# Applied Statistical Methods

## 1st Edition

**Authors:**Irving W. Burr

**Editors:**J. William Schmidt

**eBook ISBN:**9781483277868

**Imprint:**Academic Press

**Published Date:**1st January 1974

**Page Count:**500

## Description

Applied Statistical Methods covers the fundamental understanding of statistical methods necessary to deal with a wide variety of practical problems. This 14-chapter text presents the topics covered in a manner that stresses clarity of understanding, interpretation, and method of application.

The introductory chapter illustrates the importance of statistical analysis. The next chapters introduce the methods of data summarization, including frequency distributions, cumulative frequency distributions, and measures of central tendency and variability. These topics are followed by discussions of the fundamental principles of probability, the concepts of sample spaces, outcomes, events, probability, independence of events, and the characterization of discrete and continuous random variables. Other chapters explore the distribution of several important statistics; statistical tests of hypotheses; point and interval estimation; and simple linear regression. The concluding chapters review the elements of single- and two-factor analysis of variance and the design of analysis of variance experiments.

This book is intended primarily for advanced undergraduate and graduate students in the mathematical, physical, and engineering sciences, as well as in economics, business, and related areas. Researchers and line personnel in industry and government will find this book useful in self-study.

## Table of Contents

Preface

Acknowledgments

Chapter 1 Introduction

1.1 Why Statistical Methods?

1.2 Advice to the Student

Chapter 2 The Frequency Distribution—A Tool and a Concept

2.1 Introduction

2.2 An Example of a Frequency Distribution

2.3 Frequency Class Nomenclature and Tabulation

2.4 Discrete versus Continuous Data

2.5 Graphical Representation of a Frequency Distribution

2.6 The Cumulative Frequency Graph

2.7 What a Frequency Distribution Shows

2.8 Some Examples of Use of Frequency Tables and Graphs

2.9 Sample versus Population

2.10 Summary

Problems

Chapter 3 Summarization of Data by Objective Measures

3.1 Introduction

3.2 Some Averages

3.3 Some Measures of Variability

3.4 Efficient Calculation of Averages and Standard Deviations

3.4.1 Calculations for Frequency Data

3.5 Further Descriptive Measures of Frequency Distributions, Third and Fourth Moments

3.6 Summary

Problems

Chapter 4 Some Elementary Probability

4.1 Introduction

4.2 Sample Spaces of Outcomes

4.3 Events

4.3.1 Relations of Events

4.3.2 Combinations of Events

4.4 Probabilities of Events

4.5 Probabilities on Discrete Sample Spaces

4.5.1 Countably Infinite Spaces

4.5.2 Events over Discrete Spaces

4.6 Independent and Dependent Events

4.6.1 Conditional Probabilities

4.6.2 Repeated Trials

4.7 Discrete Probabilities

4.7.1 Permutations and Combinations

4.7.2 Discrete Probability Examples

4.8 Probabilities on Continuous Spaces

4.9 Applied Bayes' Probabilities—Posterior Probabilities

4.10 Interpretation of a Probability

4.11 Random Variables

4.12 Summary

Problems

Chapter 5 Some Discrete Probability Distributions

5.1 Theoretical Populations

5.2 Discrete Probability Distributions in General

5.2.1 Expected Values for Y and Functions of Y

5.2.2 Population Curve-Shape Characteristics

5.2.3 Algebra of Expectations

5.2.4 Further on Population Moments

5.3 The Binomial Distribution

5.3.1 Examples of the Binomial Distribution

5.3.2 Population Moments for Binomial Distributions

5.3.3 Use of Binomial Tables

5.3.4 Calculation of a Binomial Distribution

5.3.5 Approximations to the Binomial Distribution

5.3.6 Conditions of Applicability of the Binomial Distribution

5.4 The Poisson Distribution

5.4.1 A Derivation of the Poisson Probability Function

5.4.2 Examples of the Poisson Distribution

5.4.3 Tables of the Poisson Distribution

5.4.4 Using the Poisson Distribution to Approximate the Binomial

5.4.5 Derivation of the Poisson as a Limit of the Binomial

5.4.6 Conditions of Applicability of the Poisson Distribution

5.5 The Hypergeometric Distribution

5.5.1 Tables for the Hypergeometric Distribution

5.5.2 Examples of the Hypergeometric Distribution

5.5.3 Moments for the Hypergeometric Distribution

5.5.4 Binomial Approximations to the Hypergeometric Distribution

5.5.5 Poisson Approximations to the Hypergeometric Distribution

5.5.6 Approximations to Sums of Terms of the Hypergeometric Distribution

5.5.7 Conditions of Applicability of the Hypergeometric Distribution

5.5.8 Applications of the Hypergeometric Distribution

5.6 The Uniform Distribution

5.7 The Geometric Distribution

5.8 The Negative Binomial Distribution

5.9 Generating Samples from Discrete Distributions

5.10 Summary

References

Problems

Chapter 6 Some Continuous Probability Distributions

6.1 Continuous Probability Distributions

6.2 Some General Properties of Continuous Distributions

6.2.1 Moments for a Continuous Distribution

6.3 The Normal Curve

6.3.1 Properties of the Normal Distribution

6.3.2 The General Normal Curve

6.3.3 Sketching a Normal Curve

6.3.4 Approximating Probabilities by a Normal Distribution

6.4 The Rectangular Distribution

6.5 The Exponential Distribution

6.6 The Gamma Distribution

6.6.1 Tables of the Gamma Distribution

6.6.2 Relation to the Normal Distribution

6.6.3 Use of the Gamma Distribution to Approximate Discrete Distributions

6.7 The Beta Distribution

6.8 The Weibull Distribution

6.9 The Pearson System of Distributions

6.10 An Easily Fitted General System of Frequency Curves

6.11 Sums and Averages and a Central Limit Theorem

6.12 Tchebycheff's Theorem

6.13 Summary

6.14 Proofs of Some Relations in Section 6.11

References

Problems

Chapter 7 Some Sampling Distributions

7.1 Distribution of Sample Statistics from Populations

7.2 Choice of Sample

7.2.1 Sampling from a Probability Distribution

7.2.2 Machine Generation of Random Samples

7.3 Sampling Distributions of a Sample Statistic

7.4 Distribution of Sample Means

7.4.1 Standardized Distribution for Means

7.4.2 Distribution of Means when Standard Deviation Is Unknown

7.4.3 Areas for the t Distribution

7.4.4 Interpolation Note

7.4.5 Distribution of Means from Nonnormal Populations

7.5 Distribution of Sample Variances

7.5.1 Distribution of Sample Standard Deviation

7.5.2 Population of y's Nonnormal

7.5.3 Tables of Chi-Square

7.6 Joint Distribution of ȳ and s from a Normal Population

7.7 Two Normal Populations, Independent Samples

7.7.1 Sum and Difference of Two Means, Standard Deviations Known

7.7.2 Sums and Differences of Two Means, Standard Deviations Unknown but Equal

7.7.3 Two Variances, F Distribution

7.7.4 Two Variances, Large Samples

7.8 Sampling Aspects of the Binomial and Poisson Distributions

7.9 Sum of Two Independent Chi-Square Variables

7.10 Noncentral Distributions

7.11 Summary

References

Problems

Chapter 8 Statistical Tests of Hypotheses—General and One Sample

8.1 Introduction

8.2 An Example

8.2.1 Approach 1 Given n, Set Significance Level α

8.2.2 Approach 2 Set Two Risks: α and α βμ, and Find n

8.3 Summary of the Elements in Tests of Hypotheses on One Parameter

8.4 Summary of Significance Testing for One Mean with σ Unknown

8.5 Interpretation of Decisions in Hypothesis Testing

8.6 Nonnormal Populations of y's

8.7 Significance Testing for Mean μ, with σ Unknown

8.7.1 Example

8.7.2 Operating Characteristic Curve

8.8 Significance Tests for Variability

8.8.1 An Example of First Approach

8.8.2 The Second Approach of 4, in Section 8.8

8.8.3 Operating Characteristic Curves for Variability Tests

8.8.4 Large Samples

8.9 Significance Testing for Attributes

8.9.1 The Binomial Tests

8.9.2 The Poisson Tests

8.9.3 Other Attribute Distributions

8.9.4 Operating Characteristic Curves

8.10 Relation of Significance Testing to Decision Theory

8.11 Summary

References

Problems

Chapter 9 Significance Tests—Two Samples

9.1 The General Problem

9.2 Tests on Two Variances—The F Test

9.2.1 An Example

9.2.2 Large Sample Tests

9.2.3 A Large Sample Example

9.3 Differences between Means

9.3.1 Standard Deviations Known

9.3.2 Standard Deviations Equal but Unknown

9.3.3 Standard Deviations Unknown and Possibly Unequal

9.4 Significance of Differences—Binomial Data

9.5 Significance of Differences—Poisson Data

9.5.1 Unequal Areas of Opportunity

9.6 Matched Pair Data. Importance of Experimental Design

9.6.1 Matched-Pair Model

9.7 Sample Sizes Needed for Tests of Two Means

9.8 Summary

References

Problems

Chapter 10 Estimation of Population Characteristics

10.1 Point Estimates—General Idea

10.2 Which Estimator to Use—Characteristics of Estimation

10.2.1 Consistency and Sufficiency

10.3 How to Find a Desirable Estimator

10.4 Point Estimates—Common Cases

10.5 Interval Estimation in General

10.5.1 Geometrical Argument for Confidence Intervals

10.6 Confidence Intervals for μ

10.7 Confidence Intervals for σ

10.7.1 Large Confidence Limits for σ

10.8 How to Have Narrower Confidence Intervals

10.9 Confidence Intervals for Functions of Two Parameters—Two Samples

10.9.1 Confidence Limits on the Difference of Means

10.9.2 Confidence Limits on the Ratio of σ's

10.9.3 Paired Differences

10.10 Confidence Limits for Attribute Data

10.10.1 Exact Method for Binomial Population

10.10.2 Normal Approximation for Confidence Limits for the Binomial

10.10.3 Exact Method for Poisson Population

10.10.4 Normal Approximation for Confidence Limits for the Poisson

10.10.5 Tables of Confidence Limits

10.10.6 Confidence Limits for Two Samples of Attribute Data

10.11 Relation between Interval Estimation and Significance Testing

10.12 Summary

References

Problems

Chapter 11 Simple Regression

11.1 Regression, A Study of Relationship

11.2 The Scatter Diagram

11.3 Line of Best Fit to "Linear" Data

11.3.1 Least Squares Fitting

11.3.2 Calculational Aspects

11.3.3 The Linear Model and Its Parameters

11.4 Sampling Distributions for Estimates

11.5 Significance Tests and Confidence Intervals for Parameters in Linear Regression

11.5.1 Slope

11.5.2 Intercept

11.5.3 Mean of Y's: μY

11.5.4 Error Variance σε2, and σε

11.5.5 Regression Line Mean: μY.x = μY + β(X - X̄)

11.6 Correlational Aspects

11.7 Grouped Bivariate Data

11.8 Special Case μY.x = β1X

11.9 Significance of Differences between Two Slopes

11.10 Nonlinearity Test

11.11 Use of Least Squares Fitting for Other Trends

11.11.1 Functions Linear in the Parameters

11.11.2 Least Squares after a Transformation

11.11.3 Intrinsically Nonlinear Cases

11.12 Applications to Industry and the Laboratory

11.13 Summary

References

Problems

Chapter 12 Simple Analysis of Variance

12.1 General Concept of Analysis of Variance

12.2 One-Factor Analysis of Variance

12.2.1 The Model

12.2.2 The Formulas and Test

12.2.3 The Case of Unequal Sample Sizes

12.2.4 Orthogonal Contrasts

12.3 Orthogonal Polynomials and Tests

12.4 A Method of Multiple Contrasts

12.4.1 The Newman-Keuls Multiple Range Test

12.4.2 Example

12.4.3 Interpretation of Risk α

12.5 Testing Homogeneity of Variances

12.5.1 An Example

12.5.2 Q Test with Unequal Degrees of Freedom

12.5.3 Q Test for Ranges

12.6 Types of Factors

12.7 Analysis of Variance for Two Factors

12.7.1 An Example

12.7.2 The Models and Assumptions

12.7.3 Expected Mean Squares and Significance Tests

12.7.4 Interpretation of Significant Factors

12.7.5 Case of Unreplicated Two-Factor Experiments

12.7.6 Example of an Unreplicated Two-Factor Completely Randomized Design

12.8 Other Models

12.9 Summary

References

Problems

Chapter 13 Multiple Regression

13.1 Introduction

13.2 First Approach

13.2.1 Data Table Format

13.2.2 Fitting the Equation

13.2.3 Alternative Forms of Normal Equations and Regression

13.2.4 Describing Goodness of Fit

13.2.5 Systematic Solution of the Normal Equations

13.2.6 Significance Tests on the Explained Variation

13.2.7 Simple Example

13.2.8 Second Example

13.3 Second Approach

13.3.1 Vectors and Matrices

13.3.2 The Matrix Approach

13.3.3 Selection of a Set of Predictors

13.3.4 Calculation of an Inverse

13.4 Summary of Approach

13.5 Adequacy of Regression Model

13.6 Comments and Precautions

References

Problems

Chapter 14 Goodness of Fit Tests, Contingency Tables

14.1 Introduction

14.2 The Chi-square Test for Cell Frequencies, Observed versus Theoretical

14.2.1 An Example

14.3 Testing Goodness of Fit of Theoretical Distributions

14.3.1 A Binomial Example

14.3.2 Examples of Tests of Normality

14.3.3 Example of a Gamma Distribution Fit

14.3.4 Example of a Poisson Distribution

14.4 Other Goodness of Fit Tests

14.5 Contingency Tables

14.5.1 An Example

14.5.2 The General Setup of a Contingency Table

14.5.3 A 2 ∙ 2 Contingency Table

14.5.4 Case of a 2 • b Contingency Table

14.6 The Sign Test

14.7 Summary

References

Problems

Appendix

Answers to Odd-Numbered Problems

Index

## Details

- No. of pages:
- 500

- Language:
- English

- Copyright:
- © Academic Press 1974

- Published:
- 1st January 1974

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483277868