| |

| |

Preface | |

| |

| |

Note to the Student | |

| |

| |

| |

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 | |

| |

| |

| |

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 Method | |

| |

| |

| |

The Law of Total Probability and Bayes' Rule | |

| |

| |

| |

Numerical Events and Random Variables | |

| |

| |

| |

Random Sampling | |

| |

| |

| |

Summary | |

| |

| |

| |

Discrete Random Variables and Their Probability Distributions | |

| |

| |

| |

Basic Definition | |

| |

| |

| |

The Probability Distribution for a Discrete Random Variable | |

| |

| |

| |

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

| |

| |

| |

The Binomial Probability Distribution | |

| |

| |

| |

The Geometric Probability Distribution | |

| |

| |

| |

The Negative Binomial Probability Distribution (Optional) | |

| |

| |

| |

The Hypergeometric Probability Distribution | |

| |

| |

| |

The Poisson Probability Distribution | |

| |

| |

| |

Moments and Moment-Generating Functions | |

| |

| |

| |

Probability-Generating Functions (Optional) | |

| |

| |

| |

Tchebysheff's Theorem | |

| |

| |

| |

Summary | |

| |

| |

| |

Continuous Random Variables and Their Probability Distributions | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Probability Distribution for a Continuous Random Variable | |

| |

| |

| |

Expected Values for Continuous Random Variables | |

| |

| |

| |

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 | |

| |

| |

| |

Multivariate Probability Distributions | |

| |

| |

| |

Introduction | |

| |

| |

| |

Bivariate and Multivariate Probability Distributions | |

| |

| |

| |

Marginal and Conditional 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 | |

| |

| |

| |

Functions of Random Variables | |

| |

| |

| |

Introduction | |

| |

| |

| |

Finding the Probability Distribution of a Function of Random Variables | |

| |

| |

| |

The Method of Distribution Functions | |

| |

| |

| |

The Method of Transformations | |

| |

| |

| |

The Method of Moment-Generating Functions | |

| |

| |

| |

Multivariable Transformations Using Jacobians (Optional) | |

| |

| |

| |

Order Statistics | |

| |

| |

| |

Summary | |

| |

| |

| |

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 Distribution | |

| |

| |

| |

Summary | |

| |

| |

| |

Estimation | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Bias and Mean Square Error of Point Estimators | |

| |

| |

| |

Some Common Unbiased Point Estimators | |

| |

| |

| |

Evaluating the Goodness of a Point Estimator | |

| |

| |

| |

Confidence Intervals | |

| |

| |

| |

Large-Sample Confidence Intervals | |

| |

| |

| |

Selecting the Sample Size | |

| |

| |

| |

Small-Sample Confidence Intervals for [mu] and [mu subscript 1] - [mu subscript 2] | |

| |

| |

| |

Confidence Intervals for [sigma superscript 2] | |

| |

| |

| |

Summary | |

| |

| |

| |

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 | |

| |

| |

| |

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 [mu] and [mu subscript 1] - [mu subscript 2] | |

| |

| |

| |

Testing Hypotheses Concerning Variances | |

| |

| |

| |

Power of Tests and the Neyman-Pearson Lemma | |

| |

| |

| |

Likelihood Ratio Tests | |

| |

| |

| |

Summary | |

| |

| |

| |

Linear Models and Estimation by Least Squares | |

| |

| |

| |

Introduction | |

| |

| |

| |

Linear Statistical Models | |

| |

| |

| |

The Method of Least Squares | |

| |

| |

| |

Properties of the Least Squares Estimators: Simple Linear Regression | |

| |

| |

| |

Inferences Concerning the Parameters [beta subscript i] | |

| |

| |

| |

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 | |

| |

| |

| |

Linear Functions of the Model Parameters: Multiple Linear Regression | |

| |

| |

| |

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

| |

| |

| |

Predicting a Particular Value of Y Using Multiple Regression | |

| |

| |

| |

A Test for H[subscript 0]: [beta subscript g+1] = [beta subscript g+2] = ... = [beta subscript k] = 0 | |

| |

| |

| |

Summary and Concluding Remarks | |

| |

| |

| |

Considerations in Designing Experiments | |

| |

| |

| |

The Elements Affecting the Information in a Sample | |

| |

| |

| |

Designing Experiments to Increase Accuracy | |

| |

| |

| |

The Matched Pairs Experiment | |

| |

| |

| |

Some Elementary Experimental Designs | |

| |

| |

| |

Summary | |

| |

| |

| |

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 for 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 | |

| |

| |

| |

Analysis of Categorical Data | |

| |

| |

| |

A Description of the Experiment | |

| |

| |

| |

The Chi-Square Test | |

| |

| |

| |

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

| |

| |

| |

Contingency Tables | |

| |

| |

| |

r [times] c Tables with Fixed Row or Column Totals | |

| |

| |

| |

Other Applications | |

| |

| |

| |

Summary and Concluding Remarks | |

| |

| |

| |

Nonparametric Statistics | |

| |

| |

| |

Introduction | |

| |

| |

| |

A General Two-Sample Shift Model | |

| |

| |

| |

The 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 the 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 Tests | |

| |

| |

| |

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 | |

| |

| |

| |

Common Probability Distributions, Means, Variances, and Moment-Generating Functions | |

| |

| |

| |

Discrete Distributions | |

| |

| |

| |

Continuous Distributions | |

| |

| |

| |

Tables | |

| |

| |

| |

Binomial Probabilities | |

| |

| |

| |

Table of e[superscript -x] | |

| |

| |

| |

Poisson Probabilities | |

| |

| |

| |

Normal Curve Areas | |

| |

| |

| |

Percentage Points of the t Distributions | |

| |

| |

| |

Percentage Points of the x[superscript 2] Distributions | |

| |

| |

| |

Percentage Points of the F Distributions | |

| |

| |

| |

Distribution Function of U | |

| |

| |

| |

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

| |

| |

| |

Distribution of the Total Number of Runs R in Samples of Size (n[subscript 1], n[subscript 2]); P(R [less than or equal] a) | |

| |

| |

| |

Critical Values of Spearman's Rank Correlation Coefficient | |

| |

| |

| |

Random Numbers | |

| |

| |

Answers to Exercises | |

| |

| |

Index | |