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Tables of Contents for Statistical Reasoning in Psychology and Education
Chapter/Section Title
Page #
Page Count
CHAPTER 1 INTRODUCTION
1
14
1.1 Descriptive Statistics
2
1
1.2 Inferential Statistics
3
1
1.3 Relationship and Prediction
4
1
1.4 Our Concern: Applied Statistics
4
1
1.5 The Role of Applied Statistics
5
2
1.6 Do Statistics Lie?
7
1
8
1
Point of Controversy: Are Statistics Necessary?
9
1
1.8 Some Tips on Studying Statistics
10
1
1.9 Summary
11
4
CHAPTER 2 PRELIMINARY CONCEPTS
15
12
2.1 Random Samples
16
1
2.2 Variables and Constants
17
1
2.3 Scales of Measurement
18
3
2.4 Scales of Measurement and Problems of Statistical Treatment
21
1
2.5 Computational Accuracy with Continuous Variables
22
1
2.6 Summary
23
4
CHAPTER 3 FREQUENCY DISTRIBUTIONS, PERCENTILES, AND PERCENTILE RANKS
27
22
3.1 Organizing Qualitative Data
29
1
3.2 Grouped Scores
29
2
3.3 How to Construct a Grouped Frequency Distribution
31
2
3.4 Apparent versus Real Limits
33
1
3.5 The Relative Frequency Distribution
34
2
3.6 Stem-and-Leaf Displays
36
1
3.7 The Cumulative Frequency Distribution
37
1
3.8 Percentiles and Percentile Ranks
38
3
3.9 Computing Percentiles from Grouped Data
41
3
3.10 Computation of Percentile Rank
44
1
3.11 Summary
44
5
CHAPTER 4 GRAPHIC REPRESENTATION OF FREQUENCY DISTRIBUTIONS
49
22
4.1 Basic Procedures
49
1
4.2 The Histogram
50
2
4.3 The Frequency Polygon
52
1
4.4 Choosing Between a Histogram and a Polygon
53
3
4.5 The Bar Diagram and the Pie Chart
56
2
4.6 The Cumulative Percentage Curve
58
2
4.7 Factors Affecting the Shape of Graphs
60
2
4.8 Characteristics of Frequency Distributions
62
4
4.9 Summary
66
5
CHAPTER 5 CENTRAL TENDENCY
71
16
5.1 The Mode
72
1
5.2 The Median
72
1
5.3 The Arithmetic Mean
73
2
5.4 Properties of the Mode
75
1
5.5 Properties of the Mean
75
2
Point of Controversy: Is It Permissible to Calculate the Mean for Psychological and Educational Tests?
77
1
5.6 Properties of the Median
78
2
5.7 Measures of Central Tendency in Symmetrical and Asymmetrical Distributions
80
2
5.8 The Effects of Score Transformations
82
1
5.9 Summary
82
5
CHAPTER 6 VARIABILITY
87
22
6.1 The Range
89
1
6.2 The Semi-Interquartile Range
89
1
6.3 Deviation Scores
90
1
6.4 Deviational Measures: The Variance
91
1
6.5 Deviational Measures: The Standard Deviation
92
1
Point of Controversy: Calculating the Sample Variance: Should We Divide by n or (n -- 1)?
93
1
6.6 Calculation of the Variance and Standard Deviation: Raw-Score Method
94
2
6.7 Properties of the Range
96
1
6.8 Properties of the Semi-Interquartile Range
96
1
6.9 Properties of the Standard Deviation
97
2
6.10 Score Transformations and Measures of Variability
99
1
6.11 Standard Scores (z Scores)
99
2
6.12 Measures of Variability and the Normal Distribution
101
1
6.13 Comparing the Means of Two Distributions
102
1
6.14 Summary
103
6
CHAPTER 7 THE NORMAL CURVE
109
18
7.1 Historical Aspects of the Normal Curve
109
3
7.2 The Nature of the Normal Curve
112
2
7.3 Standard Scores and the Normal Curve
114
1
7.4 The Standard Normal Curve: Finding Areas When the Score is Known
114
4
7.5 The Standard Normal Curve: Finding Scores When the Area is Known
118
2
7.6 The Normal Curve as a Model for Real Variables
120
1
7.7 The Normal Curve as a Model for Sampling Distributions
121
1
Point of Controversy: How Normal Is the Normal Curve?
122
1
7.8 Summary
123
4
CHAPTER 8 DERIVED SCORES
127
16
8.1 The Need for Derived Scores
127
1
8.2 Standard Scores
128
2
8.3 Translating Raw Scores to Standard Scores
130
2
8.4 Standard Scores as Linear Transformations of Raw Scores
132
1
8.5 Percentile Scores
133
1
8.6 Comparability of Scores
134
2
8.7 Normalized Standard Scores
136
1
8.8 Combining Measures from Different Distributions
137
1
8.9 Summary
138
5
CHAPTER 9 CORRELATION
143
30
9.1 Some History
144
2
9.2 Graphing Bivariate Distributions: The Scatter Diagram
146
2
9.3 Correlation: A Matter of Direction
148
2
9.4 Correlation: A Matter of Degree
150
2
9.5 Understanding the Meaning of Degree of Correlation
152
1
9.6 Formulas for Pearson's Coefficient of Correlation
153
4
9.7 Calculating r from Raw Scores
157
1
9.8 Correlation Does Not Establish Causation
158
2
9.9 The Effects of Score Transformations
160
1
9.10 Cautions Concerning Correlation Coefficients
161
3
9.11 Other Ways to Measure Association
164
1
9.12 Summary
165
8
CHAPTER 10 PREDICTION
173
20
10.1 The Problem of Prediction
173
2
10.2 The Criterion of Best Fit
175
2
Point of Controversy: Least-Squares Regression versus the Resistant Line
177
1
10.3 The Regression Equation: Standard-Score Form
177
1
10.4 The Regression Equation: Raw-Score Form
178
2
10.5 Error of Prediction: The Standard Error of Estimate
180
3
10.6 An Alternative (and Preferred) Formula for S(YX)
183
1
10.7 Error in Estimating Y from X
184
2
10.8 Cautions Concerning Estimation of Predictive Error
186
1
10.9 Summary
187
6
CHAPTER 11 INTERPRETIVE ASPECTS OF CORRELATION AND REGRESSION
193
22
11.1 Factors Influencing r: Range of Talent
193
2
11.2 The Correlation Coefficient in Discontinuous Distributions
195
1
11.3 Factors Influencing r: Heterogeneity of Samples
196
1
11.4 Interpretation of r: The Regression Equation I
197
2
11.5 Interpretation of r: The Regression Equation II
199
2
11.6 Regression Problems in Research
201
1
11.7 An Apparent Paradox in Regression
202
2
11.8 Interpretation of r: Proportion of Variation in Y Not Associated with Variation in X
204
2
11.9 Interpretation of r: Proportion of Variance in Y Associated with Variance in X
206
2
11.10 Interpretation of r: Proportion of Correct Placements
208
2
11.11 Summary
210
5
CHAPTER 12 PROBABILITY
215
16
12.1 Defining Probability
215
2
12.2 A Mathematical Model of Probability
217
1
12.3 Two Theorems in Probability
218
1
12.4 An Example of a Probability Distribution: The Binomial
219
3
12.5 Applying the Binomial
222
2
12.6 The Frequency Distribution (and Normal Curve) as a Probability Distribution
224
1
12.7 Are Amazing Coincidences Really that Amazing?
225
1
12.8 Summary
226
5
CHAPTER 13 THE BASIS OF STATISTICAL INFERENCE
231
22
13.1 A Problem in Inference: Testing Hypotheses
232
1
13.2 A Problem in Inference: Estimation
232
1
13.3 Basic Issues in Inference
233
1
13.4 Random Sampling
234
2
13.5 Using a Table of Random Numbers
236
1
13.6 The Random Sampling Distribution of the Mean: An Introduction
237
3
13.7 Characteristics of the Random Sampling Distribution of the Mean
240
3
13.8 Putting the Sampling Distribution of the Mean to Use
243
4
13.9 Summary
247
6
CHAPTER 14 TESTING HYPOTHESES ABOUT SINGLE MEANS (z and t)
253
30
14.1 Testing a Hypothesis About a Single Mean
253
1
14.2 When Do We Retain and When Do We Reject the Hypothesis?
254
1
14.3 Generality of the Procedure for Hypothesis Testing
255
1
14.4 Dr. Frost's Problem: Conclusion
255
3
14.5 Review of Assumptions in Inference about a Single Mean
258
1
14.6 Estimating the Standard Error of the Mean When s is Unknown
259
2
14.7 The t Distribution
261
1
14.8 Characteristics of Student's Distribution of t
262
2
14.9 Degrees of Freedom and Student's Distribution of t
264
1
14.10 Using Student's Distribution of t
265
1
14.11 An Example: Professor Dyett's Question
266
2
14.12 Computing t from Raw Scores
268
3
14.13 Directional and Nondirectional Alternative Hypotheses
271
1
14.14 Reading Research Reports in Behavioral Science
272
1
Point of Controversy: The Bootstrap Method of Statistical Inference
273
1
14.15 Problems in Selecting a Random Sample and in Drawing Conclusions
274
1
14.16 Summary
275
8
CHAPTER 15 FURTHER CONSIDERATIONS IN HYPOTHESIS TESTING
283
16
15.1 Statement of the Hypothesis
283
1
15.2 Choice of H(A): One-Tailed and Two-Tailed Tests
284
2
15.3 The Criterion for Rejecting or Retaining H(O)
286
2
15.4 The Statistical Decision
288
1
15.5 A Statistically Significant Difference Versus a Practically Important Difference
288
2
15.6 Errors in Hypothesis Testing
290
2
15.7 Levels of Significance Versus p-Values
292
1
15.8 Summary
293
1
Point of Controversy: Dichotomous Significance-testing Decisions
294
5
CHAPTER 16 TESTING HYPOTHESES ABOUT THE DIFFERENCE BETWEEN TWO INDEPENDENT MEANS
299
24
16.1 The Random Sampling Distribution of the Difference Between Two Sample Means
301
1
16.2 An Illustration of the Sampling Distribution of the Difference Between Means
302
2
16.3 Properties of the Sampling Distribution of the Difference Between Means
304
1
16.4 Determining a Formula for t
305
3
16.5 Testing the Hypothesis of No Difference Between Two Independent Means: The Dyslexic Children Experiment
308
2
16.6 The Conduct of a One-Tailed Test
310
1
16.7 Sample Size in Inference about Two Means
311
1
16.8 Assumptions Associated with Inference about the Difference Between Two Independent Means
311
2
16.9 The Random-Sampling Model Versus the Random Assignment Model
313
1
16.10 Random Sampling and Random Assignment as Experimental Controls
314
1
16.11 The Experiment Versus the In Situ Study
315
1
16.12 Summary
316
7
CHAPTER 17 TESTING HYPOTHESES ABOUT THE DIFFERENCE BETWEEN TWO DEPENDENT MEANS
323
16
17.1 Determining a Formula for t
324
1
17.2 Degrees of Freedom for Tests of No Difference Between Dependent Means
325
1
17.3 Testing a Hypothesis about Two Dependent Means
326
2
17.4 An Alternative Approach to the Problem of Two Dependent Means
328
3
17.5 Advantages of the Dependent-Samples Design
331
1
17.6 Hazards of the Dependent-Samples Design
331
2
17.7 Summary
333
6
CHAPTER 18 ESTIMATION OF (XXX) AND (XXX)(X)--(XXX)(Y)
339
16
18.1 Two Ways of Making Estimates
340
1
18.2 Interval Estimates of (XXX)(X)
340
3
18.3 Interval Estimates of (XXX)(X)--(XXX)(Y)
343
3
18.4 Evaluating an Interval Estimate
346
1
18.5 Sample Size Required for Estimates of (XXX)(X) and (XXX)(X)--(XXX)(Y)
347
2
18.6 The Relation Between Interval Estimation and Hypothesis Testing
349
1
18.7 The Merits of Interval Estimation
350
1
18.8 Summary
351
4
CHAPTER 19 POWER AND MEASURE OF EFFECT SIZE
355
22
19.1 Type I Error and Type II Error
356
1
19.2 The Power of a Test
356
1
Point of Controversy: Failure to Publish "Nonsignificant" Results
357
1
19.3 Factors Affecting Power: Discrepancy Between the True Population Mean and the Hypothesized Mean (Size of Effect)
358
1
19.4 Factors Affecting Power: Sample Size
359
1
19.5 Factors Affecting Power: Variability of the Measure and Dependent Samples
360
1
19.6 Factors Affecting Power: Level of Significance (a)
360
1
19.7 Factors Affecting Power: One-Tailed Versus Two-Tailed Tests
361
1
19.8 Summary of Factors Affecting Power
362
1
19.9 Calculating the Power of a Test
363
1
19.10 Effect Size
364
2
19.11 Estimating Power and Sample Size for Tests of Hypotheses about Means
366
3
19.12 Some Implications of Power Curves
369
1
19.13 Reporting Inferential Statistics
369
1
Point of Controversy: Meta-Analysis
370
1
19.14 Summary
371
6
CHAPTER 20 ONE-WAY ANALYSIS OF VARIANCE (AND SOME ALTERNATIVES)
377
36
20.1 The Null Hypothesis
378
1
20.2 The Logic of One Way Analysis of Variance: Variation Within and Between Groups
379
2
20.3 Partition of Sums of Squares
381
2
20.4 Degrees of Freedom
383
1
20.5 Variance Estimates and the F Ratio
384
2
20.6 The Summary Table
386
1
20.7 An Example
387
3
20.8 Raw-Score Formulas for Analysis of Variance
390
2
20.9 Comparison of t and F
392
1
20.10 Assumptions Associated with ANOVA
392
1
20.11 ANOVA and Power
393
1
20.12 Post Hoc Comparisons
393
3
20.13 An Alternative to the F Test: Planned Comparisons
396
1
20.14 How to Construct Planned Comparisons
397
3
20.15 An Alternative for Comparing One Control Group with Several Experimental Groups: Dunnett's Test
400
1
Point of Controversy: Analysis of Variance Versus A Priori Comparisons
401
1
20.16 Analysis of Variance for Repeated Measures
402
5
20.17 Summary
407
6
CHAPTER 21 FACTORIAL ANALYSIS OF VARIANCE: THE TWO-FACTOR DESIGN
413
28
21.1 Main Effects
415
1
21.2 Interaction
416
3
21.3 The Importance of Interaction
419
1
21.4 Partition of the Sum of Squares for Two-Way ANOVA
420
4
21.5 Degrees of Freedom
424
1
21.6 Variance Estimates and F Tests
425
2
21.7 Studying the Outcome of Two-Way Analysis of Variance
427
2
21.8 Planned Comparisons
429
1
21.9 Assumptions of the Two-Factor Design and the Problem of Unequal Numbers of Scores
429
1
21.10 Mixed Two-Factor Within-Subjects Design
429
6
21.11 Summary
435
6
CHAPTER 22 INFERENCE ABOUT PEARSON CORRELATION COEFFICIENTS
441
12
22.1 The Random Sampling Distribution of r
441
1
22.2 Testing the Hypothesis that p = O
442
2
22.3 Fisher's z' Transformation
444
2
22.4 Estimating p
446
2
22.5 Testing the Hypothesis of No Difference Between p(1) and p(2): Independent Samples
448
2
450
1
22.7 Summary
450
3
CHAPTER 23 CHI-SQUARE AND INFERENCE ABOUT FREQUENCIES
453
22
23.1 A Problem in Discrepancy Between Expected and Observed Frequencies
453
2
23.2 Chi-Square (X^2) as a Measure of Discrepancy Between Expected and Observed Frequencies
455
1
23.3 The Logic of the Chi-Square Test
456
2
23.4 Interpretation of the Outcome of a Chi-Square Test
458
1
23.5 Different Hypothesized Proportions in the Test for Goodness of Fit
458
1
23.6 Assumptions in the Use of the Theoretical Distribution of Chi-Square
459
1
23.7 Hypothesis Testing When df = 1
459
1
23.8 Two Variables: Contingency Tables and the Hypothesis of Independence
460
2
23.9 Finding Expected Frequencies in a Contingency Table
462
2
23.10 Calculation of X^2 and Determination of Significance in a Contingency Table
464
3
Point of Controversy: Yates' Correction for Continuity
467
1
467
2
23.12 Other Applications of Chi-Square
469
1
23.13 Summary
470
5
CHAPTER 24 SOME (ALMOST) ASSUMPTION-FREE TESTS
475
24
24.1 Randomization Tests
476
2
24.2 How to Place Scores in Rank Order
478
1
24.3 Test of Location for Two Independent Groups: The Mann -- Whitney U Test
479
3
Point of Controversy: A Comparison of the t test and Mann -- Whitney U Test with Real-World Distributions
482
1
24.4 Test of Location Among Several Independent Groups: The Kruskal -- Wallis Test
483
1
24.5 Test of Location for Two Dependent Groups: The Sign Test
484
4
24.6 Test of Location for Two Dependent Groups: The Wilcoxon Signed-Ranks Test
488
2
24.7 Spearman's Rank-Order Correlation Coefficient
490
2
Point of Controversy: Objectivity and Subjectivity in Inferential Statistics
492
2
24.8 Summary
494
5
EPILOGUE: THE REALM OF STATISTICS
499
4
APPENDIX A REVIEW OF BASIC MATHEMATICS
503
12
APPENDIX B SUMMATION RULES
515
2
APPENDIX C LIST OF SYMBOLS
517
3
APPENDIX D ANSWERS TO ODD-NUMBERED PROBLEMS
520
17
APPENDIX E STATISTICAL ANALYSIS BY COMPUTER
537
14
APPENDIX F STATISTICAL TABLES
551
29
Table A: Areas Under the Normal Curve Corresponding to Given Values of z
552
6
Table B: The Binomial Distribution
558
3
Table C: Random Numbers
561
3
Table D: Student's t Distribution
564
2
Table E: The F Distribution
566
3
Table F: The Studentized Range Statistic
569
1
Table G: Dunnett's Test: Distribution of t Statistic in Comparing Several Treatment Means with One Control
570
2
Table H: Values of the Correlation Coefficient Required for Different Levels of Significance When H(O): p = O
572
2
Table I: Values of Fisher's z' for Values of r
574
1
Table J: The X^2 Distribution
575
2
Table K: Critical One-Tail Values of SR(X) for the Mann -- Whitney U Test
577
2
Table L: Critical Values for the Smaller of W(+) or W(-) for the Wilcoxon Signed-Ranks Tests
579
1
REFERENCES
580
5
INDEX
585