Statistical Reasoning In The Behavioral Sciences

Cover......Page 1
Title Page......Page 5
Copyright......Page 6
Preface......Page 9
About the Book and Authors......Page 12
Contents......Page 15
1 Introduction......Page 25
1.2 Inferential Statistics......Page 27
1.3 Our Concern: Applied Statistics......Page 28
1.4 Variables and Constants......Page 29
1.5 Scales of Measurement......Page 30
1.6 Scales of Measurement and Problems of Statistical Treatment......Page 32
1.7 Do Statistics Lie?......Page 33
Point of Controversy: Are Statistical Procedures Necessary?......Page 35
1.9 Statistics and Computers......Page 36
1.10 Summary......Page 37
2.1 Organizing Qualitative Data......Page 40
2.2 Grouped Scores......Page 42
2.3 How to Construct a Grouped Frequency Distribution......Page 43
2.5 The Relative Frequency Distribution......Page 45
2.6 The Cumulative Frequency Distribution......Page 46
2.7 Percentiles and Percentile Ranks......Page 48
2.8 Computing Percentiles from Grouped Data......Page 49
2.10 Summary......Page 52
3.1 Basic Procedures......Page 56
3.2 The Histogram......Page 57
3.3 The Frequency Polygon......Page 58
3.4 Choosing between a Histogram and a Polygon......Page 59
3.5 The Bar Diagram and the Pie Chart......Page 61
3.6 The Cumulative Percentage Curve......Page 63
3.7 Factors Affecting the Shape of Graphs......Page 64
3.8 Shape of Frequency Distributions......Page 66
3.9 Summary......Page 67
4.1 The Mode......Page 70
4.2 The Median......Page 71
4.3 The Mean......Page 72
4.4 Properties of the Mode......Page 73
4.5 Properties of the Mean......Page 74
Point of Controversy: Is It Permissible to Calculate the Mean for Tests in the Behavioral Sciences?......Page 75
4.6 Properties of the Median......Page 76
4.7 Measures of Central Tendency in Symmetrical and Asymmetrical Distributions......Page 77
4.8 The Effects of Score Transformations......Page 78
4.9 Summary......Page 79
5.1 The Range and Semi-Interquartile Range......Page 82
5.2 Deviation Scores......Page 84
5.3 Deviational Measures: The Variance......Page 85
5.4 Deviational Measures: The Standard Deviation......Page 86
5.5 Calculation of the Variance and Standard Deviation: Raw-Score Method......Page 87
5.6 Calculation of the Standard Deviation with SPSS......Page 88
Point of Controversy: Calculating the Sample Variance: Should We Divide by n or (n − 1)?......Page 91
5.8 Properties of the Standard Deviation......Page 92
5.10 Score Transformations and Measures of Variability......Page 93
5.11 Standard Scores (z Scores)......Page 94
5.12 A Comparison of z Scores and Percentile Ranks......Page 97
5.13 Summary......Page 98
6.1 Historical Aspects of the Normal Curve......Page 102
6.3 Standard Scores and the Normal Curve......Page 105
Case 1. Finding the Area under the Normal Curve That Falls above a Known Score......Page 107
Case 2. Finding the Area under the Normal Curve That Falls below a Known Score......Page 108
Case 3. Finding the Area under the Normal Curve That Falls between Two Known Scores......Page 109
Case 1. Finding the Score above or below Which a Certain Percentage of the Total Scores Fall......Page 110
Case 2. Finding the Limits within Which a Certain Percentage of Scores Fall Equidistant from the Mean......Page 111
6.7 The Normal Curve as a Model for Sampling Distributions......Page 112
6.8 Summary......Page 113
7 Correlation......Page 116
7.1 Some History......Page 117
7.2 Graphing Bivariate Distributions: The Scatter Diagram......Page 119
7.3 Correlation: A Matter of Direction......Page 120
7.4 Correlation: A Matter of Degree......Page 122
7.5 Understanding the Meaning of Degree of Correlation......Page 123
7.6 Formulas for Pearson's Coefficient of Correlation......Page 124
7.7 Calculating r from Raw Scores......Page 125
7.8 Calculating r with SPSS......Page 127
7.9 Spearman's Rank-Order Correlation Coefficient......Page 130
7.10 Correlation Does Not Prove Causation......Page 131
7.12 Cautions Concerning Correlation Coefficients......Page 134
7.13 Summary......Page 138
8.1 The Problem of Prediction......Page 142
8.2 The Criterion of Best Fit......Page 144
Point of Controversy: Least-Squares Regression versus the Resistant Line......Page 145
8.3 The Regression Equation: Standard-Score Form......Page 146
8.4 The Regression Equation: Raw-Score Form......Page 147
8.5 Error of Prediction: The Standard Error of Estimate......Page 149
8.6 An Alternative (and Preferred) Formula for SYX......Page 151
8.7 Calculating the "Raw-Score" Regression Equation and Standard Error of Estimate with SPSS......Page 152
8.8 Error in Estimating Y from X......Page 154
8.9 Cautions Concerning Estimation of Predictive Error......Page 156
8.11 Summary......Page 157
9.1 Factors Influencing r: Degree of Variability in Each Variable......Page 160
9.2 Interpretation of r: The Regression Equation I......Page 161
9.3 Interpretation of r: The Regression Equation II......Page 163
9.4 Interpretation of r : Proportion of Variation in Y Not Associated with Variation in X......Page 164
9.5 Interpretation of r: Proportion of Variance in Y Associated with Variation in X......Page 166
9.6 Interpretation of r: Proportion of Correct Placements......Page 168
9.7 Summary......Page 169
10 Probability......Page 171
10.1 Defining Probability......Page 172
10.2 A Mathematical Model of Probability......Page 173
10.3 Two Theorems in Probability......Page 174
10.4 An Example of a Probability Distribution: The Binomial......Page 175
10.5 Applying the Binomial......Page 177
10.7 Are Amazing Coincidences Really That Amazing?......Page 179
10.8 Summary......Page 180
11 Random Sampling and Sampling Distributions......Page 184
11.1 Random Sampling......Page 185
11.2 Using a Table of Random Numbers......Page 187
11.3 The Random Sampling Distribution of the Mean: An Introduction......Page 188
11.4 Characteristics of the Random Sampling Distribution of the Mean......Page 190
11.5 Using the Sampling Distribution of X to Determine the Probability for Different Ranges of Values of X......Page 192
11.7 Summary......Page 197
12 Introduction to Statistical Inference: Testing Hypotheses about a Single Mean (z)......Page 199
12.2 The Null and Alternative Hypotheses......Page 200
12.5 Dr. Brown's Problem: Conclusion......Page 202
12.6 The Statistical Decision......Page 204
12.7 Choice of HA: One-Tailed and Two-Tailed Tests......Page 206
12.8 Review of Assumptions in Testing Hypotheses about a Single Mean......Page 207
Point of Controversy: The Single-Subject Research Design......Page 208
12.9 Summary......Page 209
13.1 Estimating the Standard Error of the Mean When �� Is Unknown......Page 211
13.2 The t Distribution......Page 213
13.3 Characteristics of Student's Distribution of t......Page 215
13.4 Degrees of Freedom and Student's Distribution of t......Page 216
13.5 An Example: Has the Violent Content of Television Programs Increased?......Page 217
13.6 Calculating t from Raw Scores......Page 220
13.7 Calculating t with SPSS......Page 222
13.8 Levels of Significance versus p-Values......Page 224
13.9 Summary......Page 226
14.1 A Statistically Significant Difference versus a Practically Important Difference......Page 229
Point of Controversy: The Failure to Publish “Nonsignificant” Results......Page 230
14.2 Effect Size......Page 231
14.3 Errors in Hypothesis Testing......Page 234
14.5 Factors Affecting Power: Difference between the True Population Mean and the Hypothesized Mean (Size of Effect)......Page 236
14.6 Factors Affecting Power: Sample Size......Page 237
14.9 Factors Affecting Power: One-Tailed versus Two-Tailed Tests......Page 238
14.10 Calculating the Power of a Test......Page 240
Point of Controversy: Meta-Analysis......Page 241
14.11 Estimating Power and Sample Size for Tests of Hypotheses about Means......Page 242
14.12 Problems in Selecting a Random Sample and in Drawing Conclusions......Page 244
14.13 Summary......Page 245
15.1 The Null and Alternative Hypotheses......Page 248
15.2 The Random Sampling Distribution of the Difference between Two Sample Means......Page 249
15.4 Determining a Formula for t......Page 252
15.5 Testing the Hypothesis of No Difference between Two Independent Means: The Dyslexic Children Experiment......Page 255
15.7 Calculation of t with SPSS......Page 258
15.9 Effect Size......Page 261
15.10 Estimating Power and Sample Size for Tests of Hypotheses about the Difference between Two Independent Means......Page 265
15.11 Assumptions Associated with Inference about the Difference between Two Independent Means......Page 266
15.12 The Random-Sampling Model versus the Random-Assignment Model......Page 267
15.13 Random Sampling and Random Assignment as Experimental Controls......Page 268
15.14 Summary......Page 269
16 Testing for a Difference between Two Dependent (Correlated) Groups......Page 273
16.1 Determining a Formula for t......Page 274
16.3 An Alternative Approach to the Problem of Two Dependent Means......Page 275
16.4 Testing a Hypothesis about Two Dependent Means: Does Text Messaging Impair Driving?......Page 276
16.5 Calculating t with SPSS......Page 278
16.6 Effect Size......Page 281
16.7 Power......Page 282
16.9 Problems with Using the Dependent-Samples Design......Page 283
16.10 Summary......Page 285
17.1 The Random Sampling Distribution of r......Page 288
17.2 Testing the Hypothesis That �� = 0......Page 289
17.3 Fisher’s z′ Transformation......Page 291
17.5 A Note about Assumptions......Page 292
17.7 Summary......Page 293
18 An Alternative to Hypothesis Testing: Confidence Intervals......Page 295
18.1 Examples of Estimation......Page 296
18.2 Confidence Intervals for ��X......Page 297
18.4 The Advantages of Confidence Intervals......Page 300
18.5 Random Sampling and Generalizing Results......Page 301
18.6 Evaluating a Confidence Interval......Page 302
Point of Controversy: Objectivity and Subjectivity in Inferential Statistics: Bayesian Statistics......Page 303
18.7 Confidence Intervals for ��X − ��Y......Page 304
18.8 Sample Size Required for Confidence Intervals of ��X − ��Y......Page 307
18.9 Confidence Intervals for ��......Page 309
18.10 Where Are We in Statistical Reform?......Page 310
18.11 Summary......Page 311
19 Testing for Differences among Three or More Groups: One-Way Analysis of Variance (and Some Alternatives)......Page 313
19.2 The Basis of One-Way Analysis of Variance: Variation within and between Groups......Page 315
19.3 Partition of the Sums of Squares......Page 317
19.4 Degrees of Freedom......Page 319
19.5 Variance Estimates and the F Ratio......Page 320
19.6 The Summary Table......Page 321
19.7 Example: Does Playing Violent Video Games Desensitize People to Real-Life Aggression?......Page 322
19.8 Comparison of t and F......Page 325
19.9 Raw-Score Formulas for Analysis of Variance......Page 326
19.10 Calculation of ANOVA for Independent Measures with SPSS......Page 327
19.12 Effect Size......Page 330
19.13 ANOVA and Power......Page 331
19.14 Post Hoc Comparisons......Page 332
19.16 An Alternative to the F Test: Planned Comparisons......Page 334
19.17 How to Construct Planned Comparisons......Page 335
19.18 Analysis of Variance for Repeated Measures......Page 338
19.19 Calculation of ANOVA for Repeated Measures with SPSS......Page 343
19.20 Summary......Page 345
20 Factorial Analysis of Variance: The Two-Factor Design......Page 350
20.1 Main Effects......Page 351
20.2 Interaction......Page 353
20.3 The Importance of Interaction......Page 355
20.4 Partition of the Sums of Squares for Two-Way ANOVA......Page 356
20.5 Degrees of Freedom......Page 360
20.6 Variance Estimates and F Tests......Page 361
20.7 Studying the Outcome of Two-Factor Analysis of Variance......Page 362
20.8 Effect Size......Page 364
20.9 Calculation of Two-Factor ANOVA with SPSS......Page 365
20.10 Planned Comparisons......Page 366
20.11 Assumptions of the Two-Factor Design and the Problem of Unequal Numbers of Scores......Page 367
20.12 Mixed Two-Factor Within-Subjects Design......Page 368
20.13 Calculation of the Mixed Two-Factor Within-Subjects Design with SPSS......Page 372
20.14 Summary......Page 373
21.1 The Chi-Square Test for Goodness of Fit......Page 377
21.2 Chi-Square (��2) as a Measure of the Difference between Observed and Expected Frequencies......Page 379
21.3 The Logic of the Chi-Square Test......Page 380
21.5 Different Hypothesized Proportions in the Test for Goodness of Fit......Page 382
21.6 Effect Size for Goodness-of-Fit Problems......Page 383
21.8 Chi-Square as a Test for Independence between Two Variables......Page 384
21.9 Finding Expected Frequencies in a Contingency Table......Page 386
21.10 Calculation of ��2 and Determination of Significance in a Contingency Table......Page 387
21.11 Measures of Effect Size (Strength of Association) for Tests of Independence......Page 388
Point of Controversy: Yates’ Correction for Continuity......Page 389
21.12 Power and the Chi-Square Test of Independence......Page 391
21.13 Summary......Page 392
22 Some (Almost) Assumption-Free Tests......Page 395
22.2 Randomization Tests......Page 396
22.3 Rank-Order Tests......Page 398
22.4 The Bootstrap Method of Statistical Inference......Page 399
22.5 An Assumption-Freer Alternative to the t Test of a Difference Between Two Independent Groups: The Mann-Whitney U Test......Page 400
Point of Controversy: A Comparison of the t Test and the Mann–Whitney U Test with Real-World Distributions......Page 403
22.6 An Assumption-Freer Alternative to the t Test of a Difference Between Two Dependent Groups: The Sign Test......Page 404
22.7 Another Assumption-Freer Alternative to the t Test of a Difference Between Two Dependent Groups: The Wilcoxon Signed-Ranks Test......Page 406
22.8 An Assumption-Freer Alternative to the One-Way ANOVA for Independent Groups: The Kruskal-Wallis Test......Page 408
22.9 An Assumption-Freer Alternative to ANOVA for Repeated Measures: Friedman's Rank Test for Correlated Samples......Page 411
22.10 Summary......Page 413
Epilogue......Page 416
A.2 Symbols and Their Meanings......Page 420
A.4 Squares and Square Roots......Page 421
A.5 Fractions......Page 422
A.6 Operations Involving Parentheses......Page 423
A.7 Equations in One Unknown......Page 424
A.8 Summation Rules......Page 425
A.9 Test of Mathematical Skills......Page 426
A.10 Answers to Test Questions......Page 428
Greek Letter Symbols......Page 429
English Letter Symbols......Page 430
Appendix C Answers to Problems......Page 432
Table A: Areas under the Normal Curve Corresponding to Given Values of z......Page 448
Table B: The Binomial Distribution......Page 453
Table C: Random Numbers......Page 456
Table D: Student's t Distribution......Page 458
Table E: The F Distribution......Page 460
Table F: The Studentized Range Statistic......Page 464
Table G: Values of the Correlation Coefficient Required for Different Levels of Significance When H0∶ �� = 0......Page 465
Table H: Values of Fisher's z′ for Values of r......Page 467
Table I: The ��2 Distribution......Page 468
Table J: Critical One-Tail Values of ΣRX for the Mann-Whitney U Test......Page 469
Table K: Critical Values for the Smaller of R+ or R_ for the Wilcoxon Signed-Ranks Test......Page 471
References......Page 472
Index......Page 478
EULA......Page 486