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About this book

This 3rd edition of Modern Mathematical Statistics with Applications tries to strike a balance between mathematical foundations and statistical practice. The book provides a clear and current exposition of statistical concepts and methodology, including many examples and exercises based on real data gleaned from publicly available sources. Here is a small but representative selection of scenarios for our examples and exercises based on information in recent articles: Use of the “Big Mac index” by the publication The Economist as a humorous way to compare product costs across nationsVisualizing how the concentration of lead levels in cartridges varies for each of five brands of e-cigarettesDescribing the distribution of grip size among surgeons and how it impacts their ability to use a particular brand of surgical staplerEstimating the true average odometer reading of used Porsche Boxsters listed for sale on www.cars.com Comparing head acceleration after impact when wearing a football helmet with acceleration without a helmetInvestigating the relationship between body mass index and foot load while running The main focus of the book is on presenting and illustrating methods of inferential statistics used by investigators in a wide variety of disciplines, from actuarial science all the way to zoology. It begins with a chapter on descriptive statistics that immediately exposes the reader to the analysis of real data. The next six chapters develop the probability material that facilitates the transition from simply describing data to drawing formal conclusions based on inferential methodology. Point estimation, the use of statistical intervals, and hypothesis testing are the topics of the first three inferential chapters. The remainder of the book explores the use of these methods in a variety of more complex settings. This edition includes many new examples and exercises as well as an introduction to the simulation of events and probability distributions. There are more than 1300 exercises in the book, ranging from very straightforward to reasonably challenging. Many sections have been rewritten with the goal of streamlining and providing a more accessible exposition. Output from the most common statistical software packages is included wherever appropriate (a feature absent from virtually all other mathematical statistics textbooks). The authors hope that their enthusiasm for the theory and applicability of statistics to real world problems will encourage students to pursue more training in the discipline.

Table of Contents

Frontmatter

Chapter 1. Overview and Descriptive Statistics

Abstract
Statistical concepts and methods are not only useful but indeed often indispensable in understanding the world around us. They provide ways of gaining new insights into the behavior of many phenomena that you will encounter in your chosen field of specialization.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 2. Probability

Abstract
The term probability refers to the study of randomness and uncertainty. In any situation in which one of a number of possible outcomes may occur, the theory of probability provides methods for quantifying the chances, or likelihoods, associated with the various outcomes. The language of probability is constantly used in an informal manner in both written and spoken contexts. Examples include such statements as “It is likely that the Dow Jones Industrial Average will increase by the end of the year,” “There is a 50–50 chance that the incumbent will seek reelection,” “There will probably be at least one section of that course offered next year,” “The odds favor a quick settlement of the strike,” and “It is expected that at least 20,000 concert tickets will be sold.” In this chapter, we introduce some elementary probability concepts, indicate how probabilities can be interpreted, and show how the rules of probability can be applied to compute the probabilities of many interesting events. The methodology of probability will then permit us to express in precise language such informal statements as those given above.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 3. Discrete Random Variables and Probability Distributions

Abstract
Suppose a city’s traffic engineering department monitors a certain intersection during a one-hour period in the middle of the day. Many characteristics might be of interest: the number of vehicles that enter the intersection, the largest number of vehicles in the left turn lane during a signal cycle, the speed of the fastest vehicle going through the intersection, the average speed \( \bar{x} \) of all vehicles entering the intersection.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 4. Continuous Random Variables and Probability Distributions

Abstract
As mentioned at the beginning of Chapter 3, the two important types of random variables are discrete and continuous. In this chapter, we study the second general type of random variable that arises in many applied problems. Sections 4.1 and 4.2 present the basic definitions and properties of continuous random variables, their probability distributions, and their various expected values. In Section 4.3, we study in detail the normal distribution, arguably the most important and useful in probability and statistics. Sections 4.4 and 4.5 discuss some other continuous distributions that are often used in applied work. In Section 4.6, we introduce a method for assessing whether given sample data is consistent with a specified distribution. Section 4.7 presents methods for obtaining the distribution of a rv Y from the distribution of X when the two are related by some equation Y = g(X). The last section is dedicated to the simulation of continuous rvs.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 5. Joint Probability Distributions and Their Applications

Abstract
In Chapters 3 and 4, we developed probability models for a single random variable. Many problems in probability and statistics lead to models involving several random variables simultaneously.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 6. Statistics and Sampling Distributions

Abstract
This chapter helps make the transition between probability and inferential statistics. Given a sample of n observations from a population, we will be calculating estimates of the population mean, median, standard deviation, and various other population characteristics (parameters). Prior to obtaining data, there is uncertainty as to which of all possible samples will occur.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 7. Point Estimation

Abstract
Given a parameter of interest, such as a population mean μ or population proportion p, the objective of point estimation is to use a sample to compute a number that represents, in some sense, a “good guess” for the true value of the parameter. The resulting number is called a point estimate.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 8. Statistical Intervals Based on a Single Sample

Abstract
A point estimate, because it is a single number, by itself provides no information about the precision and reliability of estimation. Consider, for example, using the statistic \( \overline{X} \) to calculate a point estimate for the true average breaking strength of a certain brand of paper towels, and suppose that \( \bar{x} \) = 9322.7 g. Because of sampling variability, it is virtually never the case that \( \bar{x} = \mu \).
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 9. Tests of Hypotheses Based on a Single Sample

Abstract
A parameter can be estimated from sample data either by a single number (a point estimate) or an entire interval of plausible values (a confidence interval). Frequently, however, the objective of an investigation is not to estimate a parameter but to decide which of two contradictory claims about the parameter is correct. Methods for accomplishing this comprise the part of statistical inference called hypothesis testing. In this chapter, we first discuss some of the basic concepts and terminology in hypothesis testing and then develop decision procedures for the most frequently encountered testing problems based on a sample from a single population.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 10. Inferences Based on Two Samples

Abstract
Chapters 8 and 9 presented confidence intervals (CIs) and hypothesis-testing procedures for single parameters, such as a population mean μ and a population proportion p.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 11. The Analysis of Variance

Abstract
In studying methods for the analysis of quantitative data, we first focused on problems involving a single sample of numbers and then turned to a comparative analysis of two different samples. Now we are ready for the analysis of several samples.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 12. Regression and Correlation

Abstract
The general objective of a regression analysis is to investigate the relationship between two (or more) variables so that we can gain information about one of them through knowing values of the other(s). Much of mathematics is devoted to studying variables that are deterministically related, meaning that once we are told the value of x, the value of y is completely specified.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 13. Chi-Squared Tests

Abstract
In the simplest type of situation considered in this chapter, each observation in a sample is classified as belonging to one of a finite number of categories—for example, blood type could be one of the four categories O, A, B, or AB. With pi denoting the probability that any particular observation belongs in category i, we wish to test a null hypothesis that completely specifies the values of all the pi’s (such as H0: p1 = .45, p2 = .35, p3 = .15, p4 = .05).
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 14. Nonparametric Methods

Abstract
In this chapter we consider some inferential methods that are different in important ways from those considered earlier. Recall that many of the confidence intervals and test procedures developed in Chapters 8, 9, 10, 11 and 12 were based on some sort of a normality assumption.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Chapter 15. Introduction to Bayesian Estimation

Abstract
In this final chapter, we briefly introduce the Bayesian approach to parameter estimation.
Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton

Backmatter

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