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EXPLAINING PSYCHOLOGICAL STATISTICS FOURTH EDITION EXPLAINING PSYCHOLOGICAL STATISTICS FOURTH EDITION Barry H. Cohen Cover images: landscape image C iStockphoto.com/William Walsh abstract swoosh image C iStockphoto.com/Chung Lim Dave Cho Cover design: Andy Liefer This book is printed on acid-free paper. Copyright C 2013 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada All screen capture images featured in ‘‘Analysis by SPSS’’ sections are reprinted courtesy of International Business Machines, C International Business Machines Corporation. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with the respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for damages arising herefrom. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Library of Congress Cataloging-in-Publication Data: Cohen, Barry H. Explaining psychological statistics / Barry H. Cohen, New York University. —Fourth Edition. pages cm.—(Coursesmart) Includes bibliographical references and index. ISBN 978-1-118-43660-8 (hardback : alk. paper) ISBN 978-1-118-25950-4 (ebk.) ISBN 978-1-118-23485-3 (ebk.) ISBN 978-1-118-22110-5 (ebk.) 1. Psychometrics. 2. Psychology—Mathematical models. 3. Statistics—Study and teaching (Higher). I. Title. BF39.C56 2013 150.1′ 5195—dc23 2013028064 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 For Leona CONTENTS Preface to the Fourth Edition Acknowledgments PART One DESCRIPTIVE STATISTICS Chapter 1 INTRODUCTION TO PSYCHOLOGICAL STATISTICS A. B. C. Conceptual Foundation What Is (Are) Statistics? Statistics and Research Variables and Constants Scales of Measurement Parametric Versus Nonparametric Statistics Likert Scales and the Measurement Controversy Continuous Versus Discrete Variables Scales Versus Variables Versus Underlying Constructs Independent Versus Dependent Variables Experimental Versus Observational Research Populations Versus Samples Statistical Formulas Summary Exercises Basic Statistical Procedures Variables With Subscripts The Summation Sign Properties of the Summation Sign Rounding Off Numbers Summary Exercises Analysis by SPSS Ihno’s Data Variable View Data Coding Missing Values Computing New Variables Reading Excel Files Into SPSS Exercises Chapter 2 xxiii xxix 1 1 1 1 2 2 3 6 7 8 8 9 10 11 12 12 13 14 14 15 16 18 19 20 21 21 22 23 23 24 24 25 FREQUENCY TABLES, GRAPHS, AND DISTRIBUTIONS 27 A. 27 27 28 Conceptual Foundation Frequency Distributions The Cumulative Frequency Distribution The Relative Frequency and Cumulative Relative Frequency Distributions 29 vii viii CONTENTS B. C. The Cumulative Percentage Distribution Percentiles Graphs Real Versus Theoretical Distributions Summary Exercises Basic Statistical Procedures Grouped Frequency Distributions Apparent Versus Real Limits Constructing Class Intervals Choosing the Class Interval Width Choosing the Limits of the Lowest Interval Relative and Cumulative Frequency Distributions Cumulative Percentage Distribution Estimating Percentiles and Percentile Ranks by Linear Interpolation Graphing a Grouped Frequency Distribution Guidelines for Drawing Graphs of Frequency Distributions Summary Exercises Analysis by SPSS Creating Frequency Distributions Percentile Ranks and Missing Values Graphing Your Distribution Obtaining Percentiles The Split File Function Stem-and-Leaf Plots Exercises Chapter 3 29 30 30 34 35 37 38 38 39 39 39 40 41 41 42 43 44 46 47 48 48 50 50 52 52 53 55 MEASURES OF CENTRAL TENDENCY AND VARIABILITY 57 A. 57 57 61 69 73 75 76 76 Conceptual Foundation Measures of Central Tendency Measures of Variability Skewed Distributions Summary Exercises B. Basic Statistical Procedures Formulas for the Mean Computational Formulas for the Variance and Standard Deviation Obtaining the Standard Deviation Directly From Your Calculator Properties of the Mean Properties of the Standard Deviation Measuring Skewness Measuring Kurtosis Summary Exercises C. Analysis by SPSS Summary Statistics Using Explore to Obtain Additional Statistics Boxplots Selecting Cases Exercises Key Formulas 77 80 81 83 84 85 87 88 89 89 90 91 94 96 96 CONTENTS Chapter 4 STANDARDIZED SCORES AND THE NORMAL DISTRIBUTION 99 A. 99 99 101 101 102 103 104 Conceptual Foundation z Scores Finding a Raw Score From a z Score Sets of z Scores Properties of z Scores SAT, T, and IQ Scores The Normal Distribution Introducing Probability: Smooth Distributions Versus Discrete Events Real Distributions Versus the Normal Distribution z Scores as a Research Tool Sampling Distribution of the Mean Standard Error of the Mean Sampling Distribution Versus Population Distribution Summary Exercises B. Basic Statistical Procedures Finding Percentile Ranks Finding the Area Between Two z Scores Finding the Raw Scores Corresponding to a Given Area Areas in the Middle of a Distribution From Score to Proportion and Proportion to Score Describing Groups Probability Rules Summary Advanced Material: The Mathematics of the Normal Distribution Exercises C. Analysis by SPSS Creating z Scores Obtaining Standard Errors Obtaining Areas of the Normal Distribution Data Transformations Exercises Key Formulas PART Two ONE- AND TWO-SAMPLE HYPOTHESIS TESTS 106 107 108 109 110 111 112 113 115 115 116 118 119 119 120 122 125 127 128 130 130 130 131 131 132 132 135 Chapter 5 INTRODUCTION TO HYPOTHESIS TESTING: THE ONE-SAMPLE z TEST A. Conceptual Foundation Selecting a Group of Subjects The Need for Hypothesis Testing The Logic of Null Hypothesis Testing The Null Hypothesis Distribution The Null Hypothesis Distribution for the One-Sample Case z Scores and the Null Hypothesis Distribution Statistical Decisions 135 135 135 136 137 137 138 139 140 ix x CONTENTS The z Score as Test Statistic Type I and Type II Errors The Trade-Off Between Type I and Type II Errors One-Tailed Versus Two-Tailed Tests Summary Exercises B. Basic Statistical Procedures Step 1: State the Hypothesis Step 2: Select the Statistical Test and the Significance Level Step 3: Select the Sample and Collect the Data Step 4: Find the Region of Rejection Step 5: Calculate the Test Statistic Step 6: Make the Statistical Decision Interpreting the Results Assumptions Underlying the One-Sample z Test Varieties of the One-Sample Test Why the One-Sample Test Is Rarely Performed Publishing the Results of One-Sample Tests Summary Exercises Advanced Material: Correcting Null Hypothesis Testing Fallacies Advanced Exercises C. Analysis by SPSS The One-Sample z Test Testing the Normality Assumption Exercises Key Formulas Chapter 6 INTERVAL ESTIMATION AND THE t DISTRIBUTION A. B. Conceptual Foundation The Mean of the Null Hypothesis Distribution When the Population Standard Deviation Is Not Known Calculating a Simple Example The t Distribution Degrees of Freedom and the t Distribution Critical Values of the t Distribution Calculating the One-Sample t Test Sample Size and the One-Sample t Test Uses for the One-Sample t Test Cautions Concerning the One-Sample t Test Estimating the Population Mean Summary Exercises Advanced Material: A Note About Estimators Basic Statistical Procedures Step 1: Select the Sample Size Step 2: Select the Level of Confidence Step 3: Select the Random Sample and Collect the Data Step 4: Calculate the Limits of the Interval Relationship Between Interval Estimation and Null Hypothesis Testing 141 142 143 144 147 147 148 149 150 150 151 152 153 154 155 157 158 159 160 162 163 168 169 169 170 171 172 173 173 174 174 175 175 177 178 179 179 180 180 182 183 184 185 185 186 186 186 186 190 CONTENTS Assumptions Underlying the One-Sample t Test and the Confidence Interval for the Population Mean Use of the Confidence Interval for the Population Mean Publishing the Results of One-Sample t Tests Summary Exercises C. Analysis by SPSS Performing a One-Sample t Test Confidence Intervals for the Population Mean Bootstrapping Exercises Key Formulas Chapter 7 THE t TEST FOR TWO INDEPENDENT SAMPLE MEANS A. B. Conceptual Foundation Null Hypothesis Distribution for the Differences of Two Sample Means Standard Error of the Difference Formula for Comparing the Means of Two Samples Null Hypothesis for the Two-Sample Case The z Test for Two Large Samples Separate-Variances t Test The Pooled-Variances Estimate The Pooled-Variances t Test Formula for Equal Sample Sizes Calculating the Two-Sample t Test Interpreting the Calculated t Limitations of Statistical Conclusions Summary Exercises Basic Statistical Procedures Step 1: State the Hypotheses Step 2: Select the Statistical Test and the Significance Level Step 3: Select the Samples and Collect the Data Step 4: Find the Region of Rejection Step 5: Calculate the Test Statistic Step 6: Make the Statistical Decision Interpreting the Results Confidence Intervals for the Difference Between Two Population Means Assumptions of the t Test for Two Independent Samples HOV Tests and the Separate-Variances t Test Random Assignment and the Separate-Variances t Test When to Use the Two-Sample t Test When to Construct Confidence Intervals Heterogeneity of Variance as an Experimental Result Publishing the Results of the Two-Sample t Test Summary Exercises Advanced Material: Finding the Degrees of Freedom for the Separate-Variances t Test Advanced Exercises 191 193 194 194 195 196 196 198 198 200 200 203 203 204 205 206 207 208 209 209 210 211 211 212 213 213 214 215 215 216 216 217 217 218 218 219 221 223 224 225 226 226 226 227 228 230 231 xi xii CONTENTS C. Analysis by SPSS Performing the Two-Independent-Samples t Test Confidence Interval for the Difference of Two Population Means Bootstrapping Exercises Key Formulas Chapter 8 STATISTICAL POWER AND EFFECT SIZE A. Conceptual Foundation The Alternative Hypothesis Distribution The Expected t Value (Delta) The Effect Size Power Analysis The Interpretation of t Values Estimating Effect Size Manipulating Power Summary Exercises B. Basic Statistical Procedures Using Power Tables The Relationship Between Alpha and Power Power Analysis With Fixed Sample Sizes Sample Size Determination The Case of Unequal Sample Sizes The Power of a One-Sample Test Constructing Confidence Intervals for Effect Sizes Calculating Power Retrospectively Meta-Analysis Summary Exercises Advanced Material: When Is Null Hypothesis Testing Useful? C. Analysis by SPSS Power Calculations in SPSS G*Power 3 Exercises Key Formulas PART 237 237 237 239 241 242 243 244 246 246 247 248 248 249 250 251 252 253 254 255 256 257 258 259 265 265 267 268 269 Three HYPOTHESIS TESTS INVOLVING TWO MEASURES EACH SUBJECT ON Chapter 9 LINEAR CORRELATION A. 232 232 233 233 233 234 Conceptual Foundation Perfect Correlation Negative Correlation The Correlation Coefficient Linear Transformations 271 271 271 271 272 272 274 CONTENTS Graphing the Correlation Dealing With Curvilinear Relationships Problems in Generalizing From Sample Correlations Correlation Does Not Imply Causation True Experiments Involving Correlation Summary Exercises B. Basic Statistical Procedures The Covariance The Unbiased Covariance An Example of Calculating Pearson’s r Which Formula to Use Testing Pearson’s r for Significance Understanding the Degrees of Freedom Assumptions Associated With Pearson’s r Uses of the Pearson Correlation Coefficient Publishing the Results of Correlational Studies The Power Associated With Correlational Tests Summary Exercises C. Analysis by SPSS Creating Scatterplots Computing Pearson’s r The Listwise Option Using the Syntax Window for More Options Using the Keyword ‘‘With’’ to Reduce the Size of Your Correlation Matrix Bootstrapping Exercises Key Formulas Chapter 10 LINEAR REGRESSION A. B. Conceptual Foundation Perfect Predictions Predicting With z Scores Calculating an Example Regression Toward the Mean Graphing Regression in Terms of z Scores The Raw-Score Regression Formula The Slope and the Y Intercept Predictions Based on Raw Scores Interpreting the Y Intercept Quantifying the Errors Around the Regression Line The Variance of the Estimate Explained and Unexplained Variance The Coefficient of Determination The Coefficient of Nondetermination Calculating the Variance of the Estimate Summary Exercises Basic Statistical Procedures Life Insurance Rates Regression in Terms of Sample Statistics 274 275 277 279 280 280 281 283 283 284 284 285 285 287 288 289 290 291 293 294 296 296 296 298 298 299 300 301 302 303 303 303 304 304 305 305 306 307 308 309 309 310 311 312 312 312 313 313 314 314 315 xiii