains the values of one variable for all the individuals. Translated from the U.S. Census Bureau's abbreviations, the variables are user-F452 c01PicturingDistributionsWithGra5 Page 5 11/15/11 4:59:20 PM user-s163 user-F452 • Individuals and Variables 5 FIGU R E 1. 1 A 1 SERIALNO 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 B C PWGTP 283 283 323 346 346 370 370 370 487 487 511 511 515 515 515 515 D AGEP 187 158 176 339 91 234 181 155 233 146 236 131 213 194 221 193 E JWMNP 66 66 54 37 27 53 46 18 26 23 53 53 38 40 18 11 F SCHL 10 10 10 10 15 20 G SEX 6 9 12 11 10 13 10 9 14 12 9 11 11 9 9 3 1 2 2 1 2 1 2 2 2 2 2 1 2 1 1 1 WAGP 24000 0 11900 6000 30000 83000 74000 0 800 8000 0 0 12500 800 2500 A spreadsheet displaying data from the American Community Survey, for Example 1.1. Each row in the spreadsheet contains data on one individual. eg01-01 SERIALNO PWGTP AGEP JWMNP SCHL SEX WAGP An identifying number for the household. Weight in pounds. Age in years. Travel time to work in minutes. Highest level of education. The numbers designate categories, not specific grades. For example, 9 ⫽ high school graduate, 10 ⫽ some college but no degree, and 13 ⫽ bachelor's degree. Sex, designated by 1 ⫽ male and 2 ⫽ female. Wage and salary income last year, in dollars. Look at the highlighted row in Figure 1.1. This individual is a 53-year-old man who weighs 234 pounds, travels 10 minutes to work, has a bachelor's degree, and earned $83,000 last year. In addition to the household serial number, there are six variables. Education and sex are categorical variables. The values for education and sex are stored as numbers, but these numbers are just labels for the categories and have no units of measurement. The other four variables are quantitative. Their values do have units. These variables are weight in pounds, age in years, travel time in minutes, and income in dollars. The purpose of the American Community Survey is to collect data that represent the entire nation in order to guide government policy and business decisions. To do this, the households contacted are chosen at random from all households in the country. We will see in Chapter 8 why choosing at random is a good idea. ■ Most data tables follow this format—each row is an individual, and each column is a variable. The data set in Figure 1.1 appears in a spreadsheet program that has rows and columns ready for your use. Spreadsheets are commonly used to enter and transmit data and to do simple calculations. spreadsheet c01PicturingDistributionsWithGra6 Page 6 8/17/11 9:41:15 PM user-s163 6 CHAP TER 1 • user-F452 Picturing Distributions with Graphs A P P LY Y O U R K N O W L E D G E 1.1 Fuel economy. Here is a small part of a data set that describes the fuel economy (in miles per gallon) of model year 2010 motor vehicles: Make and model Vehicle type Transmission type Number of cylinders City mpg Highway mpg Carbon footprint (tons) Aston Martin Vantage Two-seater Manual 8 12 19 13.1 Honda Civic Subcompact Automatic 4 25 36 6.3 Toyota Prius Midsize Automatic 4 51 48 3.7 Chevrolet Impala Large Automatic 6 18 29 8.3 M M The carbon footprint measures a vehicle's impact on climate change in tons of carbon dioxide emitted annually. (a) What are the individuals in this data set? (b) For each individual, what variables are given? Which of these variables are categorical and which are quantitative? 1.2 Students and exercise. You are preparing to study the exercise habits of college students. Describe two categorical variables and two quantitative variables that you might measure for each student. Give the units of measurement for the quantitative variables. CATEGORICAL VARIABLES: Pie Charts and Bar Graphs exploratory data analysis Statistical tools and ideas help us examine data in order to describe their main features. This examination is called exploratory data analysis. Like an explorer crossing unknown lands, we want first to simply describe what we see. Here are two principles that help us organize our exploration of a set of data. EXPLORING DATA 1. Begin by examining each variable by itself. Then move on to study the relationships among the variables. 2. Begin with a graph or graphs. Then add numerical summaries of specific aspects of the data. We will follow these principles in organizing our learning. Chapters 1 to 3 present methods for describing a single variable. We study relationships among c01PicturingDistributionsWithGra7 Page 7 8/17/11 9:41:16 PM user-s163 user-F452 • Categorical Variables: Pie Charts and Bar Graphs several variables in Chapters 4 to 6. In each case, we begin with graphical displays, then add numerical summaries for more complete description. The proper choice of graph depends on the nature of the variable. To examine a single variable, we usually want to display its distribution. DISTRIBUTION OF A VARIABLE The distribution of a variable tells us what values it takes and how often it takes these values. The values of a categorical variable are labels for the categories. The distribution of a categorical variable lists the categories and gives either the count or the percent of individuals who fall in each category. E X A M P L E 1 . 2 Which major? About 1.8 million students enroll in colleges and universities each year. What do they plan to study? Here are data from 2008 on the percents of first-year students who plan to major in several discipline areas:1 Field of study Arts and humanities Biological sciences Business Percent of students 13.5 9.3 16.8 Education 8.2 Engineering 9.4 Physical sciences 3.2 Professional 13.8 Social science 11.5 Technical MAJORS 1.0 Other majors and undeclared 13.2 Total 99.9 It's a good idea to check data for consistency. The percents should add to 100%. In fact, they add to 99.9%. What happened? Each percent is rounded to the nearest tenth. The exact percents would add to 100, but the rounded percents only come close. This is roundoff error. Roundoff errors don't point to mistakes in our work, just to the effect of rounding off results. ■ Columns of numbers take time to read. You can use a pie chart or a bar graph to display the distribution of a categorical variable more vividly. Figures 1.2 and 1.3 illustrate these displays for the distribution of intended college majors. roundoff error 7 c01PicturingDistributionsWithGra8 Page 8 8/17/11 9:41:16 PM user-s163 8 CHAP TER 1 • user-F452 Picturing Distributions with Graphs FIGURE 1.2 Arts/humanities Other You can use a pie chart to display the distribution of a categorical variable. Here is a pie chart of the distribution of intended majors of students entering college. Technical Biological sciences Social science Business Professional Physical sciences Engineering 20 15 10 es s P Ar rof. ts /h um . Ot he r So c . En sc gi ne i. er in g Bi o. Ed sci. uc at io Ph n ys . Te sci. ch ni ca l Percent of students who plan to major 5 sin Bu sc i. ch ni ca l Ot he r Te So c. of . i. Pr sc Ph ys . g n rin io gi ne e at En Ed uc es s ci. sin Bu .s Bi o /h u m . 0 5 10 15 This bar has height 16.8% because 16.8% of students plan to major in business. 0 20 bar graph Ar ts This slice occupies 16.8% of the pie because 16.8% of students plan to major in business. Pie charts show the distribution of a categorical variable as a "pie'' whose slices are sized by the counts or percents for the categories. Pie charts are awkward to make by hand, but software will do the job for you. A pie chart must include all the categories that make up a whole. Use a pie chart only when you want to emphasize each category's relation to the whole. We need the "Other majors'' category in Example 1.2 to complete the whole (all intended majors) and allow us to make the pie chart in Figure 1.