## Table of Contents

1. The Role of Statistics in Engineering

1.1 The Engineering Method and Statistical Thinking

1.1.1 Variability

1.1.2 Populations and Samples

1.2 Collecting Engineering Data

1.2.1 Basic Principles

1.2.2 Retrospective Study

1.2.3 Observational Study

1.2.4 Designed Experiments

1.2.5 Observing Processes Over Time

1.3 Mechanistic and Empirical Models

1.4 Probability and Probability Models

2. Probability

2.1.1 Random Experiments

2.1.2 Sample Spaces

2.1.3 Events

2.2 Counting Techniques

2.3 Interpretations and Axioms of Probability

2.4 Unions of Events and Addition Rules

2.5 Conditional Probability

2.6 Intersections of Events and Multiplication and Total Probability Rules

2.7 Independence

2.8 Bayes’ Theorem

2.9 Random Variables

2.10 Exercises

3. Discrete Random Variables and Probability Distributions

3.1 Probability Distributions and Probability Mass Functions

3.2 Cumulative Distribution Functions

3.3 Mean and Variance of a Discrete Random Variable

3.4 Discrete Uniform Distribution

3.5 Binomial Distribution

3.6 Geometric and Negative Binomial Distributions

3.7 Hypergeometric Distribution

3.8 Poisson Distribution

3.9 Exercises

4. Continuous Random Variables and Probability Distributions

4.1 Probability Distributions and Probability Density Functions

4.2 Cumulative Distribution Functions

4.3 Mean and Variance of a Continuous Random Variable

4.4 Continuous Uniform Distribution

4.5 Normal Distribution

4.6 Normal Approximation to the Binomial and Poisson Distributions

4.7 Exponential Distribution

4.8 Erlang and Gamma Distributions

4.9 Weibull Distribution

4.10 Lognormal Distribution

4.11 Beta Distribution

4.12 Exercises

5. Joint Probability Distributions

5.1 Joint Probability Distributions for Two Random Variables

5.2 Conditional Probability Distributions and Independence

5.3 Joint Probability Distributions for More Than Two Random Variables

5.4 Covariance and Correlation

5.5 Common Joint Distributions

5.5.1 Multinomial Probability Distribution

5.5.2 Bivariate Normal Distribution

5.6 Linear Functions of Random Variables

5.7 General Functions of Random Variables

5.8 Moment-Generating Functions

5.9 Exercises

6. Descriptive Statistics

6.1 Numerical Summaries of Data

6.2 Stem-and-Leaf Diagrams

6.3 Frequency Distributions and Histograms

6.4 Box Plots

6.5 Time Sequence Plots

6.6 Scatter Diagrams

6.7 Probability Plots

6.8 Exercises

7. Point Estimation of Parameters and Sampling Distributions

7.1 Point Estimation

7.2 Sampling Distributions and the Central Limit Theorem

7.3 General Concepts of Point Estimation

7.3.1 Unbiased Estimators

7.3.2 Variance of a Point Estimator

7.3.3 Standard Error: Reporting a Point Estimate

7.3.4 Bootstrap Standard Error

7.3.5 Mean Squared Error of an Estimator

7.4 Methods of Point Estimation

7.4.1 Method of Moments

7.4.2 Method of Maximum Likelihood

7.4.3 Bayesian Estimation of Parameters

7.5 Exercises

8. Statistical Intervals for a Single Sample

8.1 Confidence Interval on the Mean of a Normal Distribution, Variance Known

8.1.1 Development of the Confidence Interval and Its Basic Properties

8.1.2 Choice of Sample Size

8.1.3 One-Sided Confidence Bounds

8.1.4 General Method to Derive a Confidence Interval

8.1.5 Large-Sample Confidence Interval for μ

8.2 Confidence Interval on the Mean of a Normal Distribution, Variance Unknown

8.2.1 t Distribution

8.2.2 t Confidence Interval on μ

8.3 Confidence Interval on the Variance and Standard Deviation of a Normal Distribution

8.4 Large-Sample Confidence Interval for a Population Proportion

8.5 Guidelines for Constructing Confidence Intervals

8.6 Bootstrap Confidence Interval

8.7 Tolerance and Prediction Intervals

8.7.1 Prediction Interval for a Future Observation

8.7.2 Tolerance Interval for a Normal Distribution

8.8 Exercises

9. Tests of Hypotheses for a Single Sample

9.1 Hypothesis Testing

9.1.1 Statistical Hypotheses

9.1.2 Tests of Statistical Hypotheses

9.1.3 One-Sided and Two-Sided Hypotheses

9.1.4 p-Values in Hypothesis Tests

9.1.5 Connection between Hypothesis Tests and Confidence Intervals

9.1.6 General Procedure for Hypothesis Tests

9.2 Tests on the Mean of a Normal Distribution, Variance Known

9.2.1 Hypothesis Tests on the Mean

9.2.2 Type II Error and Choice of Sample Size

9.2.3 Large-Sample Test

9.3 Tests on the Mean of a Normal Distribution, Variance Unknown

9.3.1 Hypothesis Tests on the Mean

9.3.2 Type II Error and Choice of Sample Size

9.4 Tests on the Variance and Standard Deviation of a Normal Distribution

9.4.1 Hypothesis Tests on the Variance

9.4.2 Type II Error and Choice of Sample Size

9.5 Tests on a Population Proportion

9.5.1 Large-Sample Tests on a Proportion

9.5.2 Type II Error and Choice of Sample Size

9.6 Summary Table of Inference Procedures for a Single Sample

9.7 Testing for Goodness of Fit

9.8 Contingency Table Tests

9.9 Nonparametric Procedures

9.9.1 The Sign Test

9.9.2 The Wilcoxon Signed-Rank Test

9.9.3 Comparison to the t-Test

9.10 Equivalence Testing

9.11 Combining p-Values

9.12 Exercises

10. Statistical Inference for Two Samples

10.1 Inference on the Difference in Means of Two Normal Distributions, Variances Known

10.1.1 Hypothesis Tests on the Difference in Means, Variances Known

10.1.2 Type II Error and Choice of Sample Size

10.1.3 Confidence Interval on the Difference in Means, Variances Known

10.2 Inference on the Difference in Means of Two Normal Distributions, Variances Unknown

10.2.1 Hypotheses Tests on the Difference in Means, Variances Unknown

10.2.2 Type II Error and Choice of Sample Size

10.2.3 Confidence Interval on the Difference in Means, Variances Unknown

10.3 A Nonparametric Test for the Difference in Two Means

10.3.1 Description of the Wilcoxon Rank-Sum Test

10.3.2 Large-Sample Approximation

10.3.3 Comparison to the t-Test

10.4 Paired t-Test

10.5 Inference on the Variances of Two Normal Distributions

10.5.1 F Distribution

10.5.2 Hypothesis Tests on the Equity of Two Variances

10.5.3 Type II Error and Choice of Sample Size

10.5.4 Confidence Interval on the Ratio of Two Variances

10.6 Inference on Two Population Proportions

10.6.1 Large-Sample Tests on the Difference in Population Proportions

10.6.2 Type II Error and Choice of Sample Size

10.6.3 Confidence Interval on the Difference in Population Proportions

10.7 Summary Table and Road Map for Inference Procedures for Two Samples

10.8 Exercises

11. Simple Linear Regression and Correlation

11.1 Empirical Models

11.2 Simple Linear Regression

11.3 Properties of the Least Squares Estimators

11.4 Hypothesis Tests in Simple Linear Regression

11.4.1 Use of t-Tests

11.4.2 Analysis of Variance Approach to Test Significance of Regression

11.5 Confidence Intervals

11.5.1 Confidence Intervals on the Slope and Intercept

11.5.2 Confidence Interval on the Mean Response

11.6 Prediction of New Observations

11.7 Adequacy of the Regression Model

11.7.1 Residual Analysis

11.7.2 Coefficient of Determination (R2)

11.8 Correlation

11.9 Regression on Transformed Variables

11.10 Logistic Regression

11.11 Exercises

12. Multiple Linear Regression

12.1 Multiple Linear Regression Model

12.1.1 Introduction

12.1.2 Least Squares Estimation of the Parameters

12.1.3 Matrix Approach to Multiple Linear Regression

12.1.4 Properties of the Least Squares Estimators

12.2 Hypothesis Tests in Multiple Linear Regression

12.2.1 Test for Significance of Regression

12.2.2 Tests on Individual Regression Coefficients and Subsets of Coefficients

12.3 Confidence Intervals in Multiple Linear Regression

12.3.1 Confidence Intervals on Individual Regression Coefficients

12.3.2 Confidence Interval on the Mean Response

12.4 Prediction of New Observations

12.5 Model Adequacy Checking

12.5.1 Residual Analysis

12.5.2 Influential Observations

12.6 Aspects of Multiple Regression Modeling

12.6.1 Polynomial Regression Models

12.6.2 Categorical Regressors and Indicator Variables

12.6.3 Selection of Variables and Model Building

12.6.4 Multicollinearity

12.7 Exercises

13. Design and Analysis of Single-Factor Experiments: The Analysis of Variance

13.1 Designing Engineering Experiments

13.2 Completely Randomized Single-Factor Experiment

13.2.1 Example: Tensile Strength

13.2.2 Analysis of Variance

13.2.3 Multiple Comparisons Following the ANOVA

13.2.4 Residual Analysis and Model Checking

13.2.5 Determining Sample Size

13.3 The Random-Effects Model

13.3.1 Fixed Versus Random Factors

13.3.2 ANOVA and Variance Components

13.4 Randomized Complete Block Design

13.4.1 Design and Statistical Analysis

13.4.2 Multiple Comparisons

13.4.3 Residual Analysis and Model Checking

13.5 Exercises

14. Design of Experiments with Several Factors

14.1 Introduction

14.2 Factorial Experiments

14.3 Two-Factor Factorial Experiments

14.3.1 Statistical Analysis

14.3.2 Model Adequacy Checking

14.3.3 One Observation per Cell

14.4 General Factorial Experiments

14.5 2k Factorial Designs

14.5.1 22 Design

14.5.2 2k Design for k ≥ 3 Factors

14.6 Single Replicate of the 2k Design

14.7 Addition of Center Points to a 2k Design

14.8 Blocking and Confounding in the 2k Design

14.9 One-Half Fraction of the 2k Design

14.10 Smaller Fractions: The 2k−p Fractional Factorial

14.11 Response Surface Methods and Designs

14.12 Exercises

15. Statistical Quality Control

15.1 Quality Improvement and Statistics

15.1.1 Statistical Quality Control

15.1.2 Statistical Process Control

15.2 Introduction to Control Charts

15.2.1 Basic Principles

15.2.2 Design of a Control Chart

15.2.3 Rational Subgroups

15.2.4 Analysis of Patterns on Control Charts

15.3 X and R or S Control Charts

15.4 Control Charts for Individual Measurements

15.5 Process Capability

15.6 Attribute Control Charts

15.6.1 P Chart (Control Chart for Proportions)

15.6.2 U Chart (Control Chart for Defects per Unit)

15.7 Control Chart Performance

15.8 Time-Weighted Charts

15.8.1 Exponentially Weighted Moving-Average Control Chart

15.8.2 Cumulative Sum Control Chart

15.9 Other SPC Problem-Solving Tools

15.10 Decision Theory

15.10.1 Decision Models

15.10.2 Decision Criteria

15.11 Implementing SPC

15.12 Exercises

16. Appendix: Statistical Tables and Charts

16.1 Stats zyTools and Resources

Normal Distribution Calculator

t-Distribution Calculator

Binomial Distribution Calculator

Visualizing Probability Distributions (graph generator)

16.2 Statistical Tables and Charts

Summary of Common Probability Distributions

Cumulative Binomial Probabilities P(X ≤ x)

Cumulative Standard Normal Distribution

Percentage Points χ2α,v of the Chi-Squared Distribution

Percentage Points tα,v of the t Distribution

Percentage Points fα,v1,v2 of the F Distribution

Operating Characteristic Curves

Critical Values for the Sign Test

Critical Values for the Wilcoxon Signed-Rank Test

Critical Values for the Wilcoxon Rank-Sum Test

Factors for Constructing Variables Control Charts

Factors for Tolerance Intervals

17. Appendix: Bibliography

18 Appendix: Summary of Confidence Intervals and Hypothesis Testing Equations for One and Two Sample Applications

## The interactive zyBooks version of a classic introduction to statistics and probability for engineers

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## Authors

**Douglas C. Montgomery, PhD**

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Professor. School of Engineering, Arizona State University

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