Mathematical Statistics With Applications

by Wackerly, Dennis; Mendenhall, William; Scheaffer, Richard L.

• ISBN13: 9780534377410
• ISBN10: 0534377416
• Edition: 6th
• Format: Hardcover
• Copyright: 2001-05-30
• Publisher: Duxbury Press

Summary

This is the most widely used mathematical statistics book at the top 200 universities in the United States. Premiere authors Dennis Wackerly, William Mendenhall, and Richard L. Scheaffer present a solid foundation in statistical theory while conveying the relevance and importance of the theory in solving practical problems in the real world. The authors' use of practical applications and excellent exercises helps readers discover the nature of statistics and understand its essential role in scientific research.

Table of Contents

1. WHAT IS STATISTICS? Introduction

Characterizing a Set of Measurements: Graphical Methods

Characterizing a Set of Measurements: Numerical Methods

How Inferences Are Made

Theory and Reality

Summary

2. PROBABILITY

Introduction

Probability and Inference

A Review of Set Notation

A Probabilistic Model for an Experiment: The Discrete Case

Calculating the Probability of an Event: The Sample-Point Method

Tools for Counting Sample Points

Conditional Probability and the Independence of Events

Two Laws of Probability

Calculating the Probability of an Event: The Event-Composition Methods

The Law of Total Probability and Bayes's Rule

Numerical Events and Random Variables

Random Sampling

Summary

3. DISCRETE RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS

Basic Definition

The Probability Distribution for Discrete Random Variable

The Expected Value of Random Variable or a Function of Random Variable

The Binomial Probability Distribution

The Geometric Probability Distribution

The Negative Binomial Probability Distribution (Optional)

The Hypergeometric Probability Distribution

Moments and Moment-Generating Functions

Probability-Generating Functions (Optional)

Tchebysheff's Theorem

Summary

4. CONTINUOUS RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS

Introduction

The Probability Distribution for Continuous Random Variable

The Expected Value for Continuous Random Variable

The Uniform Probability Distribution

The Normal Probability Distribution

The Gamma Probability Distribution

The Beta Probability Distribution

Some General Comments

Other Expected Values

Tchebysheff's Theorem

Expectations of Discontinuous Functions and Mixed Probability Distributions (Optional)

Summary

5. MULTIVARIATE PROBABILITY DISTRIBUTIONS

Introduction

Bivariate and Multivariate Probability Distributions

Independent Random Variables

The Expected Value of a Function of Random Variables

Special Theorems

The Covariance of Two Random Variables

The Expected Value and Variance of Linear Functions of Random Variables

The Multinomial Probability Distribution

The Bivariate Normal Distribution (Optional)

Conditional Expectations

Summary

6. FUNCTIONS OF RANDOM VARIABLES

Introductions

Finding the Probability Distribution of a Function of Random Variables

The Method of Distribution Functions

The Methods of Transformations

Multivariable Transformations Using Jacobians

Order Statistics

Summary

7. SAMPLING DISTRIBUTIONS AND THE CENTRAL LIMIT THEOREM

Introduction

Sampling Distributions Related to the Normal Distribution

The Central Limit Theorem

A Proof of the Central Limit Theorem (Optional)

The Normal Approximation to the Binomial Distributions

Summary

8. ESTIMATION

Introduction

The Bias and Mean Square Error of Point Estimators

Some Common Unbiased Point Estimators

Evaluating the Goodness of Point Estimator

Confidence Intervals

Large-Sample Confidence Intervals Selecting the Sample Size

Small-Sample Confidence Intervals for u and u1-u2. Confidence Intervals for o2

Summary

9. PROPERTIES OF POINT ESTIMATORS AND METHODS OF ESTIMATION

Introduction

Relative Efficiency

Consistency

Sufficiency

The Rao-Blackwell Theorem and Minimum-Variance Unbiased Estimation

The Method of Moments

The Method of Maximum Likelihood

Some Large-Sample Properties of MLEs (Optional)

Summary

10. HYPOTHESIS TESTING

Introduction

Elements of a Statistical Test

Common Large-Sample Tests

Calculating Type II Error Probabilities and Finding the Sample Size for the Z Test

Relationships Between Hypothesis Testing Procedures and Confidence Intervals

Another Way to Report the Results of a Statistical Test: Attained Significance Levels or p-Values

Some Comments on the Theory of Hypothesis Testing

Small-Sample Hypothesis Testing for u and u1-u2. Testing Hypotheses Concerning Variances

Power of Test and the Neyman-Pearson Lemma

Likelihood Ration Test

Summary

11. LINEAR MODELS AND ESTIMATION BY LEAST SQUARES

Introduction

Linear Statistical Models

The Method of Least Squares

Properties of the Least Squares Estimators for the Simple Linear Regression Model

Inference Concerning the Parameters BI

Inferences Concerning Linear Functions of the Model Parameters: Simple Linear Regression

Predicting a Particular Value of Y Using Simple Linear Regression

Correlation

Some Practical Examples

Fitting the Linear Model by Using Matrices

Properties of the Least Squares Estimators for the Multiple Linear Regression Model

Inferences Concerning Linear Functions of the Model Parameters: Multiple Linear Regression

Prediction a Particular Value of Y Using Multiple Regression

A Test for H0: Bg+1 + Bg+2 = … = Bk = 0. Summary and Concluding Remarks

12. CONSIDERATIONS IN DESIGNING EXPERIMENTS

The Elements Affecting the Information in a Sample

Designing Experiment to Increase Accuracy

The Matched Pairs Experiment

Some Elementary Experimental Designs

Summary

13. THE ANALYSIS OF VARIANCE

Introduction

The Analysis of Variance Procedure

Comparison of More than Two Means: Analysis of Variance for a One-way Layout

An Analysis of Variance Table for a One-Way Layout

A Statistical Model of the One-Way Layout

Proof of Additivity of the Sums of Squares and E (MST) for a One-Way Layout (Optional)

Estimation in the One-Way Layout

A Statistical Model for the Randomized Block Design

The Analysis of Variance for a Randomized Block Design

Estimation in the Randomized Block Design

Selecting the Sample Size

Simultaneous Confidence Intervals for More than One Parameter

Analysis of Variance Using Linear Models

Summary

14. ANALYSIS OF CATEGORICAL DATA

A Description of the Experiment

The Chi-Square Test

A Test of Hypothesis Concerning Specified Cell Probabilities: A Goodness-of-Fit Test

Contingency Tables

r x c Tables with Fixed Row or Column Totals

Other Applications

Summary and Concluding Remarks

15. NONPARAMETRIC STATISTICS

Introduction

A General Two-Sampling Shift Model

A Sign Test for a Matched Pairs Experiment

The Wilcoxon Signed-Rank Test for a Matched Pairs Experiment

The Use of Ranks for Comparing Two Population Distributions: Independent Random Samples

The Mann-Whitney U Test: Independent Random Samples

The Kruskal-Wallis Test for One-Way Layout

The Friedman Test for Randomized Block Designs

The Runs Test: A Test for Randomness

Rank Correlation Coefficient

Some General Comments on Nonparametric Statistical Test

APPENDIX 1. MATRICES AND OTHER USEFUL MATHEMATICAL RESULTS

Matrices and Matrix Algebra

Addition of Matrices

Multiplication of a Matrix by a Real Number

Matrix Multiplication

Identity Elements

The Inverse of a Matrix

The Transpose of a Matrix

A Matrix Expression for a System of Simultaneous Linear Equations

Inverting a Matrix

Solving a System of Simultaneous Linear Equations

Other Useful Mathematical Results

APPENDIX 2. COMMON PROBABILITY DISTRIBUTIONS, MEANS, VARIANCES, AND MOMENT-GENERATING FUNCTIONS

Discrete Distributions

Continuous Distributions

APPENDIX 3. TABLES

Binomial Probabilities

Table of e-x

Poisson Probabilities

Normal Curve Areas

Percentage Points of the t Distributions

Percentage Points of the F Distributions

Distribution of Function U

Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test