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Table of Contents
Preface X
The Statistical Imagination 1
Introduction 1
The Statistical Imagination 3
Linking the Statistical Imagination to the Sociological Imagination 4
Statistical Norms and Social Norms 4
Statistical Ideals and Social Values 5
Statistics and Science: Tools for Proportional Thinking 7
Descriptive and Inferential Statistics 7
What Is Science? 8
Scientific Skepticism and the Statistical Imagination 9
Conceiving of Data 10
The Research Process 13
Proportional Thinking: Calculating Proportions, Percentages, and Rates 15
How to Succeed in This Course and Enjoy It 20
Statistical Follies and Fallacies: The Problem of Small Denominators 21
Organizing Data to Minimize Statistical Error 36
Introduction 36
Controlling Sampling Error 37
Careful Statistical Estimation versus Hasty Guesstimation 40
Sampling Error and Its Management with Probability Theory 41
Controlling Measurement Error 42
Levels of Measurement: Careful Selection of Statistical Procedures 42
Measurement 42
Nominal Variables 43
Ordinal Variables 44
Interval Variables 44
Ratio Variables 45
Improving the Level of Measurement 47
Distinguishing Level of Measurement and Unit of Measure 47
Coding and Counting Observations 48
Frequency Distributions 50
Standardizing Score Distributions 51
Coding and Counting Interval/Ratio Data 52
Rounding Interval/Ratio Observations 53
The Real Limits of Rounded Scores 53
Proportional and Percentage Frequency Distributions for Interval/Ratio Variables 55
Cumulative Percentage Frequency Distributions 56
Percentiles and Quartiles 58
Grouping Interval/Ratio Data 60
Statistical Follies and Fallacies: The Importance of Having a Representative Sample 61
Charts and Graphs: A Picture Says a Thousand Words 78
Introduction: Pictorial Presentation of Data 78
Graphing and Table Construction Guidelines 79
Graphing Nominal/Ordinal Data 80
Pie Charts 80
Bar Charts 83
Graphing Interval/Ratio Variables 86
Histograms 86
Polygons and Line Graphs 89
Using Graphs with Inferential Statistics and Research Applications 93
Statistical Follies and Fallacies: Graphical Distortion 94
Measuring Averages 107
Introduction 107
The Mean 108
Proportional Thinking about the Mean 109
Potential Weaknesses of the Mean: Situations Where Reporting It Alone May Mislead 111
The Median 112
Potential Weaknesses of the Median: Situations Where Reporting It Alone May Mislead 114
The Mode 115
Potential Weaknesses of the Mode: Situations Where Reporting It Alone May Mislead 116
Central Tendency Statistics and the Appropriate Level of Measurement 117
Frequency Distribution Curves: Relationships Among the Mean, Median, and Mode 118
The Normal Distribution 118
Skewed Distributions 119
Using Sample Data to Estimate the Shape of a Score Distribution in a Population 120
Organizing Data for Calculating Central Tendency Statistics 122
Spreadsheet Format for Calculating Central Tendency Statistics 122
Frequency Distribution Format for Calculating the Mode 123
Statistical Follies and Fallacies: Mixing Subgroups in the Calculation of the Mean 124
Measuring Dispersion or Spread in a Distribution of Scores 136
Introduction 136
The Range 138
Limitations of the Range: Situations Where Reporting It Alone May Mislead 139
The Standard Deviation 139
Proportional and Linear Thinking about the Standard Deviation 140
Limitations of the Standard Deviation 145
The Standard Deviation as an Integral Part of Inferential Statistics 147
Why Is It Called the "Standard" Deviation? 148
Standardized Scores (ZScores) 148
The Standard Deviation and the Normal Distribution 150
Tabular Presentation of Results 153
Statistical Follies and Fallacies: What Does It Indicate When the Standard Deviation Is Larger than the Mean? 154
Probability Theory and the Normal Probability Distribution 168
Introduction: The Human Urge to Predict the Future 168
What Is a Probability? 170
Basic Rules of Probability Theory 172
Probabilities Always Range Between 0 and 1 172
The Addition Rule for Alternative Events 172
Adjust for Joint Occurrences 173
The Multiplication Rule for Compound Events 174
Account for Replacement with Compound Events 174
Using the Normal Curve as a Probability Distribution 176
Proportional Thinking about a Group of Cases and Single Cases 176
Partitioning Areas Under the Normal Curve 179
Sample Problems Using the Normal Curve 181
Computing Percentiles for Normally Distributed Populations 191
The Normal Curve as a Tool for Proportional Thinking 193
Statistical Follies and Fallacies: The Gambler's Fallacy: Independence of Probability Events 194
Using Probability Theory to Produce Sampling Distributions 206
Introduction: Estimating Parameters 206
Point Estimates 207
Predicting Sampling Error 207
Sampling Distributions 209
Sampling Distributions for Interval/Ratio Variables 209
The Standard Error 211
The Law of Large Numbers 212
The Central Limit Theorem 212
Sampling Distributions for Nominal Variables 215
Rules Concerning a Sampling Distribution of Proportions 218
Bean Counting as a Way of Grasping the Statistical Imagination 219
Distinguishing Among Populations, Samples, and Sampling Distributions 221
Statistical Follies and Fallacies: Treating a Point Estimate as Though It Were Absolutely True 222
Parameter Estimation Using Confidence Intervals 237
Introduction 237
Confidence Interval of a Population Mean 240
Calculating the Standard Error for a Confidence Interval of a Population Mean 241
Choosing the Critical ZScore, Z[subscript Alpha] 242
Calculating the Error Term 243
Calculating the Confidence Interval 243
The Five Steps for Computing a Confidence Interval of a Population Mean, Mu[subscript x] 245
Proper Interpretation of Confidence Intervals 247
Common Misinterpretations of Confidence Intervals 249
The Chosen Level of Confidence and the Precision of the Confidence Interval 249
Sample Size and the Precision of the Confidence Interval 250
LargeSample Confidence Interval of a Population Proportion 252
Choosing a Sample Size for Polls, Surveys, and Research Studies 256
Sample Size for a Confidence Interval of a Population Proportion 256
Statistical Follies and Fallacies: It Is Plus and Minus the Error Term 258
Hypothesis Testing I: The Six Steps of Statistical Inference 267
Introduction: Scientific Theory and the Development of Testable Hypotheses 267
Making Empirical Predictions 268
Statistical Inference 269
The Importance of Sampling Distributions for Hypothesis Testing 272
The Six Steps of Statistical Inference for a Large SingleSample Means Test 274
Test Preparation 276
The Six Steps 276
Special Note on Symbols 287
Understanding the Place of Probability Theory in Hypothesis Testing 287
A Focus on pValues 287
The Level of Significance and Critical Regions of the Sampling Distribution Curve 288
The Level of Confidence 295
Study Hints: Organizing Problem Solutions 295
Solution Boxes Using the Six Steps 297
Interpreting Results When the Null Hypothesis Is Rejected: The Hypothetical Framework of Hypothesis Testing 301
Selecting Which Statistical Test to Use 301
Statistical Follies and Fallacies: Informed Common Sense: Going Beyond Common Sense by Observing Data 302
Hypothesis Testing II: SingleSample Hypothesis Tests: Establishing the Representativeness of Samples 315
Introduction 315
The Small SingleSample Means Test 317
The "Students' t" Sampling Distribution 317
Selecting the Critical Probability Score, t[subscript Alpha], from the tdistribution Table 321
Special Note on Symbols 321
What Are Degrees of Freedom? 322
The Six Steps of Statistical Inference for a Small SingleSample Means Test 324
Gaining a Sense of Proportion About the Dynamics of a Means Test 330
Relationships among Hypothesized Parameters, Observed Sample Statistics, Computed Test Statistics, pValues, and Alpha Levels 330
Using SingleSample Hypothesis Tests to Establish Sample Representativeness 340
Target Values for Hypothesis Tests of Sample Representativeness 340
Large SingleSample Proportions Test 344
The Six Steps of Statistical Inference for a Large SingleSample Proportions Test 346
What to Do If a Sample Is Found Not to Be Representative? 349
Presentation of Data from SingleSample Hypothesis Tests 350
A Confidence Interval of the Population Mean When n Is Small 351
Statistical Follies and Fallacies: Issues of Sample Size and Sample Representativeness 353
Bivariate Relationships: tTest for Comparing the Means of Two Groups 368
Introduction: Bivariate Analysis 368
Difference of Means Tests 369
Joint Occurrences of Attributes 370
Correlation 371
TwoGroup Difference of Means Test (tTest) for Independent Samples 371
The Standard Error and Sampling Distribution for the tTest of the Difference Between Two Means 374
The Six Steps of Statistical Inference for the TwoGroup Difference of Means Test 378
When the Population Variances (or Standard Deviations) Appear Radically Different 380
The TwoGroup Difference of Means Test for Nonindependent or MatchedPair Samples 383
The Six Steps of Statistical Inference for the TwoGroup Difference of Means Test for Nonindependent or MatchedPair Samples 388
Practical versus Statistical Significance 389
The Four Aspects of Statistical Relationships 390
Existence of a Relationship 390
Direction of the Relationship 390
Strength of the Relationship, Predictive Power, and Proportional Reduction in Error 391
Practical Applications of the Relationship 392
When to Apply the Various Aspects of Relationships 393
Relevant Aspects of a Relationship for TwoGroup Difference of Means Tests 393
Statistical Follies and Fallacies: Fixating on Differences of Means While Ignoring Differences in Variances 395
Analysis of Variance: Differences Among Means of Three or More Groups 414
Introduction 414
Calculating Main Effects 415
The General Linear Model: Testing the Statistical Significance of Main Effects 418
Determining the Statistical Significance of Main Effects Using ANOVA 421
The FRatio Test Statistic 428
How the FRatio Turns Out When Group Means Are Not Significantly Different 429
The FRatio as a Sampling Distribution 430
Relevant Aspects of a Relationship for ANOVA 432
Existence of the Relationship 432
Direction of the Relationship 432
Strength of the Relationship 433
Practical Applications of the Relationship 434
The Six Steps of Statistical Inference for OneWay ANOVA 437
Tabular Presentation of Results 442
Multivariate Applications of the General Linear Model 442
Similarities Between the tTest and the FRatio Test 443
Statistical Follies and Fallacies: Individualizing Group Findings 444
Nominal Variables: The ChiSquare and Binomial Distributions 464
Introduction: Proportional Thinking About Social Status 464
Crosstab Tables: Comparing the Frequencies of Two Nominal/Ordinal Variables 466
The ChiSquare Test: Focusing on the Frequencies of Joint Occurrences 468
Calculating Expected Frequencies 470
Differences Between Observed and Expected Cell Frequencies 470
Degrees of Freedom for the ChiSquare Test 472
The ChiSquare Sampling Distribution and Its Critical Regions 474
The Six Steps of Statistical Inference for the ChiSquare Test 475
Relevant Aspects of a Relationship for the ChiSquare Test 478
Using ChiSquare as a Difference of Proportions Test 479
Tabular Presentation of Data 481
Small SingleSample Proportions Test: The Binomial Distribution 483
The Binomial Distribution Equation 484
Shortcut Formula for Expanding the Binomial Equation 486
The Six Steps of Statistical Inference for a Small SingleSample Proportions Test: The Binomial Distribution Test 489
Statistical Follies and Fallacies: Low Statistical Power When the Sample Size Is Small 492
Bivariate Correlation and Regression: Part 1: Concepts and Calculations 509
Introduction: Improving Best Estimates of a Dependent Variable 509
A Correlation Between Two Interval/Ratio Variables 510
Identifying a Linear Relationship 511
Drawing the Scatterplot 513
Identifying a Linear Pattern 513
Using the Linear Regression Equation to Measure the Effects of X on Y 516
Pearson's r Bivariate Correlation Coefficient 518
Computational Spreadsheet for Calculating Bivariate Correlation and Regression Statistics 519
Characteristics of the Pearson's r Bivariate Correlation Coefficient 521
Understanding the Pearson's r Formulation 522
Regression Statistics 524
The Regression Coefficient or Slope, b 525
The Yintercept, a 525
Calculating the Terms of the Regression Line Formula 527
For the Especially Inquisitive: The Mathematical Relationship Between Pearson's r Correlation Coefficient and the Regression Coefficient, b 529
Statistical Follies and Fallacies The Failure to Observe a Scatterplot Before Calculating Pearson's r 531
Linear Equations Work Only with a Linear Pattern in the Scatterplot 531
Outlier Coordinates and the Attenuation and Inflation of Correlation Coefficients 532
Bivariate Correlation and Regression: Part 2: Hypothesis Testing and Aspects of a Relationship 552
Introduction: Hypothesis Test and Aspects of a Relationship Between Two Interval/Ratio Variables 552
Organizing Data for the Hypothesis Test 553
The Six Steps of Statistical Inference and the Four Aspects of a Relationship 555
Existence of a Relationship 556
Direction of the Relationship 561
Strength of the Relationship 561
Practical Applications of the Relationship 565
Careful Interpretation of Correlation and Regression Statistics 567
Correlations Apply to a Population, Not to an Individual 567
Careful Interpretation of the Slope, b 568
Distinguishing Statistical Significance from Practical Significance 568
Tabular Presentation: Correlation Tables 570
Statistical Follies and Fallacies: Correlation Does Not Always Indicate Causation 571
Review of Basic Mathematical Operations 586
Statistical Probability Tables 595
Statistical Table ARandom Number Table 595
Statistical Table BNormal Distribution Table 596
Statistical Table CtDistribution Table 598
Statistical Table DCritical Values of the FRatio Distribution at the .05 Level of Significance 599
Statistical Table ECritical Values of the FRatio Distribution at the .01 Level of Significance 600
Statistical Table FqValues of Range Tests at the .05 and .01 Levels of Significance 601
Statistical Table GCritical Values of the ChiSquare Distribution 602
Answers to Selected Chapter Exercises 603
Guide to SPSS for Windows 620
References 649
Index 654