# Statistical Inference: A Short Course

## Books A concise, easily accessible introduction to descriptive and inferential techniques

Statistical Inference: A Short Course offers a concise presentation of the essentials of basic statistics for readers seeking to acquire a working knowledge of statistical concepts, measures, and procedures.

The author conducts tests on the assumption of randomness and normality, provides nonparametric methods when parametric approaches might not work. The book also explores how to determine a confidence interval for a population median while also providing coverage of ratio estimation, randomness, and causality. To ensure a thorough understanding of all key concepts, Statistical Inference provides numerous examples and solutions along with complete and precise answers to many fundamental questions, including:

• How do we determine that a given dataset is actually a random sample?
• With what level of precision and reliability can a population sample be estimated?
• How are probabilities determined and are they the same thing as odds?
• How can we predict the level of one variable from that of another?
• What is the strength of the relationship between two variables?

The book is organized to present fundamental statistical concepts first, with later chapters exploring more advanced topics and additional statistical tests such as Distributional Hypotheses, Multinomial Chi-Square Statistics, and the Chi-Square Distribution. Each chapter includes appendices and exercises, allowing readers to test their comprehension of the presented material.

Statistical Inference: A Short Course is an excellent book for courses on probability, mathematical statistics, and statistical inference at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for researchers and practitioners who would like to develop further insights into essential statistical tools.

Preface xv

1 The Nature of Statistics 1

1.1 Statistics Defined 1

1.2 The Population and the Sample 2

1.3 Selecting a Sample from a Population 3

1.4 Measurement Scales 4

Exercises 7

2 Analyzing Quantitative Data 9

2.1 Imposing Order 9

2.2 Tabular and Graphical Techniques: Ungrouped Data 9

2.3 Tabular and Graphical Techniques: Grouped Data 11

Exercises 16

Appendix 2.A Histograms with Classes of Different Lengths 18

3 Descriptive Characteristics of Quantitative Data 22

3.1 The Search for Summary Characteristics 22

3.2 The Arithmetic Mean 23

3.3 The Median 26

3.4 The Mode 27

3.5 The Range 27

3.6 The Standard Deviation 28

3.7 Relative Variation 33

3.8 Skewness 34

3.9 Quantiles 36

3.10 Kurtosis 38

3.11 Detection of Outliers 39

3.12 So What Do We Do with All This Stuff? 41

Exercises 47

Appendix 3.A Descriptive Characteristics of Grouped Data 51

3.A.1 The Arithmetic Mean 52

3.A.2 The Median 53

3.A.3 The Mode 55

3.A.4 The Standard Deviation 57

3.A.5 Quantiles (Quartiles, Deciles, and Percentiles) 58

4 Essentials of Probability 61

4.1 Set Notation 61

4.2 Events within the Sample Space 63

4.3 Basic Probability Calculations 64

4.4 Joint, Marginal, and Conditional Probability 68

4.5 Sources of Probabilities 73

Exercises 75

5 Discrete Probability Distributions and Their Properties 81

5.1 The Discrete Probability Distribution 81

5.2 The Mean, Variance, and Standard Deviation of a Discrete Random Variable 85

5.3 The Binomial Probability Distribution 89

5.3.1 Counting Issues 89

5.3.2 The Bernoulli Probability Distribution 91

5.3.3 The Binomial Probability Distribution 91

Exercises 96

6 The Normal Distribution 101

6.1 The Continuous Probability Distribution 101

6.2 The Normal Distribution 102

6.3 Probability as an Area Under the Normal Curve 104

6.4 Percentiles of the Standard Normal Distribution and Percentiles of the Random Variable X 114

Exercises 116

Appendix 6.A The Normal Approximation to Binomial Probabilities 120

7 Simple Random Sampling and the Sampling Distribution of the Mean 122

7.1 Simple Random Sampling 122

7.2 The Sampling Distribution of the Mean 123

7.3 Comments on the Sampling Distribution of the Mean 127

7.4 A Central Limit Theorem 130

Exercises 132

Appendix 7.A Using a Table of Random Numbers 133

Appendix 7.B Assessing Normality via the Normal Probability Plot 136

Appendix 7.C Randomness, Risk, and Uncertainty 139

7.C.1 Introduction to Randomness 139

7.C.2 Types of Randomness 142

7.C.2.1 Type I Randomness 142

7.C.2.2 Type II Randomness 143

7.C.2.3 Type III Randomness 143

7.C.3 Pseudo-Random Numbers 144

7.C.4 Chaotic Behavior 145

7.C.5 Risk and Uncertainty 146

8 Confidence Interval Estimation of m 152

8.1 The Error Bound on X as an Estimator of m 152

8.2 A Confidence Interval for the Population Mean m (s Known) 154

8.3 A Sample Size Requirements Formula 159

8.4 A Confidence Interval for the Population Mean m (s Unknown) 160

Exercises 165

Appendix 8.A A Confidence Interval for the Population Median MED 167

9 The Sampling Distribution of a Proportion and its Confidence Interval Estimation 170

9.1 The Sampling Distribution of a Proportion 170

9.2 The Error Bound on ^p as an Estimator for p 173

9.3 A Confidence Interval for the Population Proportion (of Successes) p 174

9.4 A Sample Size Requirements Formula 176

Exercises 177

Appendix 9.A Ratio Estimation 179

10 Testing Statistical Hypotheses 184

10.1 What is a Statistical Hypothesis? 184

10.2 Errors in Testing 185

10.3 The Contextual Framework of Hypothesis Testing 186

10.3.1 Types of Errors in a Legal Context 188

10.3.2 Types of Errors in a Medical Context 188

10.3.3 Types of Errors in a Processing or Control Context 189

10.3.4 Types of Errors in a Sports Context 189

10.4 Selecting a Test Statistic 190

10.5 The Classical Approach to Hypothesis Testing 190

10.6 Types of Hypothesis Tests 191

10.7 Hypothesis Tests for m (s Known) 194

10.8 Hypothesis Tests for m (s Unknown and n Small) 195

10.9 Reporting the Results of Statistical Hypothesis Tests 198

10.10 Hypothesis Tests for the Population Proportion (of Successes) p 201

Exercises 204

Appendix 10.A Assessing the Randomness of a Sample 208

Appendix 10.B Wilcoxon Signed Rank Test (of a Median) 210

Appendix 10.C Lilliefors Goodness-of-Fit Test for Normality 213

11 Comparing Two Population Means and Two Population Proportions 217

11.1 Confidence Intervals for the Difference of Means when Sampling from Two Independent Normal Populations 217

11.1.1 Sampling from Two Independent Normal Populations with Equal and Known Variances 217

11.1.2 Sampling from Two Independent Normal Populations with Unequal but Known Variances 218

11.1.3 Sampling from Two Independent Normal Populations with Equal but Unknown Variances 218

11.1.4 Sampling from Two Independent Normal Populations with Unequal and Unknown Variances 219

11.2 Confidence Intervals for the Difference of Means when Sampling from Two Dependent Populations: Paired Comparisons 224

11.3 Confidence Intervals for the Difference of Proportions when Sampling from Two Independent Binomial Populations 227

11.4 Statistical Hypothesis Tests for the Difference of Means when Sampling from Two Independent Normal Populations 228

11.4.1 Population Variances Equal and Known 229

11.4.2 Population Variances Unequal but Known 229

11.4.3 Population Variances Equal and Unknown 229

11.4.4 Population Variances Unequal and Unknown (an Approximate Test) 230

11.5 Hypothesis Tests for the Difference of Means when Sampling from Two Dependent Populations: Paired Comparisons 234

11.6 Hypothesis Tests for the Difference of Proportions when Sampling from Two Independent Binomial Populations 236

Exercises 239

Appendix 11.A Runs Test for Two Independent Samples 243

Appendix 11.B Mann–Whitney (Rank Sum) Test for Two Independent Populations 245

Appendix 11.C Wilcoxon Signed Rank Test when Sampling from Two Dependent Populations: Paired Comparisons 249

12 Bivariate Regression and Correlation 253

12.1 Introducing an Additional Dimension to our Statistical Analysis 253

12.2 Linear Relationships 254

12.2.1 Exact Linear Relationships 254

12.3 Estimating the Slope and Intercept of the Population Regression Line 257

12.4 Decomposition of the Sample Variation in Y 262

12.5 Mean, Variance, and Sampling Distribution of the Least Squares Estimators ^b0 and ^b1 264

12.6 Confidence Intervals for b0 and b1 266

12.7 Testing Hypotheses about b0 and b1 267

12.8 Predicting the Average Value of Y given X 269

12.9 The Prediction of a Particular Value of Y given X 270

12.10 Correlation Analysis 272

12.10.1 Case A: X and Y Random Variables 272

12.10.1.1 Estimating the Population Correlation Coefficient r 274

12.10.1.2 Inferences about the Population Correlation Coefficient r 275

12.10.2 Case B: X Values Fixed, Y a Random Variable 277

Exercises 278

Appendix 12.A Assessing Normality (Appendix 7.B Continued) 280

Appendix 12.B On Making Causal Inferences 281

12.B.1 Introduction 281

12.B.2 Rudiments of Experimental Design 282

12.B.3 Truth Sets, Propositions, and Logical Implications 283

12.B.4 Necessary and Sufficient Conditions 285

12.B.5 Causality Proper 286

12.B.6 Logical Implications and Causality 287

12.B.7 Correlation and Causality 288

12.B.8 Causality from Counterfactuals 289

12.B.9 Testing Causality 292

12.B.10 Suggestions for Further Reading 294

13 An Assortment of Additional Statistical Tests 295

13.1 Distributional Hypotheses 295

13.2 The Multinomial Chi-Square Statistic 295

13.3 The Chi-Square Distribution 298

13.4 Testing Goodness of Fit 299

13.5 Testing Independence 304

13.6 Testing k Proportions 309

13.7 A Measure of Strength of Association in a Contingency Table 311

13.8 A Confidence Interval for s2 under Random Sampling from a Normal Population 312

13.9 The F Distribution 314

13.10 Applications of the F Statistic to Regression Analysis 316

13.10.1 Testing the Significance of the Regression Relationship Between X and Y 316

13.10.2 A Joint Test of the Regression Intercept and Slope 317

Exercises 318

Appendix A 323

Table A.1 Standard Normal Areas [Z is N(0,1)] 323

Table A.2 Quantiles of the t Distribution (T is tv) 325

Table A.3 Quantiles of the Chi-Square Distribution (X is w2v) 327

Table A.4 Quantiles of the F Distribution (F is Fv1;v2 ) 329

Table A.5 Binomial Probabilities P(X;n,p) 334

Table A.6 Cumulative Binomial Probabilities 338

Table A.7 Quantiles of Lilliefors’ Test for Normality 342

Solutions to Exercises 343

References 369

Index 373

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