statistics using spss: an integrative approach second...

STATISTICS USING SPSS: AN INTEGRATIVE APPROACHSECOND EDITION

This is an introductory applied statistics text that can be used for a one- or two-semester course at either the undergraduate or graduate level. Central features areits hands-on approach; the use of real data; the wealth of exercises and illustratedexamples using these data; the complete set of detailed answers to exercises in anappendix; the presentation of statistical methods with a clear, conceptual empha-sis that includes an historical account of each method; and the integration of SPSSin a way that reflects statistical practice. Step-by-step instructions for using SPSSare provided as each new analytic procedure is introduced. A data CD is includedwith the text so that students may conduct their own statistical analyses and learnfirsthand how statistics is used in practice.

Sharon Lawner Weinberg is Professor of Quantitative Methods and Psychologyand former Vice Provost for Faculty Affairs at New York University. She is widelypublished, with more than 50 publications in her field, including books, bookchapters, journal articles, and major reports. She is the recipient of numerousgrants and author with Kenneth P. Goldberg of Statistics for the Behavioral Sciences(Cambridge, 1990).

Sarah Knapp Abramowitz is Associate Professor of Mathematics and ComputerScience at Drew University. She received her Ph.D. from New York University andis an Associate Editor of the Journal of Statistics Education.

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Cambridge University Press978-0-521-89922-2 - Statistics Using SPSS: An Integrative Approach, Second EditionSharon Lawner Weinberg and Sarah Knapp AbramowitzFrontmatterMore information

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Statistics Using SPSS

AN INTEGRATIVE APPROACHSecond Edition

SHARON LAWNER WEINBERG

New York University

SARAH KNAPP ABRAMOWITZ

Drew University






CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paolo, Delhi

Cambridge University Press32 Avenue of the Americas, New York, NY 10013-2473, USA

www.cambridge.orgInformation on this title: www.cambridge.org/9780521899222

© Cambridge University Press 2008

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2008

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication Data

Weinberg, Sharon L.Statistics using SPSS 2nd ed. / Sharon L. Weinberg, Sarah Knapp Abramowitz. –

p. cm.Includes bibliographical references and index.ISBN-13: 978-0-521-89922-2 (hardback)ISBN-13: 978-0-521-67637-3 (pbk.)ISBN-10: 0-521-67637-1 (pbk.)1. Mathematical statistics – Data processing. 2. SPSS (Computer file)

I. Abramowitz, Sarah Knapp, 1967– II. Title.QA276.W4423 2008519.50285–dc22

2007031423

ISBN 978-0-521-89922-2 hardbackISBN 978-0-521-67637-3 paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.






Contents

Preface page xiiiAcknowledgments xv

1 INTRODUCTION 1

The Role of the Computer in Data Analysis 1Statistics: Descriptive and Inferential 2Variables and Constants 2The Measurement of Variables 3Discrete and Continuous Variables 8Setting a Context with Real Data 11Exercises 12

2 EXAMINING UNIVARIATE DISTRIBUTIONS 20

Counting the Occurrence of Data Values 20When Variables Are Measured at the Nominal Level 20

Bar Graphs, 21 • Pie Graphs, 23When Variables Are Measured at the Ordinal, Interval, or Ratio Level 24

Frequency and Percent Distribution Tables, 24 •Stem-and-Leaf Displays, 26 • Histograms, 29 •Line Graphs, 31

Describing the Shape of a Distribution 33Accumulating Data 35

Cumulative Percent Distributions 35Ogive Curves 35Percentile Ranks 36Percentiles 37Five-Number Summaries and Boxplots 40

Exercises 45

3 MEASURES OF LOCATION, SPREAD, AND SKEWNESS 60

Characterizing the Location of a Distribution 60The Mode 60The Median 63The Arithmetic Mean 65Comparing the Mode, Median, and Mean 67

Characterizing the Spread of a Distribution 70The Range and Interquartile Range 72The Variance 74The Standard Deviation 76

v






Characterizing the Skewness of a Distribution 77Selecting Measures of Location and Spread 78Applying What We Have Learned 79Exercises 82

4 RE-EXPRESSING VARIABLES 91

Linear and Nonlinear Transformations 91Linear Transformations: Addition, Subtraction, Multiplication, and Division 91

The Effect on the Shape of a Distribution 93The Effect on Summary Statistics of a Distribution 95Common Linear Transformations 95Standard Scores 97z-Scores 98

Using z-Scores to Detect Outliers, 100 • Using z-Scores to Compare Scores in Different Distributions, 101 •Relating z-Scores to Percentile Ranks, 102

Nonlinear Transformations: Square Roots and Logarithms 103Nonlinear Transformations: Ranking Variables 110Other Transformations: Recoding and Combining Variables 111

Recoding Variables 111Combining Variables 113

Exercises 114

5 EXPLORING RELATIONSHIPS BETWEEN TWO VARIABLES 121

When Both Variables Are at Least Interval-Leveled 121Scatterplots 122The Pearson Product Moment Correlation Coefficient 126

Interpreting the Pearson Correlation Coefficient, 130 •The Effect of Linear Transformations, 132 • Restriction of Range, 132 • The Shape of the Underlying Distributions, 133 •The Reliability of the Data, 133

When at Least One Variable Is Ordinal and the Other Is at Least Ordinal:The Spearman Rank Correlation Coefficient 133

When at Least One Variable Is Dichotomous: Other Special Cases of the Pearson Correlation Coefficient 135The Point Biserial Correlation Coefficient: The Case of One at-Least-Interval and One

Dichotomous Variable 135The Phi Coefficient: The Case of Two Dichotomous Variables 140

Other Visual Displays of Bivariate Relationships 144Selection of Appropriate Statistic/Graph to Summarize a Relationship 147Exercises 148

6 SIMPLE LINEAR REGRESSION 158

The “Best-Fitting” Linear Equation 158The Accuracy of Prediction Using the Linear Regression Model 164The Standardized Regression Equation 165R as a Measure of the Overall Fit of the Linear Regression Model 165Simple Linear Regression When the Independent Variable Is Dichotomous 169Using r and R as Measures of Effect Size 172

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Emphasizing the Importance of the Scatterplot 172Exercises 174

7 PROBABILITY FUNDAMENTALS 182

The Discrete Case 182The Complement Rule of Probability 184The Additive Rules of Probability 184

First Additive Rule of Probability 185Second Additive Rule of Probability 186

The Multiplicative Rule of Probability 187The Relationship between Independence and Mutual Exclusivity 189Conditional Probability 190The Law of Large Numbers 191Exercises 192

8 THEORETICAL PROBABILITY MODELS 195

The Binomial Probability Model and Distribution 195The Applicability of the Binomial Probability Model 200

The Normal Probability Model and Distribution 204Using the Normal Distribution to Approximate the Binomial Distribution 210Exercises 210

9 THE ROLE OF SAMPLING IN INFERENTIAL STATISTICS 217

Samples and Populations 217Random Samples 218

Obtaining a Simple Random Sample 219Sampling with and without Replacement 221Sampling Distributions 223Describing the Sampling Distribution of Means Empirically 223Describing the Sampling Distribution of Means Theoretically: The Central

Limit Theorem 226Central Limit Theorem (CLT) 227

Estimators and BIAS 230Exercises 231

10 INFERENCES INVOLVING THE MEAN OF A SINGLE POPULATION WHEN � IS KNOWN 234

Estimating the Population Mean � When the Population Standard Deviation � Is Known 234

Interval Estimation 236Relating the Length of a Confidence Interval, the Level of Confidence, and

the Sample Size 239Hypothesis Testing 239The Relationship between Hypothesis Testing and Interval Estimation 247Effect Size 248Type II Error and the Concept of Power 249

Increasing the Level of Significance, � 253Increasing the Effect Size, � 253Decreasing the Standard Error of the Mean, � _x- 253

CONTENTS vii






Closing Remarks 254Exercises 254

11 INFERENCES INVOLVING THE MEAN WHEN � IS NOT KNOWN: ONE- AND TWO-SAMPLE DESIGNS 259

Single Sample Designs When the Parameter of Interest Is the Mean and � Is Not Known 259The t-Distribution 260Degrees of Freedom for the One-Sample t-Test 261Violating the Assumption of a Normally Distributed Parent Population in the

One-Sample t-Test 262Confidence Intervals for the One-Sample t-Test 263Hypothesis Tests: The One-Sample t-Test 267Effect Size for the One-Sample t-Test 269

Two-Sample Designs When the Parameter of Interest Is �, and � IsNot Known 273Independent (or Unrelated) and Dependent (or Related) Samples 274Independent Samples t-Test and Confidence Interval 275The Assumptions of the Independent Samples t-Test 277

Effect Size for the Independent Samples t-Test, 285Paired Samples t-Test and Confidence Interval 288The Assumptions of the Paired Samples t-Test 289Effect Size for the Paired Samples t-Test 293

Summary 294The Standard Error of the Mean Difference for Independent Samples: A More

Complete Account (Optional) 295Case 1: � Known 295Case 2: � Not Known 297

Step 1: Estimating � 2 Using the Variance Estimators �̂ 21 and �̂ 2

2, 299 •Step 2: Estimating the Standard Error of the Mean Difference,Using � 2, 299

Exercises 300

12 ONE-WAY ANALYSIS OF VARIANCE 315

The Disadvantage of Multiple t-Tests 315The One-Way Analysis of Variance 317

A Graphical Illustration of the Role of Variance in Tests on Means 317ANOVA as an Extension of the Independent Samples t-Test 318Developing an Index of Separation for the Analysis of Variance 319Carrying Out the ANOVA Computation 319

The Between-Group Variance (MSB), 320 • The Within-Group Variance (MSW), 321

The Assumptions of the One-Way ANOVA 321Testing the Equality of Population Means: The F-Ratio 322How to Read the Tables and to Use the SPSS Compute Statement for

the F-Distribution 324ANOVA Summary Table 327

sX 1-X 2

viii CONTENTS






Measuring the Effect Size 328Post-HOC Multiple Comparison Tests 330The Bonferroni Adjustment: Testing Planned Comparisons 340The Bonferroni Tests on Multiple Measures 343Exercises 345

13 TWO-WAY ANALYSIS OF VARIANCE 350

The Two-Factor Design 350The Concept of Interaction 353The Hypotheses That Are Tested by a Two-Way Analysis of Variance 358Assumptions of the Two-Way ANOVA 358Balanced versus Unbalanced Factorial Designs 360Partitioning the Total Sum of Squares 360Using the F-Ratio to Test the Effects in Two-Way ANOVA 361Carrying Out the Two-Way ANOVA Computation by Hand 362Decomposing Score Deviations about the Grand Mean 366Modeling Each Score as a Sum of Component Parts 367Explaining the Interaction as a Joint (or Multiplicative) Effect 368Measuring Effect Size 368Fixed versus Random Factors 372Post-Hoc Multiple Comparison Tests 373Summary of Steps to Be Taken in a Two-Way ANOVA Procedure 379Exercises 383

14 CORRELATION AND SIMPLE REGRESSION AS INFERENTIAL TECHNIQUES 391

The Bivariate Normal Distribution 391Testing Whether the Population Pearson Product Moment Correlation

Equals Zero 394Using a Confidence Interval to Estimate the Size of the Population

Correlation Coefficient, � 397Revisiting Simple Linear Regression for Prediction 400

Estimating the Population Standard Error of Prediction, �Y |X 400Testing the b-Weight for Statistical Significance 401Explaining Simple Regression Using an Analysis

of Variance Framework 405Measuring the Fit of the Overall Regression Equation: Using R and R2 407Relating R2 to � 2

Y|X 408Testing R2 for Statistical Significance 409Estimating the True Population R2: The Adjusted R2 409

Exploring the Goodness of Fit of the Regression Equation: Using Regression Diagnostics 410

Residual Plots: Evaluating the Assumptions Underlying Regression, 413 •Detecting Influential Observations: Discrepancy and Leverage, 415 •Using SPSS to Obtain Leverage, 417 • Using SPSS to Obtain Discrepancy, 417 • Using SPSS to Obtain Influence, 418

Using the Prediction Model to Predict Ice Cream Sales 422Simple Regression When the Predictor Is Dichotomous 422

Exercises 424

CONTENTS ix






15 AN INTRODUCTION TO MULTIPLE REGRESSION 435

The Basic Equation with Two Predictors 436Equations for b, �, and RY.12 When the Predictors Are Not Correlated 437Equations for b, �, and RY.12 When the Predictors Are Correlated 438Summarizing and Expanding on Some Important Principles of Multiple Regression 440Testing the b-Weights for Statistical Significance 444Assessing the Relative Importance of the Independent Variables in the Equation 445Measuring the Decrease in R2 Directly: An Alternative to the Squared

Part Correlation 446Evaluating the Statistical Significance of the Change in R2 446The b-Weight as a Partial Slope in Multiple Regression 448Multiple Regression When One of the Two Independent Variables Is Dichotomous 450The Concept of Interaction between Two Variables That Are at Least Interval-Leveled 454Testing the Statistical Significance of an Interaction Using SPSS 456Centering First-Order Effects to Achieve Meaningful Interpretations of b-Weights 460Understanding the Nature of a Statistically Significant Two-Way Interaction 460Interaction When One of the Independent Variables Is Dichotomous and the

Other Is Continuous 463Putting It All Together: A Student Project Reprinted 466Measuring the Variables 467Examining the Variables Individually and in Pairs 468Examining the Variables Multivariately with Mathematics Achievement as the Criterion 471Exercises 475

16 NONPARAMETRIC METHODS 485

Parametric versus Nonparametric Methods 485Nonparametric Methods When the Dependent Variable Is at the Nominal Level 486The Chi-Square Distribution (�2) 486

The Chi-Square Goodness-of-Fit Test 489The Chi-Square Test of Independence 493

Assumptions of the Chi-Square Test of Independence, 497Fisher’s Exact Test 499

Calculating the Fisher Exact Test by Hand Using the Hypergeometric Distribution, 501

Nonparametric Methods When the Dependent Variable Is Ordinal-Leveled 505Wilcoxon Sign Test 505The Mann-Whitney U-Test 508The Kruskal–Wallis Analysis of Variance 512

Exercises 514

APPENDIX A. DATA SET DESCRIPTIONS 519

Anscombe 519Basket 519Blood 519Brainsz 520Currency 520Framingham 520Hamburg 522

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Ice Cream 522Impeach 522Learndis 523Mandex.sav 524Marijuan 524NELS 524Skulls 528States 529Stepping 529Temp 530Wages 530

APPENDIX B. GENERATING DISTRIBUTIONS FOR CHAPTERS 8 AND 9 USING SPSS SYNTAX 531

(1) Creating a New Data Set File with ID Values for 75 Cases 531(2) The SPSS Syntax Program (Called, in General, a Macro) to Generate the

Set of 50,000 Sample Means Used to Form the Sampling Distribution of Means Graphed as the Histogram of Figure 9.2 532

(3) The SPSS Syntax Program to Generate the Set of 1,000 Normally Distributed Scores with Mean � 15 and SD � 3 as Illustrated by the Histogram of Figure 9.3 533

(4) The SPSS Syntax Program to Generate the Set of 1,000 Normally Distributed Scores with Mean � 15 and SD � 3 as Illustrated by the Histogram of Figure 9.4 534

(5) The SPSS Syntax Program to Generate the Set of 1,000 Positively Skewed Distributed Scores with Mean � 8 and SD � 4 as Illustrated by the Histogram ofFigure 9.5 534

(6) The SPSS Syntax Program, Sampdisver2.Sps, to Generate the Set of 5,000 Sample Means Used to Form the Sampling Distribution of Means Graphed as the Histogram of Figure 9.6. 535

APPENDIX C. STATISTICAL TABLES 537

Table 1. Areas under the Standard Normal Curve (to the Right of the z-Score) 537Table 2. Distribution of t-Values for Right-Tail Areas 538Table 3. Distribution of F-Values for Right-Tail Areas 539Table 4. Binomial Distribution Table 543Table 5. Chi-Square Distribution Values for Right-Tailed Areas 548Table 6. The Critical q-Values 549Table 7. The Critical U-Values 550

APPENDIX D. REFERENCES 554

APPENDIX E. SOLUTIONS TO EXERCISES 557

Chapter 1. Solutions 557Chapter 2. Solutions 559Chapter 3. Solutions 579Chapter 4. Solutions 597Chapter 5. Solutions 607Chapter 6. Solutions 626Chapter 7. Solutions 640Chapter 8. Solutions 641

CONTENTS xi






Chapter 9. Solutions 644Chapter 10. Solutions 648Chapter 11. Solutions 649Chapter 12. Solutions 673Chapter 13. Solutions 689Chapter 14. Solutions 703Chapter 15. Solutions 715Chapter 16. Solutions 743

Index 752

xii CONTENTS






xiii

Preface

Although based loosely on Basic Statistics for Education and the Behavioral Sciences bySharon Lawner Weinberg and Kenneth P. Goldberg, published by Cambridge UniversityPress (1990), this text, in its second edition, capitalizes on the widespread availability ofmenu-driven software packages to create a course of study that links good statistical prac-tice to the analysis of real data. Several important guiding principles continue to motivateour presentation.

First, and perhaps most importantly, we believe that a good data analytic plan mustserve to uncover the story behind the numbers, what the data tell us about the phenome-non under study. To begin, a good data analyst must know his data well and have confi-dence in it. Accordingly, we emphasize the usefulness of diagnostics in both graphical andstatistical form to expose anomalous cases, which might unduly influence results, and tohelp in the selection of appropriate assumption-satisfying transformations so that ulti-mately we may have confidence in our findings. We also emphasize the importance ofusing more than one method of analysis to answer fully the question posed. Seeing a three-dimensional sculpture in its entirety requires viewing that sculpture from many vantagepoints. Likewise, fully understanding the phenomenon under study often requires delvinginto data from more than one vantage point via the application of more than one methodof analysis.

Second, because we believe that data are central to the study of good statistical practice,this text comes with an accompanying disk that contains several data sets used throughoutthe text. One is a large set of real data containing 48 variables and 500 cases that we makerepeated use of in both worked-out examples and end-of-chapter exercises. By posinginteresting questions about variables in this large, real data set (e.g., Is there a gender dif-ference in expected income at age 30 of eighth graders?), we are able to employ a moremeaningful and contextual approach to the introduction of statistical methods and toengage students more actively in the learning process. The repeated use of this data set alsocontributes to creating a more cohesive presentation of statistics, one that links differentmethods of analysis to each other and avoids the perception that statistics is an often-confusing array of so many separate and distinct methods of analysis with no bearing orrelationship to one another.

Third, we believe that the result of a null hypothesis test (to determine whether aneffect is real or merely apparent) is only a means to an end (to determine whether theeffect being studied is important or useful), rather than an end in itself. Accordingly, in ourpresentation of null hypothesis testing, we stress the importance of evaluating the magni-tude of the effect if it is deemed to be real, and of drawing clear distinctions between sta-tistically significant and substantively significant results. Toward this end, we introduce the






computation of standardized measures of effect size as common practice following a sta-tistically significant result. While we provide guidelines for evaluating, in general, the mag-nitude of an effect, we encourage readers to think more subjectively about the magnitudeof an effect, bringing into the evaluation their own knowledge and expertise in a particu-lar area.

Fourth, a course in applied statistics should not only provide students with a sound sta-tistical knowledge base but also with a set of data analytic skills. Accordingly, we haveincorporated SPSS, a popularly used statistical software package, into the presentation ofstatistical material using a highly integrative approach. SPSS is used to provide studentswith a platform for actively engaging in the learning process associated with what it meansto be a good data analyst by allowing them to apply their newly learned knowledge to thereal world of applications. This approach serves also to enhance the conceptual under-standing of material and the ability to interpret output and communicate findings.

Finally, we believe that a key ingredient of an introductory statistics text is a clear, con-ceptual, yet rigorous approach. We emphasize conceptual understanding through anexploration of both the mathematical principles underlying statistical methods and real-world applications. We use an easygoing, informal style of writing that we have foundgives readers the impression that they are involved in a personal conversation with theauthors. And we sequence concepts with concern for student readiness, reintroducing top-ics in a spiraling manner to provide reinforcement and promote transfer of learning.

New to the second edition are a description of each statistical method within a histori-cal context so that students can appreciate the development of this relatively new field ofstudy as a twentieth-century phenomenon; the inclusion of Fisher’s Exact Test; the updat-ing of all SPSS commands to be consistent with SPSS version 15; a bibliography of refer-ences to relevant books and journal articles; several new real data sets from a variety offields, including health; an expanded appendix of SPSS syntax programs for generatingsimulated data; and many more end-of-chapter exercises along with detailed answers in anappendix.

The book is intended for use in a one- or two-semester introductory applied statisticscourse for the behavioral, social, or health sciences at either the graduate or undergraduatelevel, or as a reference text as well. It is not intended for readers who wish to acquire a moretheoretical understanding of mathematical statistics. The book consists of 16 chapters. Inaddition to topics traditionally found in introductory applied statistics texts in the behav-ioral, social, or health sciences, the book covers such topics as data transformations, diag-nostic tools for the analysis of model fit, the logic of null hypothesis testing, assessing themagnitude of effects, interaction and its interpretation in two-way analysis of variance andmultiple regression, and non-parametric statistics. This broad coverage of topics gives theinstructor flexibility in curriculum planning and provides students with more advancedmaterial for future work in statistics.

xiv PREFACE






xv

Acknowledgments

Each of the editions of our book has benefited from the many helpful comments of ourNew York University and Drew University students. Each also has benefited from theinsights and suggestions of several colleagues. For their help in improving the first edition,we would like to thank (in alphabetical order) Chris Apelian, John Daws, Linda Lesniak,Kathleen Madden, Robert Norman, and Eileen Rodriguez; for their help on the second edi-tion, we would like to thank Gabriella Belli, Patricia Busk, Ellie Buteau, Michael Karchmer,Steve Kass, and Jon Kettenring. Of course, any errors or shortcomings in the second editionremain the responsibility of the authors.

Finally, and most importantly, we would like to thank our families, Martha Lawner;Steve, Allison, and Carolyn Weinberg; Jason and Danielle Barro; Philip Korn; Susan andTony Knapp; and Dave, Michelle, and Scott Abramowitz for their enduring love, patience,and support.






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