This book describes the probability theory associated with frequently used statistical procedures and the relation between probability theory and statistical inference. The first third of the book is dedicated to probability theory including topics relating to events, random variables, and the Central Limit Theorem. Statistical topics then include parameter estimation with confidence intervals, hypothesis testing, chi-square tests, t tests, and several non-parametric tests. Flow charts are frequently used to facilitate an understanding of the material considered. The examples and problems in the book all concern simple data sets which can be analyzed with a simple calculator; however, the R code required to complete many examples and problems is provided as well for those that are interested.
Author(s): Warren J. Ewens, Katherine Brumberg
Publisher: Springer
Year: 2023
Language: English
Pages: 271
City: Cham
Preface
Contents
Part I Introduction
1 Statistics and Probability Theory
1.1 What is Statistics?
1.2 The Relation Between Probability Theory and Statistics
1.3 Problems
Part II Probability Theory
2 Events
2.1 What Are Events?
2.2 Notation
2.3 Derived Events: Complements, Unions and Intersections of Events
2.4 Mutually Exclusive Events
2.5 Problems
3 Probabilities of Events
3.1 Probabilities of Derived Events
3.2 Independence of Two Events
3.3 Conditional Probabilities
3.4 Conditional Probabilities and Mutually Exclusive Events
3.5 Conditional Probabilities and Independence
Flowchart: Events
3.6 Problems
4 Probability: One Discrete Random Variable
4.1 Random Variables
4.2 Random Variables and Data
4.3 The Probability Distribution of a Discrete Random Variable
4.4 Parameters
4.5 The Binomial Distribution
4.6 The Hypergeometric Distribution
4.7 The Mean of a Discrete Random Variable
4.8 The Variance of a Discrete Random Variable
Flowchart: Means and Variances of a Single Random Variable, X or P
4.9 Problems
5 Many Random Variables
5.1 Introduction
5.2 Notation
5.3 Independently and Identically Distributed Random Variables
5.4 The Mean and Variance of a Sum and of an Average
5.5 The Mean and the Variance of a Difference
5.6 The Proportion of Successes in n Binomial Trials
Flowchart: Sums, Averages, and Differences of Random Variables
5.7 Problems
6 Continuous Random Variables
6.1 Definition
6.2 The Mean and Variance of a Continuous Random Variable
6.3 The Normal Distribution
6.4 The Standardization Procedure
6.5 Numbers that Are Seen Often in Statistics
6.6 Using the Normal Distribution Chart in Reverse
6.7 Sums, Averages and Differences of Independent Normal Random Variables
6.8 The Central Limit Theorem
6.9 Approximating Discrete Random Variable Probabilities Using the Normal Distribution
6.9.1 The Binomial Case
6.9.2 The Die Example
6.10 A Window Into Statistics
Flowchart: Normal Random Variables and the CLT
6.11 Problems
Part III Statistics
7 Introduction
8 Estimation of a Parameter
8.1 Introduction
8.2 Estimating the Binomial Parameter θ
8.2.1 Properties of Estimates and Estimators
8.2.2 The Precision of the Estimate of θ
8.3 Estimating the Mean μ
8.3.1 The Estimate of μ
8.3.2 The Precision of the Estimate of μ
8.4 Estimating the Difference Between Two Binomial Parameters θ1-θ2
8.4.1 The Estimate of θ1 - θ2
8.4.2 The Precision of the Estimate of θ1 - θ2
8.5 Estimating the Difference Between Two Means μ1-μ2
8.5.1 The Estimate of μ1 - μ2
8.5.2 The Precision of the Estimate of μ1-μ2
Flowchart: Estimation and Confidence Intervals
8.6 Regression
Flowchart: Linear Regression
8.7 Problems
9 Testing Hypotheses About the Value of a Parameter
9.1 Introduction to Hypothesis Testing
9.2 Two Approaches to Hypothesis Testing
9.2.1 Both Approaches, Step 1
9.2.2 Both Approaches, Step 2
9.2.3 Both Approaches, Step 3
9.2.4 Steps 4 and 5
9.2.5 Approach 1, Step 4, the Medicine Example
9.2.6 Approach 1, Step 5, the Medicine Example
9.2.7 Approach 1, Step 4, the Coin Example
9.2.8 Approach 1, Step 5, the Coin Example
9.2.9 Approach 2 to Hypothesis Testing
9.2.10 Approach 2, Step 4, the Medicine and the CoinExamples
9.2.11 Approach 2, Step 5, the Medicine Example
9.2.12 Approach 2, Step 5, the Coin Example
9.3 The Hypothesis Testing Procedure and the Concepts of Deduction and Induction
9.4 Power
Flowchart: Hypothesis Testing and Power Calculations
9.5 Problems
10 Testing for the Equality of Two Binomial Parameters
10.1 Two-by-Two Tables
10.2 Simpson's Paradox and Fisher's Exact Test
10.3 Notes on Two-by-Two Tables
10.4 Two-Sided Two-by-Two Table Tests
10.5 Problems
11 Chi-Square Tests (i): Tables Bigger Than Two-by-Two
11.1 Large Contingency Tables
11.2 Problems
12 Chi-Square Tests (ii): Testing for a Specified Probability Distribution
12.1 Introduction
12.2 Generalization
12.3 A More Complicated Situation
12.4 Problems
13 Tests on Means
13.1 The One-Sample t Test
13.2 The Two-Sample t Test
13.3 The Paired Two-Sample t Test
13.4 t Tests in Regression
13.5 General Notes on t Statistics
13.6 Exact Confidence Intervals
13.7 Problems
14 Non-parametric Tests
14.1 Introduction
14.2 Non-parametric Alternative to the One-Sample t Test: The Wilcoxon Signed-Rank Test
14.3 Non-parametric Alternative to the Two-Sample t Test: The Wilcoxon Rank-Sum Test
14.4 Other Non-parametric Procedures
14.5 Permutation Methods
14.5.1 The Permutation Alternative to the Signed-Rank Test
14.5.2 The Permutation Alternative to the Rank-Sum Test
14.6 Problems
Useful Charts
Useful Charts
Solutions to Problems
Solutions to Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Index
