# Discussion 08.1: Student t-Test MA3010 – Statistics for Health Professions     Discussion 08.1: Student t-Test   Researchers conducted a s

Discussion 08.1: Student t-Test MA3010 – Statistics for Health Professions

Discussion 08.1: Student t-Test

Researchers conducted a study to determine whether magnets are effective in treating back pain. The results are shown in the table for the treatment​ (with magnets) group and the sham​ (or placebo) group. The results are a measure of reduction in back pain. Assume that the two samples are independent simple random samples selected from normally distributed​ populations, and do not assume that the population standard deviations are equal.

1.Identify the worksheet (tab) that matches the first letter of your LAST name (i.e., if your last name were “Fudd” you would use the data from the “F” tab). This will be the source data you will use to answer your remaining questions for this initial post.

2.What is the null hypothesis and alternative hypothesis?

3.Identify the test statistic and p-value from your source data.

4.Would you reject or fail to reject your null hypothesis? Explain how you came to this conclusion (i.e. either use the test-statistic or p-value to support your claim).

5.Would there be sufficient evidence (based upon your source data) to support the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment? Explain. Conceptual Understanding
Students need to be equipped with both the methods and conceptual
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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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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
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.

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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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v

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

vii

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viii Preface

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.

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.

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.

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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http://www.pearson-highered.com/triola

http://www.TriolaStats.com

http://www.TriolaStats.com

http://www.pearsonhighered.com/triola

http://www.pearsonhighered.com/triola

http://www.TriolaStats.com

Preface ix

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

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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http://www.statdisk.org

http://www.TriolaStats.com

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x Preface

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:
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-
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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xii Preface

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
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

www.mystatlab.com
xiii

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xiv Preface

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 …