# Statistics for Experimentalists

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Statistics for Experimentalists aims to provide experimental scientists with a working knowledge of statistical methods and search approaches to the analysis of data. The book first elaborates on probability and continuous probability distributions. Discussions focus on properties of continuous random variables and normal variables, independence of two random variables, central moments of a continuous distribution, prediction from a normal distribution, binomial probabilities, and multiplication of probabilities and independence. The text then examines estimation and tests of significance. Topics include estimators and estimates, expected values, minimum variance linear unbiased estimators, sufficient estimators, methods of maximum likelihood and least squares, and the test of significance method. The manuscript ponders on distribution-free tests, Poisson process and counting problems, correlation and function fitting, balanced incomplete randomized block designs and the analysis of covariance, and experimental design. The publication is a valuable reference for statisticians and researchers interested in the use of statistical methods.

## Table of Contents

Preface

Chapter 1. Introduction

1.1. Experimental Results

1.2. Experimental Design

1.3. The Power of an Experiment

1.4. Generality of the Conclusions

Chapter 2. Probability

2.1. The Classical Definition of Probability

2.2. Population and Sample

2.3. Probability Distributions

2.4. Probability Notation

2.5. Addition of Probabilities

2.6. Multiplication of Probabilities and Independence

2.7. Binomial Probabilities

2.8. Discrete and Continuous Distributions

2.9. Mean and Variance of a Discrete Probability Distribution

2.10. Crude and Central Moments of Discrete Probability Distributions

2.11. Modal Value of a Discrete Probability Distribution

Chapter 3. Continuous Probability Distributions

3.1. The Normal Probability Distribution

3.2. Mean and Variance of a Continuous Probability Distribution

3.3. Prediction from a Normal Distribution

3.4. Percentage Points and Significance Levels

3.5. Central Moments of a Continuous Distribution

3.6. Notation

3.7. Independence of Two Random Variables

3.8. Properties of Normal Variables

3.9. Properties of Continuous Random Variables

3.10. The Central Limit Theorem

3.11. Distributions Other than Normal

3.12. The Chi-Squared Distribution

Chapter 4. Estimation

4.1. The Random Sample

4.2. Estimators and Estimates

4.3. Expected Values

4.4. Unbiased Estimators

4.5. Minimum Variance Linear Unbiased Estimators

4.6. Efficiency Ratio

4.7. Consistent Estimators

4.8. Sufficient Estimators

4.9. The Method of Maximum Likelihood

4.10. The Joint Estimation of Two Parameters

4.11. The Method of Least Squares

4.12. The Methods of Maximum Likelihood and Least Squares

Appendix

Chapter 5. Tests of Significance—I

5.1. The Test of Significance Method

5.2. Is the Population Mean Equal to a Particular Value?

5.3. Is the Population Variance Equal to a Particular Value?

5.4. Is the Population Distribution of a Particular Form?

Chapter 6. Tests of Significance—II

6.1. Are the Means of Two Populations Equal?

6.2. Are the Variances of Two Populations Equal?

6.3. Are the Variances of More than Two Populations Equal?

Chapter 7. Tests of Significance—III

7.1. Are the Distributions of Several Populations Identical?

7.2. The 2 X 2 Table

7.3. A Test for Independence

7.4. Testing Particular Group Proportions

7.5. Are the Means of Several Populations Equal?

7.6. Robustness of Tests Assuming Normal Populations

7.7. Transformations and Distribution-Free Tests

Chapter 8. Analysis of Variance—I. Hierarchical Designs

8.1. Are the Means of Two or More Populations Equal?

8.2. Fixed-Effect Model for a Hierarchical Design with Three Levels

8.3. Example of a Four-Level Hierarchical Design (Mixed Model)

8.4. The General Method of Computation

8.5. Further Reading on Analysis of Variance

Chapter 9. Analysis of Variance—II. Factorial Designs

9.1. Two-Way Factorial Experiment Without Replication (Fixed-Effect Model)

9.2. Two-Way Experiment with Replication (Fixed-Effect Model)

9.3. Two-Way Factorial Experiment (Random-Effect Models)

9.4. Two-Way Factorial Experiment with Replication (Mixed Model)

9.5. Replicates or Repeats

9.6. Three-Way Factorial Experiment (Fixed-Effect Model)

9.7. An Unsymmetrical Factor

9.8. Replicated Three-Way Factorial Experiment (Random-Effect Model)

9.9. Main Effects and Interactions

9.10. Unequal Replication

9.11. Variability Estimate from Another Source

9.12. Variations of a Factorial Experiment

9.13. Polynomial Partitioning

Chapter 10. Experimental Design

10.1. Example Experiment

10.2. Complete Randomization

10.3. Randomized Blocks

10.4. The Latin Square

10.5. The Graeco-Latin Squares

10.6. Further Alphabets

Chapter 11. Balanced Incomplete Randomized Block Designs and the Analysis of Covariance

11.1. Balanced Incomplete Randomized Block Designs

11.2. A Youden Square

11.3. Other Incomplete Block Designs

11.4. Analysis of Covariance (Concomitant Observations)

Chapter 12. Correlation and Function Fitting

12.1. The Correlation Coefficient

12.2. Function Fitting

12.3. Function Fitting (Situation 1)

12.4. Function Fitting (Situation 2)

12.5. Function Fitting (Situation 3)

12.6. Function Fitting (Situation 4)

12.7. Function Fitting (Situation 5)

12.8. Fitting Other Functions

12.9. Multiple Regression

12.10. Polynomial Regression

12.11. Multiple and Polynomial Regression in Situations 2 to 5

12.12. Fitting Other Functions

Appendix. Matrix Algebra

Chapter 13. The Poisson Process and Counting Problems

13.1. The Poisson Process

13.2. A Single Count

13.3. A Number of Counts of the Same Source

13.4. Function Fitting Involving Counts

Chapter 14. Distributions-Free Tests

14.1. The Randomization Method

14.2. The Ranking Method

14.3. Two-Sample Location Rank Tests

14.4. Two-Sample Dispersion Rank Tests

14.5. Two-Sample Distribution Rank Tests

14.6. Power Comparisons for Location Tests

14.7. Location Rank Tests for Many Samples

14.8. Two-Way Factorial Analysis by Ranks

14.9. Rank Correlation

Hints and Anwers to Problems

References

Tables Section

1. The Normal Distribution

2. Significance Points for Student's t-Distribution

3. Non-Central t-Distribution

4. Significance Points for the Chi-Squared Distribution

5. Significance Points for Welch's Test

6. Significance Points for the F-Distribution (Variance Ratio)

7. Significance Points for the Sample Correlation Coefficient when ρ = 0

8. Significance Points for the Maximum F-Ratio

9. Significance Points for Wilcoxon's Test in Mann-Whitney Form

Index

## Product details

- No. of pages: 324
- Language: English
- Copyright: © Pergamon 1969
- Published: January 1, 1969
- Imprint: Pergamon
- eBook ISBN: 9781483280523

## About the Author

### B. E. Cooper

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