Statistics in Engineering

Andrew Metcalfe, David Green, Andrew Smith, and Jonathan Tuke have taught probability and statistics to students of engineering at the University of Adelaide for many years and have substantial industry experience. Their current research includes applications to water resources engineering, mining, and telecommunications. Mahayaudin Mansor worked in banking and insurance before teaching statistics and business mathematics at the Universiti Tun Abdul Razak Malaysia and is currently a researcher specializing in data analytics and quantitative research in the Health Economics and Social Policy Research Group at the Australian Centre for Precision Health, University of South Australia. Tony Greenfield, formerly Head of Process Computing and Statistics at the British Iron and Steel Research Association, is a statistical consultant. He has been awarded the Chambers Medal for outstanding services to the Royal Statistical Society; the George Box Medal by the European Network for Business and Industrial Statistics for Outstanding Contributions to Industrial Statistics; and the William G. Hunter Award by the American Society for Quality. Visit their website here: http://www.maths.adelaide.edu.au/david.green/BookWebsite/

I Foundations






Why Understand Statistics?
Introduction

Using the book

Software




Probability and Making Decisions
Introduction

Random digits

Concepts and uses

Generating random digits

Pseudo random digits

Defining probabilities

Defining probabilities {Equally likely outcomes

Defining probabilities {relative frequencies

Defining probabilities {subjective probability and expected monetary value

Axioms of Probability

The addition rule of probability

Complement

Conditional probability

Conditioning on information

Conditional probability and the multiplicative rule

Independence

Tree diagrams

Bayes' theorem

Law of total probability

Bayes' theorem for two events

Bayes' theorem for any number of events

Decision trees

Permutations and combinations

Simple random sample

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Graphical Displays of Data and Descriptive Statistics
Types of variables

Samples and populations

Displaying data

Stem-and-leaf plot

Time series plot

Pictogram

Pie chart

Bar chart

Rose plot

Line chart for discrete variables

Histogram and cumulative frequency polygon for continuous variables

Pareto chart

Numerical summaries of data

Population and sample

Measures of location

Measures of spread

Box-plots

Outlying values and robust statistics

Outlying values

Robust statistics

Grouped data

Calculation of the mean and standard deviation for discrete data

Grouped continuous data [mean and sd for grouped continuous data]

Mean as center of gravity

Case study of wave stress on offshore structure

Shape of distributions

Skewness

Kurtosis

Some contrasting histograms

Multivariate data

Scatter plot

Histogram for bivariate data

Parallel coordinates plot

Descriptive time series

Definition of time series

Missing values in time series

Decomposition of time series

Centered moving average

Additive monthly model

Multiplicative monthly model

Seasonal adjustment

Forecasting

Index numbers

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Discrete Probability Distributions
Discrete random variables

Definition of a discrete probability distribution

Expected value

Bernoulli trial

Binomial distribution

Introduction

Defining the binomial distribution

A model for conductivity

Random deviates from binomial distribution

Fitting a binomial distribution

Hypergeometric distribution

Defining the hypergeometric distribution

Random deviates from the hypergeometric distribution

Fitting the hypergeometric distribution

Negative binomial distribution

The geometric distribution

Defining the negative binomial distribution

Applications of negative binomial distribution

Fitting a negative binomial distribution

Random numbers from a negative binomial distribution

Poisson process

Defining a Poisson process in time

Superimposing Poisson processes

Spatial Poisson Process

Modifications to Poisson processes

Poisson distribution

Fitting a Poisson distribution

Times between events

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Continuous Probability Distributions
Continuous probability distributions

Definition of a continuous random variable

Definition of a continuous probability distribution

Moments of a continuous probability distribution

Median and mode of a continuous probability distribution

Parameters of probability distributions

Uniform distribution

Definition of a uniform distribution

Applications of the uniform distribution

Random deviates from a uniform distribution

Distribution of F(X) is uniform

Fitting a uniform distribution

Exponential distribution

Definition of an exponential distribution

Markov property

Poisson process

Lifetime distribution

Applications of the exponential distribution

Random deviates from an exponential distribution

Fitting an exponential distribution

Normal (Gaussian) distribution

Definition of a normal distribution

The standard normal distribution

Applications of the normal distribution

Random numbers from a normal distribution

Fitting a normal distribution

Probability plots

Quantile-quantile plots

Probability plot

Lognormal distribution

Definition of a lognormal distribution

Applications of the lognormal distribution

Random numbers from lognormal distribution

Fitting a lognormal distribution

Gamma distribution

Definition of a gamma distribution

Applications of the gamma distribution

Random deviates from gamma distribution

Fitting a gamma distribution

Gumbel distribution

Definition of a Gumbel distribution

Applications of the Gumbel distribution

Random deviates from a Gumbel distribution

Fitting a Gumbel distribution

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Correlation and Functions of Random Variables
Introduction

Sample covariance and correlation coefficient

Defining sample covariance

Bivariate distributions, population covariance and correlation coefficient

Population covariance and correlation coefficient

Bivariate distributions - discrete case

Bivariate distributions - continuous case

Marginal distributions

Bivariate histogram

Covariate and correlation

Bivariate probability distributions

Copulas

Linear combination of random variables (propagation of error)

Mean and variance of a linear combination of random variables

Bounds for correlation coefficient

Linear combination of normal random variables

Central Limit Theorem and distribution of the sample mean

Non-linear functions of random variables (propagation of error)

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Estimation and Inference
Introduction

Statistics as estimators

Population parameters

Sample statistics and sampling distributions

Bias and MSE

Accuracy and precision

Precision of estimate of population mean

Confidence interval for population mean when _ known

Confidence interval for mean when _ unknown

Construction of confidence interval and rationale for the t-distribution

The t-distribution

Robustness

Bootstrap methods

Bootstrap resampling

Basic bootstrap confidence intervals

Percentile bootstrap confidence intervals

Parametric bootstrap

Hypothesis testing

Hypothesis test for population mean when _ known

Hypothesis test for population mean when _ unknown

Relation between a hypothesis test and the confidence interval

P-value

One-sided confidence intervals and one-sided tests

Sample size

Confidence interval for a population variance and standard deviation

Comparison of means

Independent Samples

Population standard deviations differ

Population standard deviations assumed equal

Matched pairs

Comparing variances

Inference about proportions

Single sample

Comparing two proportions

McNemar's test

Prediction intervals and statistical tolerance intervals

Prediction interval

Statistical tolerance interval

Goodness of fit tests

Chi-square test

Empirical distribution function tests

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Linear Regression and Linear Relationships
Linear regression

Introduction

The model

Fitting the model

Fitting the regression line

Identical forms for the least squares estimate of the slope

Relation to correlation

Alternative form for the fitted regression line

Residuals

Identities satisfied by the residuals

Estimating the standard deviation of the errors

Checking assumptions A, A and A

Properties of the estimators

Estimator of the slope

Estimator of the intercept

Predictions

Confidence interval for mean value of Y given x

Limits of Prediction

Plotting confidence intervals and prediction limits

Summarizing the algebra

Coefficient of determination R

Regression for a bivariate normal distribution

The bivariate normal distribution

Regression towards the mean

Relationship between correlation and regression

Values of x are assumed to be measured without error and can be preselected

The data pairs are assumed to be a random sample from a bivariate normal distribution

Fitting a linear relationship when both variables are measured with error

Calibration lines

Intrinsically linear models

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises

II Developments




Multiple Regression
Introduction

Multivariate data

Multiple regression model

The linear model

Random vectors

Definition

Linear transformations of a random vector

Multivariate normal distribution

Matrix formulation of the linear model

Geometrical interpretation

Fitting the model

Principle of least squares

Multivariate calculus - three basic results

The least squares estimator of the coefficients

Estimating the coefficients

Estimating the standard deviation of the errors

Standard errors of the estimators of the coefficients

Assessing the fit

The residuals

R-squared

F-statistic

Cross validation

Predictions

Building multiple regression models

Interactions

Categorical variables

F-test for an added set of variables

Quadratic terms

Guidelines formatting regression models

Time series

Introduction

Aliasing and sampling intervals

Fitting a trend and seasonal variation with regression

Autocovariance and autocorrelation

Defining autocovariance for a stationary times series model

Defining sample autocovariance and the correlogram

Autoregressive models

AR() and AR() models

Non-linear least squares

Generalized linear model

Logistic regression

Poisson regression

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Statistical Quality Control
Continuous improvement

Defining quality

Taking measurements

Avoiding rework

Strategies for quality improvement

Quality management systems

Implementing continuous improvement

Process stability

Runs chart

Histograms and boxplots

Components of variance

Capability

Process capability index

Process performance index

One-sided process capability indices

Reliability

Introduction

Reliability of components

Reliability function and the failure rate

Weibull analysis

Definition of the Weibull distribution

Weibull quantile plot

Censored data

Maximum likelihood

Kaplan-Meier estimator of reliability

Acceptance sampling

Statistical quality control charts

Shewhart mean and range chart for continuous variables

Mean chart

Range chart

p-charts for proportions

c-charts for counts

Cumulative sum charts

Multivariate control charts

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Design of Experiments with Regression Analysis
Introduction

Factorial designs with factors at two levels

Full factorial designs

Setting up a k design

Analysis of k design

Fractional factorial designs

Central composite designs

Evolutionary operation (EVOP)

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Design of Experiments and Analysis of Variance
Introduction

Comparison of several means with one-way ANOVA

Defining the model

Multiple comparisons

One-way ANOVA

Testing HO

Follow up procedure

Two factors at multiple levels

Two factors without replication (two-way ANOVA)

Two factors with replication (three-way ANOVA)

Randomized block design

Split plot design

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Probability Models
System Reliability

Series system

Parallel system

k-out-of-n system

Modules

Duality

Paths and Cut sets

Reliability function

Redundancy

Non-repairable systems

Standby systems

Common cause failures

Reliability bounds

Markov chains

Discrete Markov chain

Equilibrium Behavior of irreducible Markov Chains

Methods for solving equilibrium equations

Absorbing Markov Chains

Markov Chains in continuous time

Simulation of systems

The simulation procedure

Drawing inference from simulation outputs

Variance reduction

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises




Sampling Strategies

Introduction

Simple random sampling from a finite population

Finite population correction

Randomization theory

Defining the simple random sample

Mean and variance of sample mean

Mean and variance of estimator of population total

Model based analysis

Sample size

Stratified sampling

Principle of stratified sampling

Estimating the population mean and total

Optimal allocation of the sample over strata

Multi-stage sampling

Quota sampling

Ratio estimators and regression estimators

Introduction

Regression estimators

Ratio estimator

Calibration of the unit cost data base

Sources of error in an AMP

Calibration factor

Summary

Notation

Summary of main results

MATLAB and R commands

Exercises

A Notation

B Glossary

C Data

D Getting started in R

E Getting started in MATLAB

F Experiments

G Mathematical explanations of key results

H MATLAB code for selected Figures

I Statistical Tables