Download Explaining Psychological Statistics

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
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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
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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
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215
216
216
217
217
218
218
219
221
223
224
225
226
226
226
227
228
230
231
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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
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296
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300
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309
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xiii