Statistics

1 INTRODUCTION TO STATISTICS 1

1 The Subject and Scope of Statistics 2

2 Statistics in Aid of Scientific Inquiry 4

3 Two Basic Concepts-Population and Sample 6

4 The Purposeful Collection of Data 11

Case Study: Statistics in Context 12

5 Objectives of Statistics 14

2 ORGANIZATION AND DESCRIPTION OF DATA 17

1 Main Types of Data 18

2 Describing Data by Tables and Graphs 19

3 Measures of Center 31

4 Measures of Variation 37

5 Checking the Stability of the Observations over Time 47

6 More on Graphics 50

Case Study: Statistics in Context 52

3 DESCRIPTIVE STUDY OF BIVARIATE DATA 64

1 Summarization of Bivariate Categorical Data 65

2 A Designed Experiment for Making a Comparison 69

3 Scatter Diagram of Bivariate Measurement Data 70

4 The Correlation Coefficient-A Measure of Linear Relation 73

5 Prediction of One Variable from another (Linear Regression) 82

4 PROBABILITY 91

1 Probability of an Event 92

2 Methods of Assigning Probability 97

3 Event Operations and Two Laws of Probability 102

4 Conditional Probability and Independence 109

5 Bayes' Theorem 117

6 Random Sampling from a Finite Population 121

Case Study: Statistics in Context 126

5 PROBABILITY DISTRIBUTIONS 134

1 Random Variables 135

2 Probability Distribution of a Discrete Random Variable 138

3 Mean (Expected Value) and Standard Deviation of a Probability Distribution 144

4 Successes and Failures-Bernoulli Trials 150

5 The Binomial Distribution 154

6 The Poisson Distribution and Rare Events 165

6 THE NORMAL DISTRIBUTION 178

1 Probability Model for a Continuous Random Variable 179

2 The Normal Distribution-Its General Features 184

3 The Standard Normal Distribution 186

4 Probability Calculations with Normal Distributions 191

5 The Normal Approximation to the Binomial 194

6 Checking the Plausibility of a Normal Model 199

7 Transforming Observations to Attain Near Normality 202

7 VARIATION IN REPEATED SAMPLES-SAMPLING DISTRIBUTIONS 210

1 The Sampling Distribution of a Statistic 212

2 Distribution of the Sample Mean and the Central Limit Theorem 218

Case Study: Statistics in Context 228

8 DRAWING INFERENCES FROM LARGE SAMPLES 235

1 Two Types of Statistical Inference: Estimation and Testing 236

2 Point Estimation of a Population Mean 238

3 Confidence Interval Estimation of a Population Mean 242

4 Testing Hypotheses about a Population Mean 250

5 Inferences about a Population Proportion 261

9 SMALL SAMPLE INFERENCES FOR NORMAL POPULATIONS 278

1 Student's t Distribution 279

2 Inferences about ;;-Small Sample Size 282

3 Relationship between Tests and Confidence Intervals 289

4 Inferences about the Standard Deviation ;; (The Chi-Square Distribution) 291

5 Robustness of Inference Procedures 296

10 COMPARING TWO TREATMENTS 303

1 Two Designs: Independent Samples and Matched Pairs Sample 305

2 Inferences about the Difference of Means-Independent Large Samples 307

3 Inferences about the Difference of Means-Independent Small Samples from Normal Populations 314

4 Randomization and Its Role in Inference 323

5 Matched Pairs Comparisons 325

6 Choosing Between Independent Samples and a Matched Pairs Sample 332

7 Comparing Two Population Proportions 333

11 REGRESSION ANALYSIS I

Simple Linear Regression 349

1 Regression with a Single Predictor 350

2 A Straight Line Regression Model 353

3 The Method of Least Squares 355

4 The Sampling Variability of the Least Squares Estimators-Tools for Inference 362

5 Important Inference Problems 363

6 The Strength of a Linear Relation 373

7 Remarks about the Straight Line Model Assumptions 377

12 REGRESSION ANALYSIS II Multiple Linear Regression and Other Topics 384

1 Nonlinear Relations and Linearizing Transformations 385

2 Multiple Linear Regression 390

3 Residual Plots to Check the Adequacy of a Statistical Model 398

13 ANALYSIS OF CATEGORICAL DATA 408

1 Formulating Testing Problems Concerning Categorical Data 409

2 Pearson's ?2 Test for Goodness of Fit 411

3 Contingency Table with One Margin Fixed (Test of Homogeneity) 415

4 Contingency Table with Neither Margin Fixed (Test of Independence) 422

14 ANALYSIS OF VARIANCE (ANOVA) 432

1 Comparison of Several Treatments-One-Way Analysis of Variance 433

2 Population Model and Inferences for a One-Way Analysis of Variance 439

3 Simultaneous Confidence Intervals 443

4 Graphical Diagnostics and Displays to Supplement ANOVA 446

5 Randomized Block Experiments for Comparing k Treatments 448

15 NONPARAMETRIC INFERENCE 460

1 The Wilcoxon Rank-Sum Test for Comparing Two Treatments 461

2 Matched Pairs Comparisons 469

3 Measure of Correlation Based on Ranks 475

4 Concluding Remarks 478

APPENDIX A1 SUMMATION NOTATION 483

APPENDIX A2 RULES FOR COUNTING 488

APPENDIX A3 EXPECTATION AND STANDARD DEVIATION-PROPERTIES 490

APPENDIX A4 THE EXPECTED VALUE AND STANDARD DEVIATION OF X 496

APPENDIX B TABLES 498

Table 1 Random Digits 498

Table 2 Cumulative Binomial Probabilities 501

Table 3 Cumulative Poisson Probabilities 507

Table 4 Standard Normal Probabilities 509

Table 5 Percentage Points t;; of t Distributions 511

Table 6 Percentage Points ?2;; of ?2 Distributions 512

Table 7 Percentage Points of F( v1, v2 ) Distributions 513

Table 8 Selected Tail Probabilities for the Null Distribution of Wilcoxon's Rank-Sum Statistic 515

Table 9 Selected Tail Probabilities for the Null Distribution of Wilcoxon's Signed-Rank Statistic 520

Table F1 General Formulas for Inferences about a Mean (;;), Difference of Two Means ( ;;1 ? ;;2 ) 522

Table F2 Inference about Proportions 523

Summary of Formulas Useful for Exams 524

DATA BANK 530

ANSWERS TO SELECTED ODD-NUMBERED EXERCISES 544

INDEX 557