Biostatistics for the Biological Health Sciences

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BIOSTATISTICS FOR THE BIOLOGICAL AND HEALTH SCIENCES

MARC M. TRIOLA, MD, FACP New York University School of Medicine

MARIO F. TRIOLA Dutchess Community College

JASON ROY, PHD University of Pennsylvania

Perelman School of Medicine

SECOND EDITION

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To Ginny Dushana and Marisa Trevor and Mitchell

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Library of Congress Cataloging-in-Publication Data

Names: Triola, Marc M. | Triola, Mario F. | Roy, Jason (Jason Allen) Title: Biostatistics for the biological and health sciences. Description: Second edition / Marc M. Triola, New York University,

Mario F. Triola, Dutchess Community College, Jason Roy, University of Pennsylvania. | Boston : Pearson, [2018] | Includes bibliographical references and index.

Identifiers: LCCN 2016016759| ISBN 9780134039015 (hardcover) | ISBN 0134039017 (hardcover)

Subjects: LCSH: Biometry. | Medical statistics. Classification: LCC QH323.5 .T75 2018 | DDC 570.1/5195–dc23 LC record available at https://lccn.loc.gov/2016016759 1 16

ISBN 13: 978-0-13-403901-5 ISBN 10: 0-13-403901-7

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iii

Marc Triola, MD, FACP is the Associate Dean for Educational Informatics at NYU School of Medicine, the founding director of the NYU Langone Medical Center Institute for Innovations in Medical Education (IIME), and an Associate Professor of Medicine. Dr. Triola’s research experience and expertise focus on the disruptive effects of the present revolution in educa- tion, driven by technological advances, big data, and learn- ing analytics. Dr. Triola has worked to create a “learning

ecosystem” that includes interconnected computer-based e-learning tools and new ways to effectively integrate growing amounts of electronic data in educational re- search. Dr. Triola and IIME have been funded by the National Institutes of Health, the Integrated Advanced Information Management Systems program, the National Science Foundation Advanced Learning Technologies program, the Josiah Macy, Jr. Foundation, the U.S. Department of Education, and the American Medical As- sociation Accelerating Change in Medical Education program. He chairs numer- ous committees at the state and national levels focused on the future of health professions educational technology development and research.

Mario F. Triola is a Professor Emeritus of Mathematics at Dutchess Community College, where he has taught statistics for over 30 years. Marty is the author of Elementary Statistics, 13th edition, Essentials of Sta- tistics, 5th edition, Elementary Statistics Using Excel, 6th edi- tion, and Elementary Statis- tics Using the TI-83>84 Plus Calculator, 4th edition, and he is a co-author of Statistical Reasoning for Everyday Life, 5th edition. Elementary Statis- tics is currently available as an

International Edition, and it has been translated into several foreign languages. Marty designed the original Statdisk statistical software, and he has written several manuals and workbooks for technology supporting statistics education.

ABOUT THE AUTHORS

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iv About the Authors

He has been a speaker at many conferences and colleges. Marty’s consulting work includes the design of casino slot machines and the design of fishing rods. He has worked with attorneys in determining probabilities in paternity lawsuits, analyz- ing data in medical malpractice lawsuits, identifying salary inequities based on gender, and analyzing disputed election results. He has also used statistical meth- ods in analyzing medical school surveys and in analyzing survey results for the New York City Transit Authority. Marty has testified as an expert witness in the New York State Supreme Court.

Jason Roy, PhD, is Associate Professor of Biostatistics in the Department of Biostatistics and Epidemiology, Perelman School of Medicine, Univer- sity of Pennsylvania. He re- ceived his PhD in Biostatistics in 2000 from the University of Michigan. He was recipi- ent of the 2002 David P. Byar Young Investigator Award from the American Statistical Asso- ciation Biometrics Section. His statistical research interests are in the areas of causal inference, missing data, and prediction

modeling. He is especially interested in the statistical challenges with analyzing data from large health care databases. He collaborates in many different disease areas, including chronic kidney disease, cardiovascular disease, and liver diseases. Dr Roy is Associate Editor of Biometrics, Journal of the American Statistical Association, and Pharmacoepidemiology & Drug Safety, and has over 90 peer- reviewed publications.

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CONTENTS

1 INTRODUCTION TO STATISTICS 1 1-1 Statistical and Critical Thinking 4 1-2 Types of Data 13 1-3 Collecting Sample Data 24

2 EXPLORING DATA WITH TABLES AND GRAPHS 40 2-1 Frequency Distributions for Organizing and Summarizing Data 42 2-2 Histograms 51 2-3 Graphs That Enlighten and Graphs That Deceive 56 2-4 Scatterplots, Correlation, and Regression 65

3 DESCRIBING, EXPLORING, AND COMPARING DATA 75 3-1 Measures of Center 77 3-2 Measures of Variation 89 3-3 Measures of Relative Standing and Boxplots 102

4 PROBABILITY 118 4-1 Basic Concepts of Probability 120 4-2 Addition Rule and Multiplication Rule 131 4-3 Complements, Conditional Probability, and Bayes’ Theorem 144 4-4 Risks and Odds 153 4-5 Rates of Mortality, Fertility, and Morbidity 162 4-6 Counting 167

5 DISCRETE PROBABILITY DISTRIBUTIONS 180 5-1 Probability Distributions 182 5-2 Binomial Probability Distributions 193 5-3 Poisson Probability Distributions 206

6 NORMAL PROBABILITY DISTRIBUTIONS 216 6-1 The Standard Normal Distribution 218 6-2 Real Applications of Normal Distributions 231 6-3 Sampling Distributions and Estimators 241 6-4 The Central Limit Theorem 252 6-5 Assessing Normality 261 6-6 Normal as Approximation to Binomial 269

7 ESTIMATING PARAMETERS AND DETERMINING SAMPLE SIZES 282 7-1 Estimating a Population Proportion 284 7-2 Estimating a Population Mean 299 7-3 Estimating a Population Standard Deviation or Variance 315 7-4 Bootstrapping: Using Technology for Estimates 324

8 HYPOTHESIS TESTING 336 8-1 Basics of Hypothesis Testing 338 8-2 Testing a Claim About a Proportion 354 8-3 Testing a Claim About a Mean 366 8-4 Testing a Claim About a Standard Deviation or Variance 377

9 INFERENCES FROM TWO SAMPLES 392 9-1 Two Proportions 394 9-2 Two Means: Independent Samples 406 9-3 Two Dependent Samples (Matched Pairs) 418 9-4 Two Variances or Standard Deviations 428

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vi Contents

10 CORRELATION AND REGRESSION 442 10-1 Correlation 444 10-2 Regression 462 10-3 Prediction Intervals and Variation 474 10-4 Multiple Regression 481 10-5 Dummy Variables and Logistic Regression 489

11 GOODNESS-OF-FIT AND CONTINGENCY TABLES 502 11-1 Goodness-of-Fit 503 11-2 Contingency Tables 514

12 ANALYSIS OF VARIANCE 531 12-1 One-Way ANOVA 533 12-2 Two-Way ANOVA 547

13 NONPARAMETRIC TESTS 560 13-1 Basics of Nonparametric Tests 562 13-2 Sign Test 564 13-3 Wilcoxon Signed-Ranks Test for Matched Pairs 575 13-4 Wilcoxon Rank-Sum Test for Two Independent Samples 581 13-5 Kruskal-Wallis Test for Three or More Samples 586 13-6 Rank Correlation 592

14 SURVIVAL ANALYSIS 603 14-1 Life Tables 604 14-2 Kaplan-Meier Survival Analysis 614

APPENDIX A TABLES 625 APPENDIX B DATA SETS 638 APPENDIX C WEBSITES AND BIBLIOGRAPHY OF BOOKS 645 APPENDIX D ANSWERS TO ODD-NUMBERED SECTION EXERCISES 646

(and all Quick Quizzes, all Review Exercises, and all Cumulative Review Exercises)

Credits 683 Index 685

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PREFACE

Statistics permeates nearly every aspect of our lives, and its role has become partic- ularly important in the biological, life, medical, and health sciences. From opinion polls to clinical trials in medicine and analysis of big data from health applications, statistics inf luences and shapes the world around us. Biostatistics for the Health and Biological Sciences forges the relationship between statistics and our world through extensive use of a wide variety of real applications that bring life to theory and methods.

Goals of This Second Edition ■ Incorporate the latest and best methods used by professional statisticians.

■ Include features that address all of the recommendations included in the Guide- lines for Assessment and Instruction in Statistics Education (GAISE) as recom- mended by the American Statistical Association.

■ Provide an abundance of new and interesting data sets, examples, and exercises.

■ Foster personal growth of students through critical thinking, use of technology, collaborative work, and development of communication skills.

■ Enhance teaching and learning with the most extensive and best set of supple- ments and digital resources.

Audience ,Prerequisites Biostatistics for the Health and Biological Sciences is written for students major- ing in the biological and health sciences, and it is designed for a wide variety of students taking their first statistics course. Algebra is used minimally, and calculus is not required. It is recommended that students have completed at least an elemen- tary algebra course or that students should learn the relevant algebra components through an integrated or co-requisite course. In many cases, underlying theory is included, but this book does not require the mathematical rigor more appropriate for mathematics majors.

Hallmark Features Great care has been taken to ensure that each chapter of Biostatistics for the Health and Biological Sciences will help students understand the concepts presented. The following features are designed to help meet that objective.

Real Data

Hundreds of hours have been devoted to finding data that are real, meaningful, and interesting to students. Fully 87% of the examples are based on real data, and 89% of the exercises are based on real data. Some exercises refer to the 18 data sets listed in Appendix B, and 12 of those data sets are new to this edition. Exercises requiring use of the Appendix B data sets are located toward the end of each exercise set and are marked with a special data set icon .

Real data sets are included throughout the book to provide relevant and interesting real-world statistical applications, including biometric security, body measurements, brain sizes and IQ scores, and data from births. Appendix B includes descriptions of

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the 18 data sets that can be downloaded from the companion website www.pearson- highered.com/triola, the author maintained www.TriolaStats.com and MyStatLab.

TriolaStats.com includes downloadable data sets in formats for technologies including Excel, Minitab, JMP, SPSS, and TI@83>84 Plus calculators. The data sets are also included in the free Statdisk software, which is also available on the website.

Readability

Great care, enthusiasm, and passion have been devoted to creating a book that is readable, understandable, interesting, and relevant. Students pursuing any major in the biological, life, medical, or health fields are sure to find applications related to their future work.

Website

This textbook is supported by www.TriolaStats.com, and www.pearsonhighered.com/ triola which are continually updated to provide the latest digital resources, including:

■ Statdisk: A free, robust statistical software package designed for this book.

■ Downloadable Appendix B data sets in a variety of technology formats.

■ Downloadable textbook supplements including Glossary of Statistical Terms and Formulas and Tables.

■ Online instructional videos created specifically for this book that provide step- by-step technology instructions.

■ Triola Blog, which highlights current applications of statistics, statistics in the news, and online resources.

Chapter Features

Chapter Opening Features

■ Chapters begin with a Chapter Problem that uses real data and motivates the chapter material.

■ Chapter Objectives provide a summary of key learning goals for each section in the chapter.

Exercises

Many exercises require the interpretation of results. Great care has been taken to ensure their usefulness, relevance, and accuracy. Exercises are arranged in order of increasing difficulty, and they begin with Basic Skills and Concepts. Most sections include additional Beyond the Basics exercises that address more difficult concepts or require a stronger mathematical background. In a few cases, these exercises introduce a new concept.

End-of-Chapter Features

■ Chapter Quick Quiz provides review questions that require brief answers.

■ Review Exercises offer practice on the chapter concepts and procedures.

■ Cumulative Review Exercises reinforce earlier material.

■ Technology Project provides an activity that can be used with a variety of technologies.

■ From Data to Decision is a capstone problem that requires critical thinking and writing.

■ Cooperative Group Activities encourage active learning in groups.

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Other Features

Margin Essays There are 57 margin essays designed to highlight real-world topics and foster student interest.

Flowcharts The text includes flowcharts that simplify and clarify more complex con- cepts and procedures. Animated versions of the text’s flowcharts are available within MyStatLab and MathXL.

Quick-Reference Endpapers Tables A-2 and A-3 (the normal and t distributions) are reproduced on the rear inside cover pages.

Detachable Formula and Table Card This insert, organized by chapter, gives students a quick reference for studying, or for use when taking tests (if allowed by the instruc- tor). It also includes the most commonly used tables. This is also available for download at www.TriolaStats.com, www.pearsonhighered.com/triola and in MyStatLab.

Technology Integration

As in the preceding edition, there are many displays of screens from technology through- out the book, and some exercises are based on displayed results from technology. Where appropriate, sections include a reference to an online Tech Center subsection that in- cludes detailed instructions for Statdisk, Minitab®, Excel®, StatCrunch, or a TI@83>84 Plus® calculator. (Throughout this text, “TI-83>84 Plus” is used to identify a TI-83 Plus or TI-84 Plus calculator). The end-of-chapter features include a Technology Project.

The Statdisk statistical software package is designed specifically for this textbook and contains all Appendix B data sets. Statdisk is free to users of this book, and it can be downloaded at www.statdisk.org.

Changes in This Edition New Features

Chapter Objectives provide a summary of key learning goals for each section in the chapter.

Larger Data Sets: Some of the data sets in Appendix B are much larger than in the previous edition. It is no longer practical to print all of the Appendix B data sets in this book, so the data sets are described in Appendix B, and they can be downloaded at www.TriolaStats.com, www.pearsonhighered.com/triola, and MyStatLab.

New Content: New examples, new exercises, and Chapter Problems provide relevant and interesting real-world statistical applications, including biometric security, drug testing, gender selection, and analyzing ultrasound images.

Number New to This Edition Use Real Data

Exercises 1600 85% 89%

Examples 200 84% 87%

Major Organization Changes

All Chapters

■ New Chapter Objectives: All chapters now begin with a list of key learning goals for that chapter. Chapter Objectives replaces the former Overview numbered sec- tions. The first numbered section of each chapter now covers a major topic.

Chapter 1

■ New Section 1-1: Statistical and Critical Thinking

■ New Subsection 1-3, Part 2: Big Data and Missing Data: Too Much and Not Enough

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Chapters 2 and 3

■ Chapter Partitioned: Chapter 2 (Describing, Exploring, and Comparing Data) from the first edition has been partitioned into Chapter 2 (Summarizing and Graph- ing) and Chapter 3 (Statistics for Describing, Exploring, and Comparing Data).

■ New Section 2-4: Scatterplots, Correlation, and Regression This new section includes scatterplots in Part 1, the linear correlation coefficient r in Part 2, and linear regression in Part 3. These additions are intended to greatly facilitate cover- age for those professors who prefer some early coverage of correlation and regres- sion concepts. Chapter 10 includes these topics discussed with much greater detail.

Chapter 4

■ Combined Sections: Section 3-3 (Addition Rule) and Section 3-4 (Multiplication Rule) from the first edition are now combined into one section: 4-2 (Addition Rule and Multiplication Rule).

■ New Subsection 4-3, Part 3: Bayes’ Theorem

Chapter 5

■ Combined Sections: Section 4-3 (Binomial Probability Distributions) and Section 4-4 (Mean, Variance, and Standard Deviation for the Binomial Distribu- tion) from the first edition are now combined into one section: 5-2 (Binomial Probability Distributions).

Chapter 6

■ Switched Sections: Section 6-5 (Assessing Normality) now precedes Section 6-6 (Normal as Approximation to Binomial).

Chapter 7

■ Combined Sections: Sections 6-4 (Estimating a Population Mean: s Known) and 6-5 (Estimating a Population Mean: s Not Known) from the first edition have been combined into one section: 7-2 (Estimating a Population Mean). The coverage of the s known case has been substantially reduced and it is now lim- ited to Part 2 of Section 7-2.

■ New Section 7-4: Bootstrapping: Using Technology for Estimates

Chapter 8

■ Combined Sections: Sections 7-4 (Testing a Claim About a Population Mean: s Known) and 7-5 (Testing a Claim About a Population Mean: s Not Known) from the first edition have been combined into one section: 8-3 (Testing a Claim About a Mean). Coverage of the s known case has been substantially reduced and it is now limited to Part 2 of Section 8-3.

Chapter 10

■ New Section: 10-5 Dummy Variables and Logistic Regression

Chapter 11

■ New Subsection: Section 11-2, Part 2 Test of Homogeneity, Fisher’s Exact Test, and McNemar’s Test for Matched Pairs

Chapter 14

■ Combined Sections: Section 13-2 (Elements of a Life Table) and Section 13-3 (Applications of Life Tables) from the first edition have been combined into Section 14-1 (Life Tables).

■ New Section: 14-2 Kaplan-Meier Survival Analysis

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Preface xi

Flexible Syllabus This book’s organization reflects the preferences of most statistics instructors, but there are two common variations:

■ Early Coverage of Correlation and Regression: Some instructors prefer to cover the basics of correlation and regression early in the course. Section 2-4 now includes basic concepts of scatterplots, correlation, and regression without the use of formulas and greater depth found in Sections 10-1 (Correlation) and 10-2 (Regression).

■ Minimum Probability: Some instructors prefer extensive coverage of probability, while others prefer to include only basic concepts. Instructors preferring mini- mum coverage can include Section 4-1 while skipping the remaining sections of Chapter 4, as they are not essential for the chapters that follow. Many instructors prefer to cover the fundamentals of probability along with the basics of the addi- tion rule and multiplication rule (Section 4-2).

GAISE This book reflects recommendations from the American Statistical Association and its Guidelines for Assessment and Instruction in Statistics Education (GAISE). Those guidelines suggest the following objectives and strategies.

1. Emphasize statistical literacy and develop statistical thinking: Each section exercise set begins with Statistical Literacy and Critical Thinking exercises. Many of the book’s exercises are designed to encourage statistical thinking rather than the blind use of mechanical procedures.

2. Use real data: 87% of the examples and 89% of the exercises use real data.

3. Stress conceptual understanding rather than mere knowledge of procedures: Instead of seeking simple numerical answers, most exercises and examples involve conceptual understanding through questions that encourage practical interpretations of results. Also, each chapter includes a From Data to Decision project.

4. Foster active learning in the classroom: Each chapter ends with several Cooperative Group Activities.

5. Use technology for developing conceptual understanding and analyzing data: Computer software displays are included throughout the book. Special Tech Center subsections are available online, and they include instruction for using the software. Each chapter includes a Technology Project. When there are dis- crepancies between answers based on tables and answers based on technology, Appendix D provides both answers. The websites www.TriolaStats.com and www.pearsonhighered.com/triola as well as MyStatLab include free text-specific software (Statdisk), data sets formatted for several different technologies, and instructional videos for technologies.

6. Use assessments to improve and evaluate student learning: Assessment tools include an abundance of section exercises, Chapter Quick Quizzes, Review Exercises, Cumulative Review Exercises, Technology Projects, From Data to Decision projects, and Cooperative Group Activities.

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Acknowledgments We would like to thank the many statistics professors and students who have contrib- uted to the success of this book. We thank the reviewers for their suggestions for this second edition:

James Baldone, Virginia College Naomi Brownstein, Florida State University Christina Caruso, University of Guelph Erica A. Corbett, Southeastern Oklahoma State University Xiangming Fang, East Carolina University Phil Gona, UMASS Boston Sharon Homan, University of North Texas Jackie Milton, Boston University Joe Pick, Palm Beach State College Steve Rigdon, St. Louis University Brian Smith, Black Hills State University Mahbobeh Vezvaei, Kent State University David Zeitler, Grand Valley State University

We also thank Paul Lorczak, Joseph Pick and Erica Corbett for their help in checking the accuracy of the text and answers.

Marc Triola Mario Triola

Jason Roy September 2016

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MyStatLab® Online Course for Biostatistics: For the Biological and Health Sciences, 2e by Marc M. Triola, Mario F. Triola and Jason Roy (access code required) MyStatLab is available to accompany Pearson’s market leading text offerings. To give students a consistent tone, voice, and teaching method each text’s flavor and ap- proach is tightly integrated throughout the accompanying MyStatLab course, making learning the material as seamless as possible.

Real-World Data Examples – Help understand how statistics applies to everyday life through the extensive current, real-world data examples and exercises provided throughout the text.

MathXL coverage – MathXL is a market-leading text-specific autograded homework system built to improve student learning outcomes.

Enhanced video program to meet Introductory Statistics needs: • New! Tech-Specific Video Tutorials – These

short, topical videos address how to use varying technologies to complete exercises.

• Updated! Section Lecture Videos – Watch author, Marty Triola, work through examples and elaborate on key objectives of the chapter.

Resources for Success

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Supplements For the Student

Student’s Solutions Manual, by James Lapp (Colorado Mesa University) provides detailed, worked-out solutions to all odd-numbered text exercises. (ISBN-13: 978-0-13-403909-1; ISBN-10: 0-13-403909-2)

Student Workbook for the Triola Statistics Series, by Laura lossi (Broward College) offers additional exam- ples, concept exercises, and vocabulary exercises for each chapter. (ISBN-13: 978-0-13-446423-7; ISBN 10: 0-13-446423-0)

The following technology manuals, available in MyStatLab, include instructions, examples from the main text, and interpretations to complement those given in the text.

Excel Student Laboratory Manual and Workbook (Download Only), by Laurel Chiappetta (University of Pittsburgh). (ISBN-13: 978-0-13-446427-5; ISBN-10: 0-13-446427-3)

MINITAB Student Laboratory Manual and Work- book (Download Only), by Mario F. Triola. (ISBN-13: 978-0-13-446418-3; ISBN-10: 0-13-446418-4)

Graphing Calculator Manual for the TI-83 Plus, TI-84 Plus, TI-84 Plus C and TI-84 Plus CE (Down- load Only), by Kathleen McLaughlin (University of Connecticut) & Dorothy Wakefield (University of Con- necticut Health Center). (ISBN-13: 978-0-13-446414-5; ISBN 10: 0-13-446414-1)

Statdisk Student Laboratory Manual and Workbook (Download Only), by Mario F. Triola. These files are available to instructors and students through the Triola Sta- tistics Series website, www.pearsonhighered.com/triola, and MyStatLab.

SPSS Student Laboratory Manual and Workbook (Download Only), by James J. Ball (Indiana State Uni- versity). These files are available to instructors and stu- dents through the Triola Statistics Series website, www. pearsonhighered.com/triola, and MyStatLab.

For the Instructor

Instructor’s Solutions Manual (Download Only), by James Lapp (Colorado Mesa University) contains so- lutions to all the exercises. These files are available to qualified instructors through Pearson Education’s on- line catalog at www.pearsonhighered.com/irc or within MyStatLab.

Insider’s Guide to Teaching with the Triola Statistics Series, by Mario F. Triola, contains sample syllabi and tips for incorporating projects, as well as lesson overviews, extra examples, minimum outcome objectives, and recom- mended assignments for each chapter. (ISBN-13: 978-0-13-446425-1; ISBN-10: 0-13-446425-7)

TestGen® Computerized Test Bank (www.pearsoned. com/testgen) enables instructors to build, edit, print, and administer tests using a computerized bank of questions developed to cover all the objectives of the text. TestGen is algorithmically based, allowing instructors to create mul- tiple but equivalent versions of the same question or test with the click of a button. Instructors can also modify test bank questions or add new questions. The software and tes- tbank are available for download from Pearson Education’s online catalog at www.pearsonhighered.com. A test bank (Download Only) is also available from the online catalog.

Learning Catalytics: Learning Catalytics is a web-based engagement and assessment tool. As a “bring-your-own- device” direct response system, Learning Catalytics offers a diverse library of dynamic question types that allow stu- dents to interact with and think critically about statistical concepts. As a real-time resource, instructors can take ad- vantage of critical teaching moments both in the classroom or through assignable and gradeable homework.

Technology Resources The following resources can be found on the Triola Statistics Series website (http://www.pearsonhighered.com/triola), the author maintained www.triolastats.com, and MyStatLab

■ Appendix B data sets formatted for Minitab, SPSS, SAS, Excel, JMP, and as text files. Additionally, these data sets are available as an APP for the TI-83>84 Plus calculators, and supplemental programs for the TI-83>84 Plus calculator are also available.

■ Statdisk statistical software instructions for down- load. New features include the ability to directly use lists of data instead of requiring the use of their sum- mary statistics.

■ Extra data sets, an index of applications, and a sym- bols table.

Video resources have been expanded, updated and now supplement most sections of the book, with many topics presented by the author. The videos aim to support both instructors and students through lecture, reinforcing sta- tistical basics through technology, and applying concepts:

■ Section Lecture Videos

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Preface xv

■ New! Technology Video Tutorials – These short, topical videos address how to use Excel, Statdisk, and the TI graphing calculator to complete exercises.

■ StatTalk Videos: 24 Conceptual Videos to Help You Actually Understand Statistics. Fun-loving statistician Andrew Vickers takes to the streets of Brooklyn, NY, to demonstrate important statistical concepts through interesting stories and real-life events. These fun and engaging videos will help students actually understand statistical concepts. Available with an instructors user guide and assess- ment questions.

MyStatLab™ Online Course (access code required) MyStatLab is a course management system that delivers proven results in helping individual students succeed.

■ MyStatLab can be successfully implemented in any environment—lab-based, hybrid, fully online, traditional—and demonstrates the quantifiable differ- ence that integrated usage has on student retention, subsequent success, and overall achievement.

■ MyStatLab’s comprehensive online gradebook au- tomatically tracks students’ results on tests, quizzes, homework, and in the study plan. Instructors can use the gradebook to provide positive feedback or inter- vene if students have trouble. Gradebook data can be easily exported to a variety of spreadsheet programs, such as Microsoft Excel. You can determine which points of data you want to export, and then analyze the results to determine success.

MyStatLab provides engaging experiences that personal- ize, stimulate, and measure learning for each student. In addition to the resources below, each course includes a full interactive online version of the accompanying textbook.

■ Tutorial Exercises with Multimedia Learning Aids: The homework and practice exercises in MyStatLab align with the exercises in the textbook, and they regenerate algorithmically to give students unlim- ited opportunity for practice and mastery. Exercises offer immediate helpful feedback, guided solutions, sample problems, animations, videos, and eText clips for extra help at point-of-use.

■ Getting Ready for Statistics: A library of questions now appears within each MyStatLab course to offer the developmental math topics students need for the course. These can be assigned as a prerequisite to other assignments, if desired.

■ Conceptual Question Library: In addition to algo- rithmically regenerated questions that are aligned with

your textbook, there is a library of 1000 Conceptual Questions available in the assessment manager that re- quire students to apply their statistical understanding.

■ StatCrunch™: MyStatLab integrates the web-based statistical software, StatCrunch, within the online as- sessment platform so that students can easily analyze data sets from exercises and the text. In addition, MyStatLab includes access to www.StatCrunch.com, a website where users can access more than 15,000 shared data sets, conduct online surveys, perform complex analyses using the powerful statistical software, and generate compelling reports.

■ Statistical Software Support: Knowing that students often use external statistical software, we make it easy to copy our data sets, both from the ebook and the MyStatLab questions, into software such as StatCrunch, Minitab, Excel, and more. Students have access to a variety of support tools—Technology Tutorial Videos, Technology Study Cards, and Tech- nology Manuals for select titles—to learn how to effectively use statistical software.

MathXL® for Statistics Online Course (access code required) MathXL® is the homework and assessment engine that runs MyStatLab. (MyStatLab is MathXL plus a learning management system.)

With MathXL for Statistics, instructors can:

■ Create, edit, and assign online homework and tests using algorithmically generated exercises correlated at the objective level to the textbook.

■ Create and assign their own online exercises and import TestGen tests for added flexibility.

■ Maintain records of all student work, tracked in MathXL’s online gradebook.

With MathXL for Statistics, students can:

■ Take chapter tests in MathXL and receive personal- ized study plans and>or personalized homework assignments based on their test results.

■ Use the study plan and>or the homework to link directly to tutorial exercises for the objectives they need to study.

■ Students can also access supplemental animations and video clips directly from selected exercises.

■ Knowing that students often use external statistical software, we make it easy to copy our data sets, both from the ebook and the MyStatLab questions, into software like StatCrunch™, Minitab, Excel, and more.

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http://www.StatCrunch.com
xvi Preface

MathXL for Statistics is available to qualified adopters. For more information, visit our website at www.mathxl .com, or contact your Pearson representative.

StatCrunch™ StatCrunch is powerful, web-based statistical software that allows users to perform complex analyses, share data sets, and generate compelling reports. A vibrant online community offers more than 15,000 data sets for students to analyze.

■■ Collect. Users can upload their own data to StatCrunch or search a large library of publicly shared data sets, spanning almost any topic of interest. Also, an online survey tool allows users to quickly collect data via web-based surveys.

■■ Crunch. A full range of numerical and graphical methods allow users to analyze and gain insights from any data set. Interactive graphics help users understand statistical concepts and are available for export to enrich reports with visual representations of data.

■■ Communicate. Reporting options help users create a wide variety of visually appealing representations of their data.

Full access to StatCrunch is available with MyStatLab and StatCrunch is available by itself to qualified adopt- ers. StatCrunch Mobile is now available to access from your mobile device. For more information, visit our web- site at www.StatCrunch.com, or contact your Pearson representative.

Minitab® 17 and Minitab Express™ make learning sta- tistics easy and provide students with a skill-set that’s in demand in today’s data driven workforce. Bundling Minitab® software with educational materials ensures stu- dents have access to the software they need in the class- room, around campus, and at home. And having 12 month versions of Minitab 17 and Minitab Express available ensures students can use the software for the duration of their course. ISBN 13: 978-0-13-445640-9 ISBN 10: 0-13-445640-8 (Access Card only; not sold as stand alone.)

JMP Student Edition, Version 12 is an easy-to-use, stream- lined version of JMP desktop statistical discovery software from SAS Institute, Inc., and is available for bundling with the text. (ISBN-13: 978-0-13-467979-2 ISBN-10: 0-13-467979-2)

A01_TRIO9015_02_SE_FM_i-xvi.indd 16 11/9/16 3:00 PM

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Statistical and Critical Thinking

Types of Data

Collecting Sample Data

1-1

1-2

1-3

Survey Question: Do You Need Caffeine to Start Up Your Brain for the Day?CHAPTER PROBLEM

Introduction to Statistics1

Surveys provide data that enable us to improve products or

services. Surveys guide political candidates, shape business

practices, identify effective medical treatments, and affect

many aspects of our lives. Surveys give us insight into the

opinions and behaviors of others. As an example, the National

Health and Nutrition Examination Survey (NHANES) is part

1

of a research program that studies the health and nutrition of

thousands of adults and children in the United States.

Let’s consider one USA Today survey in which respondents

were asked if they need caffeine to start up their brain for the

day. Among 2,006 respondents, 74% said that they did need the

caffeine. Figure 1-1 includes graphs that depict these results.

M01_TRIO9015_02_SE_C01_001-039.indd 1 02/08/16 4:46 PM

The survey results suggest that people overwhelmingly need caffeine to start up their brains

for the day. The graphs in Figure 1-1 visually depict the survey results. One of the most impor-

tant objectives of this book is to encourage the use of critical thinking so that such results are

not blindly accepted. We might question whether the survey results are valid. Who conducted

the survey? How were respondents selected? Do the graphs in Figure 1-1 depict the results

well, or are those graphs somehow misleading?

The survey results presented here have major flaws that are among the most common, so

they are especially important to recognize. Here are brief descriptions of each of the major flaws:

Flaw 1: Misleading Graphs The bar chart in Figure 1-1(a) is very deceptive. By using a vertical scale that does not start at zero, the difference between the two percentages is grossly

exaggerated. Figure 1-1(a) makes it appear that approximately eight times as many people

need the caffeine. However, with 74% needing caffeine and 26% not needing caffeine, the

ratio is actually about 3:1, rather than the 8:1 ratio that is suggested by the graph.

The illustration in Figure 1-1(b) is also deceptive. Again, the difference between the actual

response rates of 74% (needing caffeine) and 26% (not needing caffeine) is a difference that

is grossly distorted. The picture graph (or “pictograph”) in Figure 1-1(b) makes it appear that

2 CHAPTER 1 Introduction to Statistics

FIGURE 1-1(a) Survey Results

FIGURE 1-1(b) Survey Results

People Needing Ca�eine to Start Up Brain for the Day

People Not Needing Ca�eine to Start Up Brain for the Day

M01_TRIO9015_02_SE_C01_001-039.indd 2 02/08/16 4:46 PM

the ratio of people needing caffeine to people not needing caffeine is roughly 9:1 instead of

the correct ratio of about 3:1. (Objects with area or volume can distort perceptions because

they can be drawn to be disproportionately larger or smaller than the data indicate.) Decep-

tive graphs are discussed in more detail in Section 2-3, but we see here that the illustrations in

Figure 1-1 grossly exaggerate the number of people needing caffeine.

Flaw 2: Bad Sampling Method The aforementioned survey responses are from a USA Today survey of Internet users. The survey question was posted on a website and Internet

users decided whether to respond. This is an example of a voluntary response sample—a

sample in which respondents themselves decide whether to participate. With a voluntary

response sample, it often happens that those with a strong interest in the topic are more likely

to participate, so the results are very questionable. For example, people who strongly feel that

they cannot function without their morning cup(s) of coffee might be more likely to respond to

the caffeine survey than people who are more ambivalent about caffeine or coffee. When using

sample data to learn something about a population, it is extremely important to obtain sample

data that are representative of the population from which the data are drawn. As we proceed

through this chapter and discuss types of data and sampling methods, we should focus on

these key concepts:

• Sample data must be collected in an appropriate way, such as through a process of

random selection.

• If sample data are not collected in an appropriate way, the data may be so completely

useless that no amount of statistical torturing can salvage them.

It would be easy to accept the preceding survey results and blindly proceed with calcula-

tions and statistical analyses, but we would miss the critical two flaws described above. We

could then develop conclusions that are fundamentally wrong and misleading. Instead, we

should develop skills in statistical thinking and critical thinking so that we are better prepared

to analyze such data.

Chapter Objectives 3

The single most important concept presented in this chapter is this: When using meth- ods of statistics with sample data to form conclusions about a population, it is absolutely essential to collect sample data in a way that is appropriate. Here are the main chapter objectives:

Statistical and Critical Thinking

• Analyze sample data relative to context, source, and sampling method. • Understand the difference between statistical significance and practical significance. • Define and identify a voluntary response sample and know that statistical conclu-

sions based on data from such a sample are generally not valid.

1-1

CHAPTER OBJECTIVES

> > >

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4 CHAPTER 1 Introduction to Statistics

Because populations are often very large, a common objective of the use of statis- tics is to obtain data from a sample and then use those data to form a conclusion about the population.

Types of Data

• Distinguish between a parameter and a statistic. • Distinguish between quantitative data and categorical (or qualitative or attribute) data. • Distinguish between discrete data and continuous data. • Determine whether basic statistical calculations are appropriate for a particular data set.

Collecting Sample Data

• Define and identify a simple random sample. • Understand the importance of sound sampling methods and the importance of

good design of experiments.

1-2

1-3

Types of Data

• Distinguish between a parameter and a parameter and a parameter statistic. • Distinguish between quantitative data and categorical (or categorical (or categorical qualitative or attribute) data. • Distinguish between discrete data and continuous data. • Determine whether basic statistical calculations are appropriate for a particular data set.

Collecting Sample Data

• Define and identify a simple random sample. • Understand the importance of sound sampling methods and the importance of

good design of experiments.

Key Concept In this section we begin with a few very basic definitions, and then we consider an overview of the process involved in conducting a statistical study. This process consists of “prepare, analyze, and conclude.” “Preparation” involves consid- eration of the context, the source of data, and sampling method. In future chapters we construct suitable graphs, explore the data, and execute computations required for the statistical method being used. In future chapters we also form conclusions by deter- mining whether results have statistical significance and practical significance.

Statistical thinking involves critical thinking and the ability to make sense of results. Statistical thinking demands so much more than the ability to execute complicated cal- culations. Through numerous examples, exercises, and discussions, this text will help you develop the statistical thinking skills that are so important in today’s world.

We begin with some very basic definitions.

1-1 Statistical and Critical Thinking

DEFINITIONS

Data are collections of observations, such as measurements, or survey responses. (A single data value is called a datum, a term rarely used. The term “data” is plural, so it is correct to say “data are…” not “data is…”)

Statistics is the science of planning studies and experiments; obtaining data; and organizing, summarizing, presenting, analyzing, and interpreting those data and then drawing conclusions based on them.

A population is the complete collection of all measurements or data that are be- ing considered. Typically, the population is the complete collection of data that we would like to make inferences about.

A census is the collection of data from every member of the population.

A sample is a subcollection of members selected from a population.

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1-1 Statistical and Critical Thinking 5

We now proceed to consider the process involved in a statistical study. See Figure 1-2 for a summary of this process and note that the focus is on critical thinking, not mathe- matical calculations. Thanks to wonderful developments in technology, we have power- ful tools that effectively do the number crunching so that we can focus on understanding and interpreting results.

EXAMPLE 1 Residential Carbon Monoxide Detectors

In the journal article “Residential Carbon Monoxide Detector Failure Rates in the United States” (by Ryan and Arnold, American Journal of Public Health, Vol. 101, No. 10), it was stated that there are 38 million carbon monoxide detectors installed in the United States. When 30 of them were randomly selected and tested, it was found that 12 of them failed to provide an alarm in hazardous carbon monoxide conditions. In this case, the population and sample are as follows:

Population: All 38 million carbon monoxide detectors in the United States

Sample: The 30 carbon monoxide detectors that were selected and tested

The objective is to use the sample data as a basis for drawing a conclusion about the population of all carbon monoxide detectors, and methods of statistics are helpful in drawing such conclusions.

Conclude 1. Significance

• Do the results have statistical significance? • Do the results have practical significance?

Analyze 1. Graph the Data 2. Explore the Data

• Are there any outliers (numbers very far away from almost all of the other data)? • What important statistics summarize the data (such as the mean and standard deviation described in Chapter 3)? • How are the data distributed? • Are there missing data? • Did many selected subjects refuse to respond?

3. Apply Statistical Methods • Use technology to obtain results.

Prepare 1. Context

• What do the data represent? • What is the goal of study?

2. Source of the Data • Are the data from a source with a special interest so that there is pressure to obtain results that are favorable to the source?

3. Sampling Method • Were the data collected in a way that is unbiased, or were the data collected in a

way that is biased (such as a procedure in which respondents volunteer to participate)?

FIGURE 1-2 Statistical Thinking

Survivorship Bias

In World War

II, statisti-

cian Abraham

Wald saved

many lives

with his work

on the Applied

Mathematics Panel. Military

leaders asked the panel how they

could improve the chances of

aircraft bombers returning after

missions. They wanted to add

some armor for protection, and

they recorded locations on the

bombers where damaging holes

were found. They reasoned that

armor should be placed in loca-

tions with the most holes, but

Wald said that strategy would be

a big mistake. He said that armor

should be placed where returning

bombers were not damaged. His

reasoning was this: The bombers

that made it back with damage

were survivors, so the damage

they suffered could be survived.

Locations on the aircraft that

were not damaged were the most

vulnerable, and aircraft suffer-

ing damage in those vulnerable

areas were the ones that did

not make it back. The military

leaders would have made a big

mistake with survivorship bias by

studying the planes that survived

instead of thinking about the

planes that did not survive.

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6 CHAPTER 1 Introduction to Statistics

Prepare Context Figure 1-2 suggests that we begin our preparation by considering the context of the data, so let’s start with context by considering the data in Table 1-1. (The data are from Data Set 9 “IQ and Brain Size” in Appendix B.) The data in Table 1-1 consist of measured IQ scores and measured brain volumes from 10 different subjects. The data are matched in the sense that each individual “IQ>brain volume” pair of values is from the same person. The first subject had a measured IQ score of 96 and a brain volume of 1005 cm3. The format of Table 1-1 suggests the following goal: Determine whether there is a relationship between IQ score and brain volume. This goal suggests a possible hypothesis: People with larger brains tend to have higher IQ scores.

Source of the Data The data in Table 1-1 were provided by M. J. Tramo, W. C. Loftus, T. A. Stukel, J. B. Weaver, and M. S. Gazziniga, who discuss the data in the article “Brain Size, Head Size, and IQ in Monozygotic Twins,” Neurology, Vol. 50. The researchers are from reputable medical schools and hospitals, and they would not gain by presenting the results in way that is misleading. In contrast, Kiwi Brands, a maker of shoe polish, commissioned a study that resulted in this statement, which was printed in some newspapers: “According to a nationwide survey of 250 hiring profes- sionals, scuffed shoes was the most common reason for a male job seeker’s failure to make a good first impression.”

When physicians who conduct clinical experiments on the efficacy of drugs re- ceive funding from drug companies, they have an incentive to obtain favorable results. Some professional journals, such as the Journal of the American Medical Association, now require that physicians report sources of funding in journal articles. We should be skeptical of studies from sources that may be biased.

Sampling Method Figure 1-2 suggests that we conclude our preparation by consid- ering the sampling method. The data in Table 1-1 were obtained from subjects whose medical histories were reviewed in an effort to ensure that no subjects had neurologic or psychiatric disease. In this case, the sampling method appears to be sound, but we cannot be sure of that without knowing how the subjects were recruited and whether any payments may have affected participation in the study.

Sampling methods and the use of randomization will be discussed in Section 1-3, but for now, we stress that a sound sampling method is absolutely essential for good results in a statistical study. It is generally a bad practice to use voluntary response (or self-selected) samples, even though their use is common.

TABLE 1-1 IQ Scores and Brain Volumes (cm3)

IQ 96 87 101 103 127 96 88 85 97 124

Brain Volume (cm3) 1005 1035 1281 1051 1034 1079 1104 1439 1029 1160

DEFINITION

A voluntary response sample (or self-selected sample) is one in which the respondents themselves decide whether to be included.

The following types of polls are common examples of voluntary response samples. By their very nature, all are seriously flawed because we should not make conclusions about a population on the basis of samples with a strong possibility of bias:

■ Internet polls, in which people online can decide whether to respond

■ Mail-in polls, in which people decide whether to reply

Origin of “Statistics”

The word

statistics is

derived from

the Latin word

status (mean-

ing “state”).

Early uses of

statistics involved compilations

of data and graphs describing

various aspects of a state or

country. In 1662, John Graunt

published statistical information

about births and deaths. Graunt’s

work was followed by studies

of mortality and disease rates,

population sizes, incomes, and

unemployment rates. House-

holds, governments, and busi-

nesses rely heavily on statistical

data for guidance. For example,

unemployment rates, inflation

rates, consumer indexes, and

birth and death rates are carefully

compiled on a regular basis,

and the resulting data are used

by business leaders to make

decisions affecting future hiring,

production levels, and expansion

into new markets.

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1-1 Statistical and Critical Thinking 7

■ Telephone call-in polls, in which newspaper, radio, or television announcements ask that you voluntarily call a special number to register your opinion

The Chapter Problem involves a USA Today survey with a voluntary response sample. See also the following Example 2.

EXAMPLE 2 Voluntary Response Sample

USA Today posted this question on the electronic edition of their newspaper: “Have you ever been bitten by an animal?” Internet users who saw that question then de- cided themselves whether to respond. Among the 2361 responses, 65% said “yes” and 35% said “no.” Because the 2361 subjects themselves chose to respond, they are a voluntary response sample and the results of the survey are highly question- able. It would be much better to get results through a poll in which the pollster ran- domly selects the subjects, instead of allowing the subjects to volunteer themselves.

Analyze Figure 1-2 indicates that after completing our preparation by considering the context, source, and sampling method, we begin to analyze the data.

Graph and Explore An analysis should begin with appropriate graphs and explora- tions of the data. Graphs are discussed in Chapter 2, and important statistics are dis- cussed in Chapter 3.

Apply Statistical Methods Later chapters describe important statistical methods, but application of these methods is often made easy with technology (calculators and>or statistical software packages). A good statistical analysis does not require strong computational skills. A good statistical analysis does require using common sense and paying careful attention to sound statistical methods.

Conclude Figure 1-2 shows that the final step in our statistical process involves conclusions, and we should develop an ability to distinguish between statistical significance and practi- cal significance.

Statistical Significance Statistical significance is achieved in a study when we get a result that is very unlikely to occur by chance. A common criterion is that we have statistical significance if the likelihood of an event occurring by chance is 5% or less.

■ Getting 98 girls in 100 random births is statistically significant because such an extreme outcome is not likely to result from random chance.

■ Getting 52 girls in 100 births is not statistically significant because that event could easily occur with random chance.

Practical Significance It is possible that some treatment or finding is effective, but common sense might suggest that the treatment or finding does not make enough of a difference to justify its use or to be practical, as illustrated in Example 3 which follows.

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8 CHAPTER 1 Introduction to Statistics

Analyzing Data: Potential Pitfalls Here are a few more items that could cause problems when analyzing data.

Misleading Conclusions When forming a conclusion based on a statistical analy- sis, we should make statements that are clear even to those who have no understand- ing of statistics and its terminology. We should carefully avoid making statements not justified by the statistical analysis. For example, later in this book we introduce the concept of a correlation, or association between two variables, such as smoking and pulse rate. A statistical analysis might justify the statement that there is a cor- relation between the number of cigarettes smoked and pulse rate, but it would not justify a statement that the number of cigarettes smoked causes a person’s pulse rate to change. Such a statement about causality can be justified by physical evidence, not by statistical analysis.

Correlation does not imply causation.

Sample Data Reported Instead of Measured When collecting data from people, it is better to take measurements yourself instead of asking subjects to report results. Ask people what they weigh and you are likely to get their desired weights, not their actual weights. People tend to round, usually down, sometimes way down. When asked, someone with a weight of 187 lb might respond that he or she weighs 160 lb. Accurate weights are collected by using a scale to measure weights, not by asking people what they weigh.

Loaded Questions If survey questions are not worded carefully, the results of a study can be misleading. Survey questions can be “loaded” or intentionally worded to elicit a desired response. Here are the actual rates of “yes” responses for the two dif- ferent wordings of a question:

97% yes: “Should the President have the line item veto to eliminate waste?”

57% yes: “Should the President have the line item veto, or not?”

Order of Questions Sometimes survey questions are unintentionally loaded by such factors as the order of the items being considered. See the following two

EXAMPLE 3 Statistical Significance Versus Practical Significance

ProCare Industries once supplied a product named Gender Choice that supposedly increased the chance of a couple having a baby with the gender that they desired. In the absence of any evidence of its effectiveness, the product was banned by the Food and Drug Administration (FDA) as a “gross deception of the consumer.” But suppose that the product was tested with 10,000 couples who wanted to have baby girls, and the results consist of 5200 baby girls born in the 10,000 births. This re- sult is statistically significant because the likelihood of it happening due to chance is only 0.003%, so chance doesn’t seem like a feasible explanation. That 52% rate of girls is statistically significant, but it lacks practical significance because 52% is only slightly above 50%. Couples would not want to spend the time and money to increase the likelihood of a girl from 50% to 52%. (Note: In reality, the likelihood of a baby being a girl is about 48.8%, not 50%.)

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1-1 Statistical and Critical Thinking 9

questions from a poll conducted in Germany, along with the very different response rates:

“Would you say that tra¦c contributes more or less to air pollution than indus- try?” (45% blamed tra¦c; 27% blamed industry.)

“Would you say that industry contributes more or less to air pollution than traf- fic?” (24% blamed tra¦c; 57% blamed industry.)

In addition to the order of items within a question, as illustrated above, the order of separate questions could also affect responses.

Nonresponse A nonresponse occurs when someone either refuses to respond to a survey question or is unavailable. When people are asked survey questions, some firmly refuse to answer. The refusal rate has been growing in recent years, partly be- cause many persistent telemarketers try to sell goods or services by beginning with a sales pitch that initially sounds as though it is part of an opinion poll. (This “selling under the guise” of a poll is called sugging.) In Lies, Damn Lies, and Statistics, author Michael Wheeler makes this very important observation:

People who refuse to talk to pollsters are likely to be different from those who do not. Some may be fearful of strangers and others jealous of their privacy, but their refusal to talk demonstrates that their view of the world around them is markedly different from that of those people who will let poll-takers into their homes.

Percentages Some studies cite misleading or unclear percentages. Note that 100% of some quantity is all of it, but if there are references made to percentages that exceed 100%, such references are often not justified. If a medical researcher claims that she has developed a treatment for migraine headaches and the treatment results in a 150% reduction in those headaches, that researcher cannot be correct, because totally elimi- nating all migraine headaches would be a 100% reduction. It is impossible to reduce the number of migraine headaches by more than 100%.

When working with percentages, we should know that % or “percent” really means “divided by 100.” Here is a principle used often in this book.

Percentage of: To find a percentage of an amount, replace the % symbol with division by 100, and then interpret “of” to be multiplication. The following calculation shows that 6% of 1200 is 72:

6% of 1200 responses = 6

100 * 1200 = 72

Statistical Literacy and Critical Thinking

1. Online Medical Info USA Today posted this question on its website: “How often do you seek medical information online?” Of 1072 Internet users who chose to respond, 38% of them responded with “frequently.” What term is used to describe this type of survey in which the people surveyed consist of those who decided to respond? What is wrong with this type of sampling method?

2. Reported Versus Measured In a survey of 1046 adults conducted by Bradley Corpora- tion, subjects were asked how often they wash their hands when using a public restroom, and 70% of the respondents said “always.”

a. Identify the sample and the population.

b. Why would better results be obtained by observing the hand washing instead of asking about it?

1-1 Basic Skills and Concepts

questions from a poll conducted in Germany, along with the very different response Publication Bias

There is a “pub-

lication bias”

in professional

journals. It is

the tendency to

publish positive

results (such

as showing that some treatment

is effective) much more often

than negative results (such as

showing that some treatment has

no effect). In the article “Regis-

tering Clinical Trials” (Journal of

the American Medical Asso-

ciation, Vol. 290, No. 4), authors

Kay Dickersin and Drummond

Rennie state that “the result of

not knowing who has performed

what (clinical trial) is loss and

distortion of the evidence, waste

and duplication of trials, inability

of funding agencies to plan, and

a chaotic system from which

only certain sponsors might

benefit, and is invariably against

the interest of those who offered

to participate in trials and of

patients in general.” They sup-

port a process in which all clinical

trials are registered in one central

system, so that future research-

ers have access to all previous

studies, not just the studies that

were published.

as showing that some treatment

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10 CHAPTER 1 Introduction to Statistics

3. Statistical Significance Versus Practical Significance When testing a new treatment, what is the difference between statistical significance and practical significance? Can a treat- ment have statistical significance, but not practical significance?

4. Correlation One study showed that for a recent period of 11 years, there was a strong cor- relation (or association) between the numbers of people who drowned in swimming pools and the amounts of power generated by nuclear power plants (based on data from the Centers for Disease Control and Prevention and the Department of Energy). Does this imply that increas- ing power from nuclear power plants is the cause of more deaths in swimming pools? Why or why not?

Consider the Source. In Exercises 5–8, determine whether the given source has the potential to create a bias in a statistical study.

5. Physicians Committee for Responsible Medicine The Physicians Committee for Re- sponsible Medicine tends to oppose the use of meat and dairy products in our diets, and that organization has received hundreds of thousands of dollars in funding from the Foundation to Support Animal Protection.

6. Arsenic in Rice Amounts of arsenic in samples of rice grown in Texas were measured by the Food and Drug Administration (FDA).

7. Brain Size A data set in Appendix B includes brain volumes from 10 pairs of monozygotic (identical) twins. The data were collected by researchers at Harvard University, Massachusetts General Hospital, Dartmouth College, and the University of California at Davis.

8. Chocolate An article in Journal of Nutrition (Vol. 130, No. 8) noted that chocolate is rich in flavonoids. The article notes “regular consumption of foods rich in flavonoids may reduce the risk of coronary heart disease.” The study received funding from Mars, Inc., the candy com- pany, and the Chocolate Manufacturers Association.

Sampling Method. In Exercises 9–12, determine whether the sampling method appears to be sound or is flawed.

9. Nuclear Power Plants In a survey of 1368 subjects, the following question was posted on the USA Today website: “In your view, are nuclear plants safe?” The survey subjects were Internet users who chose to respond to the question posted on the electronic edition of USA Today.

10. Clinical Trials Researchers at Yale University conduct a wide variety of clinical trials by using subjects who volunteer after reading advertisements soliciting paid volunteers.

11. NHANES Examinations In a recent year, the National Health and Nutrition Examina- tion Survey (NHANES), sponsored by the National Center for Health Statistics, selected more than 9000 subjects who were given physical exams. Subjects were selected through a somewhat complicated procedure designed to obtain results that are representative of the population.

12. Health In a survey of 3014 randomly selected U.S. adults, 45% reported that they have at least one chronic health condition, such as diabetes or high blood pressure. The survey was conducted by Princeton Survey Research Associates International.

Statistical Significance and Practical Significance. In Exercises 13–16, determine whether the results appear to have statistical significance, and also determine whether the results appear to have practical significance.

13. Diet and Exercise Program In a study of the Kingman diet and exercise program, 40 subjects lost an average of 22 pounds. There is about a 1% chance of getting such results with a program that has no effect.

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1-1 Statistical and Critical Thinking 11

14. MCAT The Medical College Admissions Test (MCAT) is commonly used as part of the decision-making process for determining which students to accept into medical schools. To test the effectiveness of the Siena MCAT preparation course, 16 students take the MCAT test, then they complete the preparatory course, and then they retake the MCAT test, with the result that the average (mean) score for this group rises from 25 to 30. There is a 0.3% chance of getting those results by chance. Does the course appear to be effective?

15. Gender Selection In a study of the Gender Aide method of gender selection used to increase the likelihood of a baby being born a girl, 2000 users of the method gave birth to 980 boys and 1020 girls. There is about a 19% chance of getting that many girls if the method had no effect.

16. Systolic Blood Pressure High systolic blood pressure is 140 mm Hg or higher. (Normal values are less than 120 mm Hg, and prehypertension levels are between 120 mm Hg and 139 mm Hg.) Subjects with high blood pressure are encouraged to take action to lower it. A pharmaceutical company develops a new medication designed to lower blood pressure, and tests on 25 subjects result in an average (mean) decrease of 2 mm Hg. Analysis of the results shows that there is a 15% chance of getting such results if the medication has no effect.

In Exercises 17–20, refer to the sample of body temperatures (degrees Fahrenheit) in the table below. (The body temperatures are recorded on the same day from a sample of five randomly selected males listed in a data set in Appendix B.)

Subject

1 2 3 4 5

8 AM 97.0 98.5 97.6 97.7 98.7

12 AM 97.6 97.8 98.0 98.4 98.4

17. Context of the Data Refer to the table of body temperatures. Is there some meaning- ful way in which each body temperature recorded at 8 AM is matched with the 12 AM temperature?

18. Source The listed body temperatures were obtained from Dr. Steven Wasserman, Dr. Philip Mackowiak, and Dr. Myron Levine, who were researchers at the University of Maryland. Is the source of the data likely to be biased?

19. Conclusion Given the body temperatures in the table, what issue can be addressed by con- ducting a statistical analysis of the data?

20. Conclusion If we analyze the listed body temperatures with suitable methods of statistics, we conclude that when the differences are found between the 8 AM body temperatures and the 12 AM body temperatures, there is a 64% chance that the differences can be explained by random results obtained from populations that have the same 8 AM and 12 AM body tempera- tures. What should we conclude about the statistical significance of those differences?

In Exercises 21–24, refer to the data in the table below. The entries are white blood cell counts (1000 cells ,ML) and red blood cell counts (million cells ,ML) from male subjects examined as part of a large health study conducted by the National Center for Health Statis- tics. The data are matched, so that the first subject has a white blood cell count of 8.7 and a red blood cell count of 4.91, and so on.

Subject

1 2 3 4 5

White 8.7 5.9 7.3 6.2 5.9

Red 4.91 5.59 4.44 4.80 5.17

continued

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12 CHAPTER 1 Introduction to Statistics

21. Context Given that the data (on the bottom of the preceding page) are matched and consid- ering the units of the data, does it make sense to use the difference between each white blood cell count and the corresponding red blood cell count? Why or why not?

22. Analysis Given the context of the data in the table (on the bottom of the preceding page), what issue can be addressed by conducting a statistical analysis of the measurements?

23. Source of the Data Considering the source of the data (on the bottom of the preceding page), does that source appear to be biased in some way?

24. Conclusion If we analyze the sample data (on the bottom of the preceding page) and conclude that there is a correlation between white and red blood cell counts, does it follow that higher white are the cause of higher red blood cell counts?

What’s Wrong? In Exercises 25–28, identify what is wrong.

25. Potatoes In a poll sponsored by the Idaho Potato Commission, 1000 adults were asked to select their favorite vegetables, and the favorite choice was potatoes, which were selected by 26% of the respondents.

26. Healthy Water In a USA Today online poll, 951 Internet users chose to respond, and 57% of them said that they prefer drinking bottled water instead of tap water.

27. Cheese and Bedsheet Deaths In recent years, there has been a strong correlation be- tween per capita consumption of cheese in the United States and the numbers of people who died from being tangled in their bedsheets. Really. Therefore, consumption of cheese causes bedsheet entanglement fatalities.

28. Smokers The electronic cigarette maker V2 Cigs sponsored a poll showing that 55% of smokers surveyed say that they feel ostracized “sometimes,” “often,” or “always.”

Percentages. In Exercises 29 and 30, answer the given questions, which are related to percentages.

29. Health It was noted in Exercise 12 “Health” that in a survey of 3014 randomly selected U.S. adults, 45% reported that they have at least one chronic health condition, such as diabetes or high blood pressure.

a. What is 45% of 3014 adults?

b. Could the result from part (a) be the actual number of survey subjects who have at least one chronic condition?

c. What is the actual number of survey subjects who have at least one chronic condition?

d. Among those surveyed, 1808 were called by landline and 1206 were called by cell phone. What percentage of the survey subjects were called by cell phone?

30. Chillax USA Today reported results from a Research Now for Keurig survey in which 1458 men and 1543 women were asked this: “In a typical week, how often can you kick back and relax?”

a. Among the women, 19% responded with “rarely, if ever.” What is the exact value that is 19% of the number of women surveyed?

b. Could the result from part (a) be the actual number of women who responded with “rarely, if ever”? Why or why not?

c. What is the actual number of women who responded with “rarely, if ever”?

d. Among the men who responded, 219 responded with “rarely, if ever.” What is the percentage of men who responded with “rarely, if ever”?

e. Consider the question that the subjects were asked. Is that question clear and unambiguous so that all respondents will interpret the question the same way? How might the survey be improved?

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1-2 Types of Data 13

If we have more than one statistic, we have “statistics.” Another meaning of “statis- tics” was given in Section 1-1, where we defined statistics to be the science of plan- ning studies and experiments; obtaining data; organizing, summarizing, presenting, analyzing, and interpreting those data; and then drawing conclusions based on them. We now have two different definitions of statistics, but we can determine which of these two definitions applies by considering the context in which the term statistics is used, as in the following example.

31. What’s Wrong with This Picture? The Newport Chronicle ran a survey by asking read- ers to call in their response to this question: “Do you support a ban on electronic cigarettes, which foster smoking among our children?” It was reported that 20 readers responded and that 87% said “no,” while 13% said “yes.” Identify four major flaws in this survey.

32. Falsifying Data A researcher at the Sloan-Kettering Cancer Research Center was once criticized for falsifying data. Among his data were figures obtained from 6 groups of mice, with 20 individual mice in each group. The following values were given for the percentage of successes in each group: 53%, 58%, 63%, 46%, 48%, 67%. What’s wrong with those values?

1-1 Beyond the Basics

DEFINITIONS

A parameter is a numerical measurement describing some characteristic of a population.

A statistic is a numerical measurement describing some characteristic of a sample.

HINT The alliteration in “population parameter” and “sample statistic” helps us remember the meanings of these terms.

EXAMPLE 1 Parameter ,Statistic There are 17,246,372 high school students in the United States. In a study of 8505 U.S. high school students 16 years of age or older, 44.5% of them said that they texted while driving at least once during the previous 30 days (based on data in

Key Concept A major use of statistics is to collect and use sample data to make con- clusions about populations. We should know and understand the meanings of the terms statistic and parameter, as defined below. In this section we describe a few different types of data. The type of data is one of the key factors that determine the statistical methods we use in our analysis.

In Part 1 of this section we describe the basics of different types of data, and then in Part 2 we consider “big data” and missing data.

PA RT 1 Basic Types of Data Parameter ,Statistic

1-2 Types of Data

continued

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14 CHAPTER 1 Introduction to Statistics

Quantitative ,Categorical Some data are numbers representing counts or measurements (such as a systolic blood pressure of 118 mm Hg), whereas others are attributes (such as eye color of green or brown) that are not counts or measurements. The terms quantitative data and cat- egorical data distinguish between these types.

“Texting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,” by Olsen, Shults, Eaton, Pediatrics, Vol. 131, No. 6).

1. Parameter: The population size of all 17,246,372 high school students is a parameter, because it is the size of the entire population of all high school students in the United States. If we somehow knew the percentage of all 17,246,372 high school students who reported they had texted while driving, that percentage would also be a parameter.

2. Statistic: The value of 44.5% is a statistic, because it is based on the sample, not on the entire population.

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