xivPrefaceone another.These integrating discussions appear most frequentlyin chapterintroductionsand conclusions.. Finally, the text is unique in its practical emphasis, both in exercises throughoutthe text and in the useful statistical methodological topics contained in Chap-ters 11-15, whose goal is to reinforce the elementary but sound theoreticalfoundation developed in the initial chapters.The book can be used in a variety of ways and adapted to the tastes of students andinstructors. The difficulty of the material can be increased or decreased by controllingthe assignment ofexercises,by eliminating some topics,and by varying the amount oftime devoted to each topic. A stronger applied flavor can be added by the eliminationof sometopics-forexample,some sectionsof Chapters6and7-andbydevotingmoretime to the applied chapters at the end.Changes in the Seventh EditionMany students are visual learners who can profit from visual reinforcement of con-cepts and results.New to this edition is the inclusion ofcomputer applets,all availablefor on line use at the Cengage Learning website, academic.cengage.com/statistics/wackerly. Some of these applets are used to demonstrate statistical concepts,otherapplets permit users to assess the impact of parameter choices on the shapes of densityfunctions,and the remainder of applets can be used tofind exact probabilities andquantiles associated with gamma-, beta-,normal-,x2-,t-,and F-distributed randomvariables—information of importance when constructing confidence intervals or per-forming tests of hypotheses.Some of theapplets provide information available viathe use of other software.Notably,the R language and environment for statisticalcomputationandgraphics(availablefreeathttp://www.r-project.org/)canbeusedtoprovide the quantiles and probabilities associated with the discrete and continuousdistributions previously mentioned. The appropriate R commands are given in therespective sections of Chapters 3 and 4.The advantage of the applets is that they are"point and shoot," provide accompanying graphics, and are considerably easier touse. However, R is vastly more powerful than the applets and can be used for manyother statistical purposes.We leave other applications of Rto the interested user orinstructor.Chapter 2 introduces the first applet, Bayes' Rule as a Tree, a demonstration thatallows users to see why sometimes surprising results occur when Bayes'rule is applied(see Figure 1).As in the sixthedition,maximum-likelihoodestimates are introduced inChapter3via examplesforthe estimates oftheparameters ofthebinomial,geometric.and negative binomial distributions based on specific observed numerical values ofrandom variables that possess these distributions. Follow-up problems at the end ofthe respective sections expand on these examples.In Chapter 4, the applet Normal Probabilities is used to compute the probabilitythat any user-specified, normally distributed random variable falls in any specifiedinterval. It also provides a graph of the selected normal density function and a visualreinforcement of thefact that probabilities associated with any normally distributedCopyright 2011 Cengage LeanAll RightChapter(s)Editoria
xiv Preface one another. These integrating discussions appear most frequently in chapter introductions and conclusions. • Finally, the text is unique in its practical emphasis, both in exercises throughout the text and in the useful statistical methodological topics contained in Chapters 11–15, whose goal is to reinforce the elementary but sound theoretical foundation developed in the initial chapters. The book can be used in a variety of ways and adapted to the tastes of students and instructors. The difficulty of the material can be increased or decreased by controlling the assignment of exercises, by eliminating some topics, and by varying the amount of time devoted to each topic. A stronger applied flavor can be added by the elimination of some topics—for example, some sections of Chapters 6 and 7—and by devoting more time to the applied chapters at the end. Changes in the Seventh Edition Many students are visual learners who can profit from visual reinforcement of concepts and results. New to this edition is the inclusion of computer applets, all available for on line use at the Cengage Learning website, academic.cengage.com/statistics/ wackerly. Some of these applets are used to demonstrate statistical concepts, other applets permit users to assess the impact of parameter choices on the shapes of density functions, and the remainder of applets can be used to find exact probabilities and quantiles associated with gamma-, beta-, normal-, χ 2 -, t-, and F-distributed random variables—information of importance when constructing confidence intervals or performing tests of hypotheses. Some of the applets provide information available via the use of other software. Notably, the R language and environment for statistical computation and graphics (available free at http://www.r-project.org/) can be used to provide the quantiles and probabilities associated with the discrete and continuous distributions previously mentioned. The appropriate R commands are given in the respective sections of Chapters 3 and 4. The advantage of the applets is that they are “point and shoot,” provide accompanying graphics, and are considerably easier to use. However, R is vastly more powerful than the applets and can be used for many other statistical purposes. We leave other applications of R to the interested user or instructor. Chapter 2 introduces the first applet, Bayes’ Rule as a Tree, a demonstration that allows users to see why sometimes surprising results occur when Bayes’ rule is applied (see Figure 1). As in the sixth edition, maximum-likelihood estimates are introduced in Chapter 3 via examples for the estimates of the parameters of the binomial, geometric, and negative binomial distributions based on specific observed numerical values of random variables that possess these distributions. Follow-up problems at the end of the respective sections expand on these examples. In Chapter 4, the applet Normal Probabilities is used to compute the probability that any user-specified, normally distributed random variable falls in any specified interval. It also provides a graph of the selected normal density function and a visual reinforcement of the fact that probabilities associated with any normally distributed Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
PrefacexvFIGURE10.00900.9000AppletillustrationofBayes'rule0.01000.1000~A0.0010P(B|A)=0.0090/(0.0090+0.0990)=0.08330.99000.100_A0.09900.9~A0.8910P(B)=0.0100P(A/B)=0.9000P(A/B)=0.1000random variable are equivalent to probabilities associated with the standard normaldistribution.The applet Normal Probabilities (OneTail)provides upper-tail areas as-sociated with any user-specified,normal distribution andcan alsobeused to establishthevaluethatcutsoffauser-specifiedareaintheuppertailforanynormallydistributedrandom variable.Probabilities and quantiles associated with standard normal randomvariables are obtained by selecting the parameter values mean = O and standard de-viation = 1. The beta and gamma distributions are more thoroughly explored in thischapter.Users can simultaneouslygraph three gamma (or beta)densities (all with userselected parametervalues)and assess the impactthat theparameter valueshave onthe shapes of gamma (or beta) density functions (see Figure 2).This is accomplishedFIGURE24.007Appletcomparisonofthreebeta densities3.002.00-T1.000.000.20.40.60.80.01.0Beta2BetalBeta372alpha:3beta:172Copyright 2011 Cengage Learning. All Rights Rhapter(s)Editorial revieyquire
Preface xv FIGURE 1 Applet illustration of Bayes’ rule random variable are equivalent to probabilities associated with the standard normal distribution. The applet Normal Probabilities (One Tail) provides upper-tail areas associated with any user-specified, normal distribution and can also be used to establish the value that cuts off a user-specified area in the upper tail for any normally distributed random variable. Probabilities and quantiles associated with standard normal random variables are obtained by selecting the parameter values mean = 0 and standard deviation = 1. The beta and gamma distributions are more thoroughly explored in this chapter. Users can simultaneously graph three gamma (or beta) densities (all with user selected parameter values) and assess the impact that the parameter values have on the shapes of gamma (or beta) density functions (see Figure 2). This is accomplished FIGURE 2 Applet comparison of three beta densities Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
xviPrefaceusingtheappletsComparisonofGammaDensityFunctionsandComparisonofBeta Density Functions,respectively.Probabilities and quantiles associated withgamma-andbeta-distributedrandomvariables areobtained using theapplets GammaProbabilities andQuantilesorBetaProbabilities and Quantiles.Sets of AppletEx-ercises are provided to guide the user to discover interesting and informative re-sults associated with normal-, beta-, and gamma- (including exponential and x)distributedrandom variables.We maintain emphasis on the x? distribution, includingsome theoretical results that are useful in the subsequent development of thet and FdistributionsIn Chapter 5, it is made clear that conditional densities are undefined for values ofthe conditioning variable where the marginal density is zero. We have also retainedthe discussion of the“conditional variance" and its use in finding the variance ofa random variable. Hierarchical models are briefly discussed. As in the previousedition,Chapter6introducesthe conceptofthesupportofa densityand emphasizesthat a transformation method can be used when the transformation is monotone on theregion of support.The Jacobian method is includedfor implementation of a bivariatetransformationIn Chapter7,the applet Comparison of Student's tand Normal Distributions per-mits visualization ofsimilarities and differences int and standardnormal densityfunc-tions,andtheapplets Chi-SquareProbabilitiesandQuantiles, Student'stProbabili-ties and Quantiles, and F-RatioProbabilities and Quantilesprovideprobabilites andquantiles associated with the respective distributions, all with user-specified degreesof freedom.TheappletDiceSample uses thefamiliar die-tossing exampleto intro-duce the concept of a sampling distribution. The results for different sample sizespermit the user to assess the impact of sample size on the sampling distribution of thesample mean.The applet also permits visualization of how the sampling distributionis affected if the die is not balanced.Under the general heading of "Sampling Dis-tributions and the Central Limit Theorem,four different applets illustrate differentconcepts:? Basic illustrates that, when sampling from a normally distributed population,the sample mean is itself normally distributed..SampleSize exhibits theeffectof the sample size on the sampling distribution ofthe sample mean.The samplingdistribution fortwo (user-selected)sample sizesare simultaneously generated and displayed side by side. Similarities and differ-ences ofthe samplingdistributions becomeapparent.Samples can begeneratedfrom populations with"normal,"uniform,U-shaped,and skewed distributions.The associated approximating normal sampling distributions can be overlayedontheresulting simulateddistributions,permitting immediatevisual assessmentof thequalityof thenormal approximation (seeFigure3).Variance simulates the sampling distribution of the sample variance when sam-pling from a population with a“normal"distribution.The theoretical (propor-tional to that of a x random variable) distribution can be overlayed with theclick of a button, again providing visual confirmation that theory really works..VarianceSizeallowsa comparison of theeffectof the sample size onthedistri-bution of the sample variance (again, sampling from a normal population).TheassociatedtheoreticaldensitycanbeoverlayedtoseethatthetheoryactuallyCopyright 2011 Cengage LeaninAll RightcaChapter(s)Editorial
xvi Preface using the applets Comparison of Gamma Density Functions and Comparison of Beta Density Functions, respectively. Probabilities and quantiles associated with gamma- and beta-distributed random variables are obtained using the applets Gamma Probabilities and Quantiles or Beta Probabilities and Quantiles. Sets of Applet Exercises are provided to guide the user to discover interesting and informative results associated with normal-, beta-, and gamma- (including exponential and χ 2 ) distributed random variables. We maintain emphasis on the χ 2 distribution, including some theoretical results that are useful in the subsequent development of the t and F distributions. In Chapter 5, it is made clear that conditional densities are undefined for values of the conditioning variable where the marginal density is zero. We have also retained the discussion of the “conditional variance” and its use in finding the variance of a random variable. Hierarchical models are briefly discussed. As in the previous edition, Chapter 6 introduces the concept of the support of a density and emphasizes that a transformation method can be used when the transformation is monotone on the region of support. The Jacobian method is included for implementation of a bivariate transformation. In Chapter 7, the applet Comparison of Student’s t and Normal Distributions permits visualization of similarities and differences in t and standard normal density functions, and the applets Chi-Square Probabilities and Quantiles, Student’s t Probabilities and Quantiles, and F-Ratio Probabilities and Quantiles provide probabilites and quantiles associated with the respective distributions, all with user-specified degrees of freedom. The applet DiceSample uses the familiar die-tossing example to introduce the concept of a sampling distribution. The results for different sample sizes permit the user to assess the impact of sample size on the sampling distribution of the sample mean. The applet also permits visualization of how the sampling distribution is affected if the die is not balanced. Under the general heading of “Sampling Distributions and the Central Limit Theorem,” four different applets illustrate different concepts: • Basic illustrates that, when sampling from a normally distributed population, the sample mean is itself normally distributed. • SampleSize exhibits the effect of the sample size on the sampling distribution of the sample mean. The sampling distribution for two (user-selected) sample sizes are simultaneously generated and displayed side by side. Similarities and differences of the sampling distributions become apparent. Samples can be generated from populations with “normal,” uniform, U-shaped, and skewed distributions. The associated approximating normal sampling distributions can be overlayed on the resulting simulated distributions, permitting immediate visual assessment of the quality of the normal approximation (see Figure 3). • Variance simulates the sampling distribution of the sample variance when sampling from a population with a “normal” distribution. The theoretical (proportional to that of a χ 2 random variable) distribution can be overlayed with the click of a button, again providing visual confirmation that theory really works. • VarianceSize allows a comparison of the effect of the sample size on the distribution of the sample variance (again, sampling from a normal population). The associated theoretical density can be overlayed to see that the theory actually Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
Preface xviiFIGURE3Population:M=16.50S=11.5216Distribution:Applet illustration ofthecentral limit12-theorem.U-Shaped8-Sample Size:A06131902632Selectablebelow3A10ASampleSizeSampleSize0) 1000 M= 16.40 S= 6.711000M=16.495=3.601607160120408040200+0-06131926320613192632Reset1Sample1000SamplesToggleNormalworks.In addition, it is seen thatfor large sample sizes the sample variance hasanapproximatenormaldistribution.TheappletNormal ApproximationtotheBinomialpermitstheusertoassessthequalityof the the(continuous)normal approximationfor (discrete)binomial probabilities.As in previous chapters,a sequence of AppletExercises leads theuser to discoverimportantandinterestinganswers andconcepts.Fromamoretheoreticalperspective,we establish the independence of the sample mean and sample variance for a sampleof size 2from a normal distribution.As before,theproof of this result for generaln is contained in an optional exercise.Exercises provide step-by-step derivations ofthemean and varianceforrandom variableswithtandFdistributions.Throughout Chapter 8, we have stressed the assumptions associated with confi-dence intervals based on the t distributions.Wehave also included a brief discussionof therobustness of thetprocedures and thelack of suchfor the intervals basedon thexandFdistributions.TheappletConfidencelntervalPillustrates propertiesof large-sample confidence intervals for a population proportion. In Chapter 9, theappletsPointSingle,PointbyPoint,andPointEstimationultimatelyleadto avery niceCopyright 2011 CengaRialEtito
Preface xvii FIGURE 3 Applet illustration of the central limit theorem. works. In addition, it is seen that for large sample sizes the sample variance has an approximate normal distribution. The appletNormal Approximation to the Binomial permits the user to assess the quality of the the (continuous) normal approximation for (discrete) binomial probabilities. As in previous chapters, a sequence of Applet Exercises leads the user to discover important and interesting answers and concepts. From a more theoretical perspective, we establish the independence of the sample mean and sample variance for a sample of size 2 from a normal distribution. As before, the proof of this result for general n is contained in an optional exercise. Exercises provide step-by-step derivations of the mean and variance for random variables with t and F distributions. Throughout Chapter 8, we have stressed the assumptions associated with confi- dence intervals based on the t distributions. We have also included a brief discussion of the robustness of the t procedures and the lack of such for the intervals based on the χ 2 and F distributions. The applet ConfidenceIntervalP illustrates properties of large-sample confidence intervals for a population proportion. In Chapter 9, the applets PointSingle, PointbyPoint, and PointEstimation ultimately lead to a very nice Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it
xviliPrefaceillustration ofconvergence inprobability.In Chapter 10,the appletHypothesis Testing(forProportions)illustrates importantconcepts associated withtest of hypothesesincludingthefollowing:.Whatdoes α reallymean?:Tests based on larger-sample sizestypicallyhavesmallerprobabilities of typeII errors if the level of the tests stays fixed.·Forafixed sample size,thepowerfunction increases as the valueofthe parametermoves further from the values specified by the null hypothesis.Once users visualize these concepts, the subsequent theoretical developments aremore relevant and meaningful. Applets for the x,t,F distributions are used toobtain exact p-values for associated tests of hypotheses.We also illustrate explicitlythat the power of a uniformly most powerful test can be smaller (although the largestpossible) than desired.In Chapter 11, the simplelinear regression modelisthoroughly discussed(includingconfidenceintervals,prediction intervals,andcorrelation)beforethematrix approachto multiple linear regression model is introduced. The applets Fitting a Line UsingLeast Squares and Removing Points from Regression illustrate what the least-squarescriterion accomplishes and that a few unusual data points can have considerableimpacton the fitted regression line.The coefficients of determination and multipledetermination are introduced, discussed, and related to the relevant t and F statistics.Exercises demonstrate that high (low) coefficients of (multiple)determination valuesdo not necessarilycorrespond to statistically significant (insignificant)results.Chapter 12 includes a separate section on the matched-pairs experiment. Althoughmany possible sets of dummy variables can be used to cast the analysis of varianceinto a regression context, in Chapter 13 we focus on the dummy variables typicallyused by SAS and other statistical analysis computing packages.The text stillfocusesprimarily on the randomized block design with fixed (nonrandom) block effects. Ifan instructor wishes,a series of supplemental exercises dealing with the randomizedblock design with random block effects can be used to illustrate the similarities anddifferences of these two versions of the randomized block design.The newChapter16providesa brief introductionto Bayesian methods ofstatisticalinference. The chapter focuses on using the data and the prior distribution to obtainthe posterior and using the posterior to produce estimates,credible intervals, and hy-pothesis tests for parameters.The applet Binomial Revisionfacilitates understandingof the process by which data are used to update the prior and obtain theposterior.Many of the posterior distributions are beta or gamma distributions, and previouslydiscussed applets are instrumental in obtaining credible intervals or computing theprobabilityofvarioushypotheses.The ExercisesThis edition contains more than350 newexercises.Manyof the newexercises use theapplets previously mentioned to guide the user through a series of steps that lead tomorethorough understanding of importantconcepts.Others usethe appletstoprovideconfidence intervals orp-valuesthatcould onlybe approximatedbyusingtables in theCopyright 2011 Cengage Learning. All Rights ReChapter(s)Editorial
xviii Preface illustration of convergence in probability. In Chapter 10, the applet Hypothesis Testing (for Proportions) illustrates important concepts associated with test of hypotheses including the following: • What does α really mean? • Tests based on larger-sample sizes typically have smaller probabilities of type II errors if the level of the tests stays fixed. • For a fixed sample size, the power function increases as the value of the parameter moves further from the values specified by the null hypothesis. Once users visualize these concepts, the subsequent theoretical developments are more relevant and meaningful. Applets for the χ 2 , t, F distributions are used to obtain exact p-values for associated tests of hypotheses. We also illustrate explicitly that the power of a uniformly most powerful test can be smaller (although the largest possible) than desired. In Chapter 11, the simple linear regression model is thoroughly discussed (including confidence intervals, prediction intervals, and correlation) before the matrix approach to multiple linear regression model is introduced. The applets Fitting a Line Using Least Squares and Removing Points from Regression illustrate what the least-squares criterion accomplishes and that a few unusual data points can have considerable impact on the fitted regression line. The coefficients of determination and multiple determination are introduced, discussed, and related to the relevant t and F statistics. Exercises demonstrate that high (low) coefficients of (multiple) determination values do not necessarily correspond to statistically significant (insignificant) results. Chapter 12 includes a separate section on the matched-pairs experiment. Although many possible sets of dummy variables can be used to cast the analysis of variance into a regression context, in Chapter 13 we focus on the dummy variables typically used by SAS and other statistical analysis computing packages. The text still focuses primarily on the randomized block design with fixed (nonrandom) block effects. If an instructor wishes, a series of supplemental exercises dealing with the randomized block design with random block effects can be used to illustrate the similarities and differences of these two versions of the randomized block design. The new Chapter 16 provides a brief introduction to Bayesian methods of statistical inference. The chapter focuses on using the data and the prior distribution to obtain the posterior and using the posterior to produce estimates, credible intervals, and hypothesis tests for parameters. The applet Binomial Revision facilitates understanding of the process by which data are used to update the prior and obtain the posterior. Many of the posterior distributions are beta or gamma distributions, and previously discussed applets are instrumental in obtaining credible intervals or computing the probability of various hypotheses. The Exercises This edition contains more than 350 new exercises. Many of the new exercises use the applets previously mentioned to guide the user through a series of steps that lead to more thorough understanding of important concepts. Others use the applets to provide confidence intervals or p-values that could only be approximated by using tables in the Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it