Advanced Statistics from an Elementary Point of View

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Advanced Statistics from an
Elementary Point of View
Michael J. Panik
University of Hartford
Contents
Preface xv
1 Introduction 1
1.1 Statistics Defined 1
1.2 Types of Statistics 1
1.3 Levels of Discourse: Sample vs. Population 2
1.4 Levels of Discourse: Target vs. Sampled Population 4
1.5 Measurement Scales 5
1.6 Sampling and Sampling Errors 7
1.7 Exercises 7
2 Elementary Descriptive Statistical Techniques 9
2.1 Summarizing Sets of Data Measured on a Ratio or Interval Scale 9
2.2 Tabular Methods 11
2.3 Quantitative Summary Characteristics 16
2.3.1 Measures of Central Location 16
2.3.2 Measures of Dispersion 21
2.3.3 Standardized Variables 26
2.3.4 Moments 29
2.3.5 Skewness and Kurtosis 31
2.3.6 Relative Variation 33
2.3.7 Comparison of the Mean, Median, and Mode 34
2.3.8 The Sample Variance and Standard Deviation 35
2.4 Correlation between Variables X and Y 38
2.5 Rank Correlation between Variables X and Y 42
2.6 Exercises 46
3 Probability Theory 53
3.1 Mathematical Foundations: Sets, Set Relations, and Functions 53
3.2 The Random Experiment, Events, Sample Space, and the Random
Variable 59
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关键词：Elementary Statistics statistic Advanced Element

3.3 Axiomatic Development of Probability Theory 62
3.4 The Occurrence and Probability of an Event 64
3.5 General Addition Rule for Probabilities 65
3.6 Joint, Marginal, and Conditional Probability 66
3.7 Classification of Events 72
3.8 Sources of Probabilities 77
3.9 Bayes’ Rule 79
3.10 Exercises 82
4 Random Variables and Probability Distributions 93
4.1 Random Variables 93
4.2 Discrete Probability Distributions 94
4.3 Continuous Probability Distributions 101
4.4 Mean and Variance of a Random Variable 106
4.5 Chebyshev’s Theorem for Random Variables 111
4.6 Moments of a Random Variable 113
4.7 Quantiles of a Probability Distribution 117
4.8 Moment-Generating Function 119
4.9 Probability-Generating Function 127
4.10 Exercises 132
5 Bivariate Probability Distributions 147
5.1 Bivariate Random Variables 147
5.2 Discrete Bivariate Probability Distributions 147
5.3 Continuous Bivariate Probability Distributions 154
5.4 Expectations and Moments of Bivariate Probability Distributions 162
5.5 Chebyshev’s Theorem for Bivariate Probability Distributions 169
5.6 Joint Moment-Generating Function 169
5.7 Exercises 174
6 Discrete Parametric Probability Distributions 187
6.1 Introduction 187
6.2 Counting Rules 188
6.3 Discrete Uniform Distribution 194
6.4 The Bernoulli Distribution 195
6.5 The Binomial Distribution 197
6.6 The Multinomial Distribution 203
6.7 The Geometric Distribution 206
6.8 The Negative Binomial Distribution 208
6.9 The Poisson Distribution 212
6.10 The Hypergeometric Distribution 218
6.11 The Generalized Hypergeometric Distribution 225
6.12 Exercises 226
7 Continuous Parametric Probability Distributions 235
7.1 Introduction 235
7.2 The Uniform Distribution 236
7.3 The Normal Distribution 238
7.3.1 Introduction to Normality 238
7.3.2 The Z Transformation 240
7.3.3 Moments, Quantiles, and Percentage Points 249
7.3.4 The Normal Curve of Error 253
7.4 The Normal Approximation to Binomial Probabilities 253
7.5 The Normal Approximation to Poisson Probabilities 257
7.6 The Exponential Distribution 258
7.6.1 Source of the Exponential Distribution 258
7.6.2 Features/Uses of the Exponential Distribution 260
7.7 Gamma and Beta Functions 264
7.8 The Gamma Distribution 266
7.9 The Beta Distribution 270
7.10 Other Useful Continuous Distributions 276
7.10.1 The Lognormal Distribution 276
7.10.2 The Logistic Distribution 279
7.11 Exercises 285
8 Sampling and the Sampling Distribution
of a Statistic 293
8.1 The Purpose of Random Sampling 293
8.2 Sampling Scenarios 294
8.2.1 Data Generating Process or Infinite Population 294
8.2.2 Drawings from a Finite Population 299
8.3 The Arithmetic of Random Sampling 301
8.4 The Sampling Distribution of a Statistic 306
8.5 The Sampling Distribution of the Mean 308
8.5.1 Sampling from an Infinite Population 309
8.5.2 Sampling from a Finite Population 311
8.6 A Weak Law of Large Numbers 316
8.7 Convergence Concepts 319
8.8 A Central Limit Theorem 322
8.9 The Sampling Distribution of a Proportion 326
8.10 The Sampling Distribution of the Variance 333
8.11 A Note on Sample Moments 338
8.12 Exercises 342
9 The Chi-Square, Student’s t, and
Snedecor’s F Distributions 349
9.1 Derived Continuous Parametric Distributions 349
9.2 The Chi-Square Distribution 350
9.3 The Sampling Distribution of the Variance When Sampling from a Normal
Population 354
9.4 Student’s t Distribution 357
9.5 Snedecor’s F Distribution 362
9.6 Exercises 368
10 Point Estimation and Properties of Point
Estimators 373
10.1 Statistics as Point Estimators 373
10.2 Desirable Properties of Estimators as Statistical Properties 375
10.3 Small Sample Properties of Point Estimators 376
10.3.1 Unbiased, Minimum Variance, and Minimum Mean Squared
Error (MSE) Estimators 376
10.3.2 Efficient Estimators 383
10.3.3 Most Efficient Estimators 385
10.3.4 Sufficient Statistics 394
10.3.5 Minimal Sufficient Statistics 398
10.3.6 On the Use of Sufficient Statistics 399
10.3.7 Completeness 401
10.3.8 Best Linear Unbiased Estimators 404
10.3.9 Jointly Sufficient Statistics 405
10.4 Large Sample Properties of Point Estimators 408
10.4.1 Asymptotic or Limiting Properties 408
10.4.2 Asymptotic Mean and Variance 410
10.4.3 Consistency 411
10.4.4 Asymptotic Efficiency 416
10.4.5 Asymptotic Normality 418
10.5 Techniques for Finding Good Point Estimators 419
10.5.1 Method of Maximum Likelihood 419
10.5.2 Method of Least Squares 430
10.6 Exercises 431
11 Interval Estimation and Confidence Interval
Estimates 439
11.1 Interval Estimators 439
11.2 Central Confidence Intervals 441
11.3 The Pivotal Quantity Method 442
11.4 A Confidence Interval for μ Under Random Sampling from a Normal
Population with Known Variance 443
11.5 A Confidence Interval for μ Under Random Sampling from a Normal
Population with Unknown Variance 446
11.6 A Confidence Interval for σ2 Under Random Sampling from a Normal
Population with Unknown Mean 447
11.7 A Confidence Interval for p Under Random Sampling from a Binomial
Population 451
11.8 Joint Estimation of a Family of Population Parameters 455
11.9 Confidence Intervals for the Difference of Means When Sampling from
Two Independent Normal Populations 458
11.9.1 Population Variances Known 461
11.9.2 Population Variances Unknown But Equal 461
11.9.3 Population Variances Unknown and Unequal 462
11.10 Confidence Intervals for the Difference of Means When Sampling from
Two Dependent Populations: Paired Comparisons 464
11.11 Confidence Intervals for the Difference of Proportions When Sampling
from Two Independent Binomial Populations 470
11.12 Confidence Interval for the Ratio of Two Variances When Sampling
from Two Independent Normal Populations 471
11.13 Exercises 473
12 Tests of Parametric Statistical Hypotheses 483
12.1 Statistical Inference Revisited 483
12.2 Fundamental Concepts for Testing Statistical Hypotheses 484
12.3 What Is the Research Question? 486
12.4 Decision Outcomes 487
12.5 Devising a Test for a Statistical Hypothesis 488
12.6 The Classical Approach to Statistical Hypothesis Testing 491
12.7 Types of Tests or Critical Regions 493
12.8 The Essentials of Conducting a Hypothesis Test 495
12.9 Hypothesis Test for μ Under Random Sampling from a Normal Population
with Known Variance 496
12.10 Reporting Hypothesis Test Results 501
12.11 Determining the Probability of a Type II Error β 504
12.12 Hypothesis Tests for μ Under Random Sampling from a Normal Population
with Unknown Variance 510
12.13 Hypothesis Tests for p Under Random Sampling from a Binomial
Population 512
12.14 Hypothesis Tests for σ2 Under Random Sampling from a Normal
Population 516
12.15 The Operating Characteristic and Power Functions of a Test 519
12.16 Determining the Best Test for a Statistical Hypothesis 528
12.17 Generalized Likelihood Ratio Tests 537
12.18 Hypothesis Tests for the Difference of Means When Sampling from
Two Independent Normal Populations 546
12.18.1 Population Variances Equal and Known 547
12.18.2 Population Variances Unequal But Known 547
12.18.3 Population Variances Equal But Unknown 548
12.18.4 Population Variances Unequal and Unknown 549
12.19 Hypothesis Tests for the Difference of Means When Sampling from Two
Dependent Populations: Paired Comparisons 553
16 Bivariate Linear Regression and Correlation 669
16.1 The Regression Model 669
16.2 The Strong Classical Linear Regression Model 670
16.3 Estimating the Slope and Intercept of the Population
Regression Line 673
16.4 Mean, Variance, and Sampling Distribution of the Least Squares Estimators
ˆ β0 and ˆ β1 676
16.5 Precision of the Least Squares Estimators ˆ β0, ˆ β1:
Confidence Intervals 679
16.6 Testing Hypotheses Concerning β0, β1 680
16.7 The Precision of the Entire Least Squares Regression Equation:
A Confidence Band 684
16.8 The Prediction of a Particular Value of Y Given X 687
16.9 Decomposition of the Sample Variation of Y 691
16.10 The Correlation Model 695
16.11 Estimating the Population Correlation Coefficient ρ 697
16.12 Inferences about the Population Correlation Coefficient ρ 698
16.13 Exercises 705
Appendix A 717
Table A.1 Standard Normal Areas 718
Table A.2 Cumulative Distribution Function Values for the Standard Normal
Distribution 719
Table A.3 Quantiles of Student’s t Distribution 721
Table A.4 Quantiles of the Chi-Square Distribution 722
Table A.5 Quantiles of Snedecor’s F Distribution 724
Table A.6 Binomial Probabilities 727
Table A.7 Cumulative Distribution Function Values for the Binomial
Distribution 733
Table A.8 Poisson Probabilities 738
Table A.9 Fisher’s ρˆ(= r) to ξ Transformation 744
Table A.10 R Distribution for the Runs Test of Randomness 745
Table A.11 W+ Distribution for the Wilcoxon Signed Rank Test 746
Table A.12 R1 Distribution for the Mann-Whitney Rank-Sum Test 747
Table A.13 Quantiles of the Lilliefors Test Statistic ˆDn 756
Table A.14 Quantiles of the Kolmogorov-Smirnov Test Statistic Dn 757
Table A.15 Quantiles of the Kolmogorov-Smirnov Test Statistic Dn,m
When n = m 758
Table A.16 Quantiles of the Kolmogorov-Smirnov Test Statistic Dn,m
When n
= m 759
Table A.17 Quantiles of the Shapiro-Wilk Test Statistic W 761
Table A.18 Coefficients for the Shapiro-Wilk Test 762
Table A.19 Durbin-Watson DW Statistic 764
Table A.20 D Distribution of the von Neumann Ratio of the Mean Square
Successive Difference to the Variance 766
Solutions to Selected Exercises 767
References and Suggested Reading 785
Index 789

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