Statistics in Engineering: With Examples in MATLAB® and R, Second Edition (Chapman & Hall/CRC Texts in Statistical Science)

Description

Engineers are expected to design structures and machines that can operate in challenging and volatile environments, while allowing for variation in materials and noise in measurements and signals. Statistics in Engineering, Second Edition: With Examples in MATLAB and R covers the fundamentals of probability and statistics and explains how to use these basic techniques to estimate and model random variation in the context of engineering analysis and design in all types of environments. The first eight chapters cover probability and probability distributions, graphical displays of data and descriptive statistics, combinations of random variables and propagation of error, statistical inference, bivariate distributions and correlation, linear regression on a single predictor variable, and the measurement error model. This leads to chapters including multiple regression; comparisons of several means and split-plot designs together with analysis of variance; probability models; and sampling strategies. Distinctive features include: All examples based on work in industry, consulting to industry, and research for industry Examples and case studies include all engineering disciplines Emphasis on probabilistic modeling including decision trees, Markov chains and processes, and structure functions Intuitive explanations are followed by succinct mathematical justifications Emphasis on random number generation that is used for stochastic simulations of engineering systems, demonstration of key concepts, and implementation of bootstrap methods for inference Use of MATLAB and the open source software R, both of which have an extensive range of statistical functions for standard analyses and also enable programing of specific applications Use of multiple regression for times series models and analysis of factorial and central composite designs Inclusion of topics such as Weibull analysis of failure times and split-plot designs that are commonly used in industry but are not usually included in introductory textbooks Experiments designed to show fundamental concepts that have been tested with large classes working in small groups Website with additional materials that is regularly updated 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. Cover Half Title Title Page Copyright Page Table of Contents Preface 1: Why understand statistics? 1.1 Introduction 1.2 Using the book 1.3 Software 2: Probability and making decisions 2.1 Introduction 2.2 Random digits 2.2.1 Concepts and uses 2.2.2 Generating random digits 2.2.3 Pseudo random digits 2.3 Defining probabilities 2.3.1 Defining probabilities − Equally likely outcomes 2.3.2 Defining probabilities − Relative frequencies 2.3.3 Defining probabilities − Subjective probability and expected monetary value 2.4 Axioms of probability 2.5 The addition rule of probability 2.5.1 Complement 2.6 Conditional probability 2.6.1 Conditioning on information 2.6.2 Conditional probability and the multiplicative rule 2.6.3 Independence 2.6.4 Tree diagrams 2.7 Bayes' theorem 2.7.1 Law of total probability 2.7.2 Bayes' theorem for two events 2.7.3 Bayes' theorem for any number of events 2.8 Decision trees 2.9 Permutations and combinations 2.10 Simple random sample 2.11 Summary 2.11.1 Notation 2.11.2 Summary of main results 2.11.3 MATLAB ® and R commands 2.12 Exercises 3: Graphical displays of data and descriptive statistics 3.1 Types of variables 3.2 Samples and populations 3.3 Displaying data 3.3.1 Stem-and-leaf plot 3.3.2 Time series plot 3.3.3 Pictogram 3.3.4 Pie chart 3.3.5 Bar chart 3.3.6 Rose plot 3.3.7 Line chart for discrete variables 3.3.8 Histogram and cumulative frequency polygon for continuous variables 3.3.9 Pareto chart 3.4 Numerical summaries of data 3.4.1 Population and sample 3.4.2 Measures of location 3.4.3 Measures of spread 3.5 Box-plots 3.6 Outlying values and robust statistics 3.6.1 Outlying values 3.6.2 Robust statistics 3.7 Grouped data 3.7.1 Calculation of the mean and standard deviation for discrete data 3.7.2 Grouped continuous data [Mean and standard deviation for grouped continuous data] 3.7.3 Mean as center of gravity 3.7.4 Case study of wave stress on offshore structure 3.8 Shape of distributions 3.8.1 Skewness 3.8.2 Kurtosis 3.8.3 Some contrasting histograms 3.9 Multivariate data 3.9.1 Scatter plot 3.9.2 Histogram for bivariate data 3.9.3 Parallel coordinates plot 3.10 Descriptive time series 3.10.1 Definition of time series 3.10.2 Missing values in time series 3.10.3 Decomposition of time series 3.10.3.1 Trend − Centered moving average 3.10.3.2 Seasonal component − Additive monthly model 3.10.3.3 Seasonal component − Multiplicative monthly model 3.10.3.4 Seasonal adjustment 3.10.3.5 Forecasting 3.10.4 Index numbers 3.11 Summary 3.11.1 Notation 3.11.2 Summary of main results 3.11.3 MATLAB and R commands 3.12 Exercises 4: Discrete probability distributions 4.1 Discrete random variables 4.1.1 Definition of a discrete probability distribution 4.1.2 Expected value 4.2 Bernoulli trial 4.2.1 Introduction 4.2.2 Defining the Bernoulli distribution 4.2.3 Mean and variance of the Bernoulli distribution 4.3 Binomial distribution 4.3.1 Introduction 4.3.2 Defining the Binomial distribution 4.3.3 A model for conductivity 4.3.4 Mean and variance of the binomial distribution 4.3.5 Random deviates from binomial distribution 4.3.6 Fitting a binomial distribution 4.4 Hypergeometric distribution 4.4.1 Defining the hypergeometric distribution 4.4.2 Random deviates from the hypergeometric distribution 4.4.3 Fitting the hypergeometric distribution 4.5 Negative binomial distribution 4.5.1 The geometric distribution 4.5.2 Defining the negative binomial distribution 4.5.3 Applications of negative binomial distribution 4.5.4 Fitting a negative binomial distribution 4.5.5 Random numbers from a negative binomial distribution 4.6 Poisson process 4.6.1 Defining a Poisson process in time 4.6.2 Superimposing Poisson processes 4.6.3 Spatial Poisson process 4.6.4 Modifications to Poisson processes 4.6.5 Poisson distribution 4.6.6 Fitting a Poisson distribution 4.6.7 Times between events 4.7 Summary 4.7.1 Notation 4.7.2 Summary of main results 4.7.3 MATLAB and R commands 4.8 Exercises 5: Continuous probability distributions 5.1 Continuous random variables 5.1.1 Definition of a continuous random variable 5.1.2 Definition of a continuous probability distribution 5.1.3 Moments of a continuous probability distribution 5.1.4 Median and mode of a continuous probability distribution 5.1.5 Parameters of probability distributions 5.2 Uniform distribution 5.2.1 Definition of a uniform distribution 5.2.2 Applications of the uniform distribution 5.2.3 Random deviates from a uniform distribution 5.2.4 Distribution of F(X) is uniform 5.2.5 Fitting a uniform distribution 5.3 Exponential distribution 5.3.1 Definition of an exponential distribution 5.3.2 Markov property 5.3.2.1 Poisson process 5.3.2.2 Lifetime distribution 5.3.3 Applications of the exponential distribution 5.3.4 Random deviates from an exponential distribution 5.3.5 Fitting an exponential distribution 5.4 Normal (Gaussian) distribution 5.4.1 Definition of a normal distribution 5.4.2 The standard normal distribution Z ~ N(0, 1) 5.4.3 Applications of the normal distribution 5.4.4 Random numbers from a normal distribution 5.4.5 Fitting a normal distribution 5.5 Probability plots 5.5.1 Quantile-quantile plots 5.5.2 Probability plot 5.6 Lognormal distribution 5.6.1 Definition of a lognormal distribution 5.6.2 Applications of the lognormal distribution 5.6.3 Random numbers from lognormal distribution 5.6.4 Fitting a lognormal distribution 5.7 Gamma distribution 5.7.1 Definition of a gamma distribution 5.7.2 Applications of the gamma distribution 5.7.3 Random deviates from gamma distribution 5.7.4 Fitting a gamma distribution 5.8 Gumbel distribution 5.8.1 Definition of a Gumbel distribution 5.8.2 Applications of the Gumbel distribution 5.8.3 Random deviates from a Gumbel distribution 5.8.4 Fitting a Gumbel distribution 5.9 Summary 5.9.1 Notation 5.9.2 Summary of main results 5.9.3 MATLAB and R commands 5.10 Exercises 6: Correlation and functions of random variables 6.1 Introduction 6.2 Sample covariance and correlation coefficient 6.2.1 Defining sample covariance 6.3 Bivariate distributions, population covariance and correlation coefficient 6.3.1 Population covariance and correlation coefficient 6.3.2 Bivariate distributions − Discrete case 6.3.3 Bivariate distributions − Continuous case 6.3.3.1 Marginal distributions 6.3.3.2 Bivariate histogram 6.3.3.3 Covariate and correlation 6.3.3.4 Bivariate probability distributions 6.3.4 Copulas 6.4 Linear combination of random variables (propagation of error) 6.4.1 Mean and variance of a linear combination of random variables 6.4.1.1 Bounds for correlation coefficient 6.4.2 Linear combination of normal random variables 6.4.3 Central Limit Theorem and distribution of the sample mean 6.5 Non-linear functions of random variables (propagation of error) 6.6 Summary 6.6.1 Notation 6.6.2 Summary of main results 6.6.3 MATLAB and R commands 6.7 Exercises 7: Estimation and inference 7.1 Introduction 7.2 Statistics as estimators 7.2.1 Population parameters 7.2.2 Sample statistics and sampling distributions 7.2.3 Bias and MSE 7.3 Accuracy and precision 7.4 Precision of estimate of population mean 7.4.1 Confidence interval for population mean when σ known 7.4.2 Confidence interval for mean when σ unknown 7.4.2.1 Construction of confidence interval and rationale for the t-distribution 7.4.2.2 The t-distribution 7.4.3 Robustness 7.4.4 Bootstrap methods 7.4.4.1 Bootstrap resampling 7.4.4.2 Basic bootstrap confidence intervals 7.4.4.3 Percentile bootstrap confidence intervals 7.4.5 Parametric bootstrap 7.5 Hypothesis testing 7.5.1 Hypothesis test for population mean when σ known 7.5.2 Hypothesis test for population mean when σ unknown 7.5.3 Relation between a hypothesis test and the confidence interval 7.5.4 p-value 7.5.5 One-sided confidence intervals and one-sided tests 7.6 Sample size 7.7 Confidence interval for a population variance and standard deviation 7.8 Comparison of means 7.8.1 Independent samples 7.8.1.1 Population standard deviations differ 7.8.1.2 Population standard deviations assumed equal 7.8.2 Matched pairs 7.9 Comparing variances 7.10 Inference about proportions 7.10.1 Single sample 7.10.2 Comparing two proportions 7.10.3 McNemar's test 7.11 Prediction intervals and statistical tolerance intervals 7.11.1 Prediction interval 7.11.2 Statistical tolerance interval 7.12 Goodness of fit tests 7.12.1 Chi-square test 7.12.2 Empirical distribution function tests 7.13 Summary 7.13.1 Notation 7.13.2 Summary of main results 7.13.3 MATLAB and R commands 7.14 Exercises 8: Linear regression and linear relationships 8.1 Linear regression 8.1.1 Introduction 8.1.2 The model 8.1.3 Fitting the model 8.1.3.1 Fitting the regression line 8.1.3.2 Identical forms for the least squares estimate of the slope 8.1.3.3 Relation to correlation 8.1.3.4 Alternative form for the fitted regression line 8.1.3.5 Residuals 8.1.3.6 Identities satisfied by the residuals 8.1.3.7 Estimating the standard deviation of the errors 8.1.3.8 Checking assumptions A3, A4 and A5 8.1.4 Properties of the estimators 8.1.4.1 Estimator of the slope 8.1.4.2 Estimator of the intercept 8.1.5 Predictions 8.1.5.1 Confidence interval for mean value of Y given x 8.1.5.2 Limits of prediction 8.1.5.3 Plotting confidence intervals and prediction limits 8.1.6 Summarizing the algebra 8.1.7 Coefficient of determination R2 8.2 Regression for a bivariate normal distribution 8.2.1 The bivariate normal distribution 8.3 Regression towards the mean 8.4 Relationship between correlation and regression 8.4.1 Values of x are assumed to be measured without error and can be preselected 8.4.2 The data pairs are assumed to be a random sample from a bivariate normal distribution 8.5 Fitting a linear relationship when both variables are measured with error 8.6 Calibration lines 8.7 Intrinsically linear models 8.8 Summary 8.8.1 Notation 8.8.2 Summary of main results 8.8.3 MATLAB and R commands 8.9 Exercises 9: Multiple regression 9.1 Introduction 9.2 Multivariate data 9.3 Multiple regression model 9.3.1 The linear model 9.3.2 Random vectors 9.3.2.1 Linear transformations of a random vector 9.3.2.2 Multivariate normal distribution 9.3.3 Matrix formulation of the linear model 9.3.4 Geometrical interpretation 9.4 Fitting the model 9.4.1 Principle of least squares 9.4.2 Multivariate calculus − Three basic results 9.4.3 The least squares estimator of the coefficients 9.4.4 Estimating the coefficients 9.4.5 Estimating the standard deviation of the errors 9.4.6 Standard errors of the estimators of the coefficients 9.5 Assessing the fit 9.5.1 The residuals 9.5.2 R-squared 9.5.3 F-statistic 9.5.4 Cross validation 9.6 Predictions 9.7 Building multiple regression models 9.7.1 Interactions 9.7.2 Categorical variables 9.7.3 F-test for an added set of variables 9.7.4 Quadratic terms 9.7.5 Guidelines for fitting regression models 9.8 Time series 9.8.1 Introduction 9.8.2 Aliasing and sampling intervals 9.8.3 Fitting a trend and seasonal variation with regression 9.8.4 Auto-covariance and auto-correlation 9.8.4.1 Defining auto-covariance for a stationary times series model 9.8.4.2 Defining sample auto-covariance and the correlogram 9.8.5 Auto-regressive models 9.8.5.1 AR(1) and AR(2) models 9.9 Non-linear least squares 9.10 Generalized linear model 9.10.1 Logistic regression 9.10.2 Poisson regression 9.11 Summary 9.11.1 Notation 9.11.2 Summary of main results 9.11.3 MATLAB and R commands 9.12 Exercises 10: Statistical quality control 10.1 Continuous improvement 10.1.1 Defining quality 10.1.2 Taking measurements 10.1.3 Avoiding rework 10.1.4 Strategies for quality improvement 10.1.5 Quality management systems 10.1.6 Implementing continuous improvement 10.2 Process stability 10.2.1 Runs chart 10.2.2 Histograms and box plots 10.2.3 Components of variance 10.3 Capability 10.3.1 Process capability index 10.3.2 Process performance index 10.3.3 One-sided process capability indices 10.4 Reliability 10.4.1 Introduction 10.4.1.1 Reliability of components 10.4.1.2 Reliability function and the failure rate 10.4.2 Weibull analysis 10.4.2.1 Definition of the Weibull distribution 10.4.2.2 Weibull quantile plot 10.4.2.3 Censored data 10.4.3 Maximum likelihood 10.4.4 Kaplan-Meier estimator of reliability 10.5 Acceptance sampling 10.6 Statistical quality control charts 10.6.1 Shewhart mean and range chart for continuous variables 10.6.1.1 Mean chart 10.6.1.2 Range chart 10.6.2 p-charts for proportions 10.6.3 c-charts for counts 10.6.4 Cumulative sum charts 10.6.5 Multivariate control charts 10.7 Summary 10.7.1 Notation 10.7.2 Summary of main results 10.7.3 MATLAB and R commands 10.8 Exercises 11: Design of experiments with regression analysis 11.1 Introduction 11.2 Factorial designs with factors at two levels 11.2.1 Full factorial designs 11.2.1.1 Setting up a 2k design 11.2.1.2 Analysis of 2k design 11.3 Fractional factorial designs 11.4 Central composite designs 11.5 Evolutionary operation (EVOP) 11.6 Summary 11.6.1 Notation 11.6.2 Summary of main results 11.6.3 MATLAB and R commands 11.7 Exercises 12: Design of experiments and analysis of variance 12.1 Introduction 12.2 Comparison of several means with one-way ANOVA 12.2.1 Defining the model 12.2.2 Limitation of multiple t-tests 12.2.3 One-way ANOVA 12.2.4 Testing H0O 12.2.5 Follow up procedure 12.3 Two factors at multiple levels 12.3.1 Two factors without replication (two-way ANOVA) 12.3.2 Two factors with replication (three-way ANOVA) 12.4 Randomized block design 12.5 Split plot design 12.6 Summary 12.6.1 Notation 12.6.2 Summary of main results 12.6.3 MATLAB and R commands 12.7 Exercises 13: Probability models 13.1 System reliability 13.1.1 Series system 13.1.2 Parallel system 13.1.3 k-out-of-n system 13.1.4 Modules 13.1.5 Duality 13.1.6 Paths and cut sets 13.1.7 Reliability function 13.1.8 Redundancy 13.1.9 Non-repairable systems 13.1.10 Standby systems 13.1.11 Common cause failures 13.1.12 Reliability bounds 13.2 Markov chains 13.2.1 Discrete Markov chain 13.2.2 Equilibrium behavior of irreducible Markov chains 13.2.3 Methods for solving equilibrium equations 13.2.4 Absorbing Markov chains 13.2.5 Markov chains in continuous time 13.3 Simulation of systems 13.3.1 The simulation procedure 13.3.2 Drawing inference from simulation outputs 13.3.3 Variance reduction 13.4 Summary 13.4.1 Notation 13.4.2 Summary of main results 13.5 Exercises 14: Sampling strategies 14.1 Introduction 14.2 Simple random sampling from a finite population 14.2.1 Finite population correction 14.2.2 Randomization theory 14.2.2.1 Defining the simple random sample 14.2.2.2 Mean and variance of sample mean 14.2.2.3 Mean and variance of estimator of population total 14.2.3 Model based analysis 14.2.4 Sample size 14.3 Stratified sampling 14.3.1 Principle of stratified sampling 14.3.2 Estimating the population mean and total 14.3.3 Optimal allocation of the sample over strata 14.4 Multi-stage sampling 14.5 Quota sampling 14.6 Ratio estimators and regression estimators 14.6.1 Introduction 14.6.2 Regression estimators 14.6.3 Ratio estimator 14.7 Calibration of the unit cost data base 14.7.1 Sources of error in an AMP 14.7.2 Calibration factor 14.8 Summary 14.8.1 Notation 14.8.2 Summary of main results 14.9 Exercises Appendix A - Notation A.1 General A.2 Probability A.3 Statistics A.4 Probability distributions Appendix B - Glossary Appendix C - Getting started in R C.1 Installing R C.2 Using R as a calculator C.3 Setting the path C.4 R scripts C.5 Data entry C.5.1 From keyboard C.5.2 From a file C.5.2.1 Single variable C.5.2.2 Several variables C.6 R vectors C.7 User defined functions C.8 Matrices C.9 Loops and conditionals C.10 Basic plotting C.11 Installing packages C.12 Creating time series objects Appendix D - Getting started in MATLAB D.1 Installing MATLAB D.2 Using MATLAB as a calculator D.3 Setting the path D.4 MATLAB scripts (m-files) D.5 Data entry D.5.1 From keyboard D.5.2 From a file D.5.2.1 Single variable D.5.2.2 Several variables D.6 MATLAB vectors D.7 User defined functions D.8 Matrices D.9 Loops and conditionals D.10 Basic plotting D.11 Creating time series objects Appendix E - Experiments E.1 How good is your probability assessment? E.1.1 Objectives E.1.2 Experiment E.1.3 Question sets E.1.4 Discussion E.1.5 Follow up questions E.2 Buffon's needle E.2.1 Objectives E.2.2 Experiment E.2.3 Questions E.2.4 Computer simulation E.2.5 Historical note E.3 Robot rabbit E.3.1 Objectives E.3.2 Experiment E.3.3 Data E.3.4 Discussion E.3.5 Follow up question E.4 Use your braking brains E.4.1 Objectives E.4.2 Experiment E.4.3 Discussion E.5 Predicting descent time from payload E.5.1 Objectives E.5.2 Experiment E.5.3 Discussion E.5.4 Follow up question E.6 Company effciency, resources and teamwork E.6.1 Objectives E.6.2 Experiment E.6.3 Discussion E.7 Factorial experiment − reaction times by distraction, dexterity and distinctness E.7.1 Aim E.7.2 Experiment E.7.3 Analysis E.7.4 Discussion E.7.5 Follow up questions E.8 Weibull analysis of cycles to failure E.8.1 Aim E.8.2 Experiment E.8.3 Weibull plot E.8.4 Discussion E.9 Control or tamper? E.10 Where is the summit? References Index