Statistical Inference: A Short Course

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

Chapter 1 The Nature of Statistics

Statistics Defined

The Population and the Sample

Selecting a Sample from a Population

Measurement Scales

Let?s Add!

Exercises

Chapter 2 Analyzing Quantitative Data

2.1 Imposing Order

2.2 Tabular and Graphical Techniques: Ungrouped Data

2.3 Tabular and Graphical Techniques: Grouped Data

2.4 Exercises

Appendix 2.A Histograms With Classes of Different Lengths

Chapter 3 Descriptive Characteristics of Quantitative Data

3.1 The Search for Summary Characteristics

3.2 Arithmetic Mean

3.3 The Median

3.4 The Mode

3.5 The Range

3.6 The Standard Deviation

3.7 Relative Variation

3.8 Skewness

3.9 Quantiles (Quartiles, Deciles, and Percentiles)

3.10 Kurtosis

3.11 Detection of Outliers

3.12 So What Do We Do With All This Stuff?

3.13 Exercises

Appendix 3.A Descriptive Characteristics of Grouped Data

The Arithmetic Mean

The Median

The Mode

The Standard Deviation

Quantiles (Quartiles, Deciles, and Percentiles)

Chapter 4 Essentials of Probability

4.1 Set Notation

4.2 Events Within the Sample Space

4.3 Basic Probability Calculations

4.4 Sources of Probabilities

4.5 Exercises

Chapter 5 Discrete Probability Distributions and Their Properties

5.1 The Discrete Probability Distribution

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

5.3 The Binomial Probability Distribution

A. Counting Issues

B. The Bernoulli Probability Distribution

C. The Binomial Probability Distribution

5.4 Exercises

Chapter 6 The Normal Distribution

6.1 The Continuous Probability Distribution

6.2 The Normal Distribution

6.3 Probability as an Area Under the Normal Curve

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

6.5 Exercises

Appendix 6.A The Normal Approximation to Binomial Probabilities

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

7.1 Simple Random Sampling

7.2 The Sampling Distribution of the Mean

7.3 Comments on the Sampling Distribution of the Mean

7.4 A Central Limit Theorem

7.5 Exercises

Appendix 7.A Using a Table of Random Numbers

Appendix 7.B Assessing Normality via the Normal Probability Plot

Appendix 7.C Randomness, Risk, and Uncertainty

Chapter 8 Confidence Interval Estimation of ?

8.1 The Error Bound on as an Estimator of ?

8.2 A Confidence Interval for the Population Mean ? (å Known)

8.3 A Sample Size Requirements Formula

8.4 A Confidence Interval for the Population Mean ? (å Unknown)

8.5 Exercises Appendix 8.A A Confidence Interval for the Population Median Chapter 9 The Sampling Distribution of a Proportion and its Confidence Interval Estimation. 9.1 The Sampling Distribution of a Proportion 9.2 The Error Bound onas an Estimator for p 9.3 A Confidence Interval for the Population Proportion (of Successes) p 9.4 A Sample Size Requirements Formula 9.5 Exercises Appendix 9.A Ratio Estimation Chapter 10 Testing Statistical Hypotheses 10.1 What is a Statistical Hypothesis 10.2 Errors in Testing 10.3 The Contextual Framework of Hypothesis Testing 10.4 Selecting a Test Statistic 10.5 The Classical Approach to Hypothesis Testing 10.6 Types of Hypothesis Tests 10.7 Hypothesis Tests for ? (å Known) 10.8 Hypothesis Tests for ? (å Unknown) 10.9 Reporting the Results of Statistical Hypothesis Tests 10.10 Hypothesis Tests for the Population Proportion (of Successes) p 10.11 Exercises

Appendix 10.A Assessing the Randomness of a Sample

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

Appendix 10.C Lilliefors? Goodness–of–Fit Test for Normality

Chapter 11 Comparing Two Population Means and Two Population Proportions

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

A. Sampling from Two Independent Normal Populations With Equal and Known Variances

B. Sampling from Two Independent Normal Populations With Unequal But Known Variances

C. Sampling from Two Independent Normal Populations With Equal But

Unknown Variances

D. Sampling from Two Independent Normal Populations With Unequal and Unknown Variances

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

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

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

A. Population Variances Equal and Known

B. Population Variances Unequal But Known

C. Population Variances Equal and Unknown

D. Population Variances Unequal and Unknown (an Approximate Test)

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

11.6 Hypothesis Test for the Difference of Proportions When Sampling from Two Independent Binomial Populations

11.7 Exercises

Appendix 11.A A Runs Test for Two Independent Samples

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

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

Chapter 12 Bivariate Regression and Correlation

12.1 Introducing an Additional Dimension to Our Statistical Analysis

12.2 Linear Relationships

12.2.1 Exact Linear Relationships

12.2.2 A Statistical Equation

12.3 Estimating the Slope and Intercept of the Population Regression Line

12.4 Decomposition of the Sample Variation in Y 12.5 Mean, Variance, and Sampling distribution of the Least Squares Estimators and

12.6 Confidence Intervals for and

12.7 Testing Hypotheses About and

12.8 The Prediction of a Particular Valve of Y Given X

12.9 Correlation Analysis

12.9.1 Case A: X and Y Random Variables

12.9.1.a Estimating the Population Correlation Coefficient ?

12.9.1.b Inferences About the Population Correlation Coefficient ?

12.9.2 Case B: X Values Fixed, Y a Random Variable

12.10 Exercises

Appendix 12.A Assessing Normality via Regression: A continuation of Appendix 7.B

Appendix 12.B On Making Causal Inferences

Chapter 13 An Assortment of Additional Statistical Tests

13.1 Distributional Hypotheses

13.2 The Multinomial Chi–Square Statistic

13.3 The Chi–Square Distribution

13.4 Testing Goodness of Fit

13.5 Testing Independence

13.6 Testing k proportions

13.7 A Measure of Strength of Association in a Contingency Table

13.8 A Confidence Interval for Under Random Sampling From a Normal Population

13.9 The F Distribution

13.10 Applications of the F Statistic to Regression Analysis

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

13.10.2 A Joint Test of the Regression Intercept and Slope

13.11 Exercises

Appendix A

Table A.1 Standard Normal Areas

Table A.2 Quantiles of the t Distribution

Table A.3 Quantiles of the Chi–Square Distribution

Table A.4 Quantiles of the F Distribution

Table A.5 Binomial Probabilities

Table A.6 Cumulative Distribution Function Values for the Binomial Distribution

Table A.7 Quantiles of Lilliefors? Test for Normality

Solutions to Exercises

References

Index

MICHAEL J. PANIK, PhD, is Professor Emeritus in the Department of Economics at the University of Hartford. He has served as a consultant to the Connecticut Department of Motor Vehicles as well as a variety of healthcare organizations. Dr. Panik has published numerous journal articles in the areas of economics, mathematics, and applied econometrics.