Prefacexixappendix.As in previous editions,some ofthe new exercises aretheoretical whereasothers contain data from documented sources that deal with research in a variety offields.Wecontinue to believe that exercises based on real data or actual experimentalscenarios permit students to see the practical uses of the various statistical and proba-bilistic methods presented in thetext.As theywork through these exercises,studentsgain insight into the real-life applications of the theoretical results developed in thetext. This insight makes learning the necessary theory more enjoyable and producesa deeper understanding of the theoretical methods. As in previous editions, the morechallenging exercises are marked with an asterisk (*). Answers to the odd-numberedexercises areprovided intheback ofthebook.TablesandAppendicesWe have maintained the use of the upper-tail normal tables because the users of thetextfindthemtobemoreconvenient.Wehavealsomaintainedtheformatofthetableof theFdistributions that we introduced in previous editions.Thistableof the Fdistributions provides critical values corresponding to upper-tail areas of.100, .050,.025,.010, and.005 in a single table.Because tests based on statistics possessingthe F distribution occurquite often,this tablefacilitates the computation of attainedsignificance levels, or p-values, associated with observed values of these statistics.Wehave also maintained our practice of providing easy access to often-usedinformation.Because the normal and t tables are the most frequently used statis-ticaltablesinthetext,copiesofthesetablesaregiveninAppendix3andinsidethefront coverof the text.Usersof previous editions haveoften remarkedfavorablyaboutthe utility of tables of the common probability distributions,means,variances,andmoment-generating functions provided in Appendix 2 and inside the back cover ofthetext.In addition,we haveincluded somefrequently usedmathematical results in asupplement toAppendix1.These results include thebinomial expansion of (x+y)"the series expansion of e', sums of geometric series, definitions of the gamma andbeta functions,and so on.As before,each chapter begins with an outline containingthe titles of the major sections in that chapter.AcknowledgmentsThe authors wish to thank themany colleagues,friends, and students who have madehelpful suggestions concerning the revisions of this text. In particular, we are indebtedto P.V.Rao, J.G. Saw, Malay Ghosh, Andrew Rosalsky, and Brett Presnell for theirtechnical comments. Gary McClelland, University of Colorado, did an outstandingjob of developing the applets used in the text. Jason Owen, University of Richmond,wrote the solutions manual.Mary Mortlock, Cal Poly,San Luis Obispo, checkedaccuracy.We wish to thank E.S.Pearson, W.H.Beyer, I. Olkin, R.A.Wilcox, C. WDunnett, and A. Hald. We profited substantially from the suggestions of the review-ers ofthe current andprevious editions of thetext:Roger Abernathy,Arkansas StateUniversity:ElizabethS.Allman,University of SouthernMaine;RobertBerk,RutgersCopyright 2011 Cengage LearsAll Rightshapter(s)Erors
Preface xix appendix. As in previous editions, some of the new exercises are theoretical whereas others contain data from documented sources that deal with research in a variety of fields. We continue to believe that exercises based on real data or actual experimental scenarios permit students to see the practical uses of the various statistical and probabilistic methods presented in the text. As they work through these exercises, students gain insight into the real-life applications of the theoretical results developed in the text. This insight makes learning the necessary theory more enjoyable and produces a deeper understanding of the theoretical methods. As in previous editions, the more challenging exercises are marked with an asterisk (*). Answers to the odd-numbered exercises are provided in the back of the book. Tables and Appendices We have maintained the use of the upper-tail normal tables because the users of the text find them to be more convenient. We have also maintained the format of the table of the F distributions that we introduced in previous editions. This table of the F distributions provides critical values corresponding to upper-tail areas of .100, .050, .025, .010, and .005 in a single table. Because tests based on statistics possessing the F distribution occur quite often, this table facilitates the computation of attained significance levels, or p-values, associated with observed values of these statistics. We have also maintained our practice of providing easy access to often-used information. Because the normal and t tables are the most frequently used statistical tables in the text, copies of these tables are given in Appendix 3 and inside the front cover of the text. Users of previous editions have often remarked favorably about the utility of tables of the common probability distributions, means, variances, and moment-generating functions provided in Appendix 2 and inside the back cover of the text. In addition, we have included some frequently used mathematical results in a supplement to Appendix 1. These results include the binomial expansion of (x + y) n , the series expansion of e x , sums of geometric series, definitions of the gamma and beta functions, and so on. As before, each chapter begins with an outline containing the titles of the major sections in that chapter. Acknowledgments The authors wish to thank the many colleagues, friends, and students who have made helpful suggestions concerning the revisions of this text. In particular, we are indebted to P. V. Rao, J. G. Saw, Malay Ghosh, Andrew Rosalsky, and Brett Presnell for their technical comments. Gary McClelland, University of Colorado, did an outstanding job of developing the applets used in the text. Jason Owen, University of Richmond, wrote the solutions manual. Mary Mortlock, Cal Poly, San Luis Obispo, checked accuracy. We wish to thank E. S. Pearson, W. H. Beyer, I. Olkin, R. A. Wilcox, C. W. Dunnett, and A. Hald. We profited substantially from the suggestions of the reviewers of the current and previous editions of the text: Roger Abernathy, Arkansas State University; Elizabeth S. Allman, University of Southern Maine; Robert Berk, Rutgers 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
xxPrefaceUniversity;AlbertBronstein,PurdueUniversity;SubhaChakraborti,UniversityofAl-abama;RitaChattopadhyay,EasternMichigan University;EricChicken,Florida StateUniversity;Charles Dunn, Linfield College;Eric Eide,Brigham Young University:Nelson Fong, Creighton University; Dr. Gail P. Greene, Indiana Wesleyan University;Barbara Hewitt, University of Texas, San Antonio;Richard Iltis, Willamette Univer-sity;K.G.Janardan,Eastern Michigan University:Mark Janeba,Willamette Univer-sity; Rick Jenison, Univeristy of Wisconsin, Madison; Jim Johnston, Concord Uni-versity;BessieH.Kirkwood, Sweet Briar College;Marc L.Komrosky,San Jose StateUniversity; Dr.Olga Korosteleva, California State University,Long Beach; Teck Ky,EvegreenValleyCollege;MatthewLebo,StonyBrook University;PhillipLestmann,Bryan College;Tamar London,Pennsylvania State University:Lisa Madsen,OregonState University; Martin Magid, Wellesley College; Hosam M. Mahmoud, GeorgeWashington University; Kim Maier, Michigan State University:David W.Matolak,Ohio University;James Edward Mays,Virginia Commonwealth University;Kather-ine McGivney, Shippensburg Univesity; Sanjog Misra, University of Rochester;Donald F.Morrison,University of Pennsylvania,Wharton;Mir A.Mortazavi,EasternNew Mexico University;Abdel-RazzagMugdadi, Southern Illinois University;OllieNanyes, Bradley University; Joshua Naranjo, Western Michigan University; SharonNavard,TheCollegeof New Jersey;Roger B.Nelsen,Lewis &ClarkCollege;DavidK.Park, Washington University;Cheng Peng,University of Southern Maine; SelwynPiramuthu,UniversityofFlorida,GainesvilleRobertMartinPrice,Jr.,EastTennesseeState University;Daniel Rabinowitz,Columbia University:Julianne Rainbolt, SaintLouis University; Timothy A.Riggle, Baldwin-WallaceCollege; Mark Rizzardi,Hum-boldt State University; Jesse Rothstein, Princeton University;Katherine Schindler,Eastern Michigan University: Michael E. Schuckers, St. Lawrence University; JeanT.Sells,Sacred HeartUniversity;Qin Shao,TheUniversityof Toledo;Alan Shuchat,Wellesley College; Laura J. Simon, Pennsylvania State University: Satyanand Singh,NewYorkCityCollegeofTechnology;RandallJ.Swift,California StatePolytechnicUniversity,Pomona;David Sze,MonmouthUniversity;BruceE.Trumbo,CaliforniaState University,East Bay;Harold Dean Victory,Jr.,Texas Tech University:ThomasO.Vinson,Washington & Lee University:Vasant Waikar,Miami University,Ohio;Bette Warren, Eastern Michigan University;SteveWhite, Jacksonville State Univer-sity; Shirley A. Wilson, North Central College; Lan Xue, Oregon State University:and Elaine Zanutto,The Wharton School, University of Pennsylvania.WealsowishtoacknowledgethecontributionsofCarolynCrockett.oureditorCatieRonquillo,assistanteditor;AshleySummers,editorial assistant;JenniferLiang,technology project manager; Mandy Jellerichs,marketingmanager; Ashley Pickering,marketing assistant; andofthose involved in the production ofthe text: Hal Humphrey.production projectmanager;BettyDuncan,copyeditor;and Merrill Peterson and SaraPlanck, production coordinators.Finally,we appreciatethe support ofourfamilies during the writing of the variouseditions of this text.DENNISD.WACKERLYWILLIAMMENDENHALLIIIRICHARDL.SCHEAFFERCopyright 2011 Cengage LearninAll RightsChapter(s)Editorial
xx Preface University; Albert Bronstein, Purdue University; Subha Chakraborti, University of Alabama; Rita Chattopadhyay, Eastern Michigan University; Eric Chicken, Florida State University; Charles Dunn, Linfield College; Eric Eide, Brigham Young University; Nelson Fong, Creighton University; Dr. Gail P. Greene, Indiana Wesleyan University; Barbara Hewitt, University of Texas, San Antonio; Richard Iltis, Willamette University; K. G. Janardan, Eastern Michigan University; Mark Janeba, Willamette University; Rick Jenison, Univeristy of Wisconsin, Madison; Jim Johnston, Concord University; Bessie H. Kirkwood, Sweet Briar College; Marc L. Komrosky, San Jose State University; Dr. Olga Korosteleva, California State University, Long Beach; Teck Ky, Evegreen Valley College; Matthew Lebo, Stony Brook University; Phillip Lestmann, Bryan College; Tamar London, Pennsylvania State University; Lisa Madsen, Oregon State University; Martin Magid, Wellesley College; Hosam M. Mahmoud, George Washington University; Kim Maier, Michigan State University; David W. Matolak, Ohio University; James Edward Mays, Virginia Commonwealth University; Katherine McGivney, Shippensburg Univesity; Sanjog Misra, University of Rochester; Donald F. Morrison, University of Pennsylvania, Wharton; Mir A. Mortazavi, Eastern New Mexico University; Abdel-Razzaq Mugdadi, Southern Illinois University; Ollie Nanyes, Bradley University; Joshua Naranjo, Western Michigan University; Sharon Navard, The College of New Jersey; Roger B. Nelsen, Lewis & Clark College; David K. Park, Washington University; Cheng Peng, University of Southern Maine; Selwyn Piramuthu, University of Florida, Gainesville; RobertMartin Price, Jr., East Tennessee State University; Daniel Rabinowitz, Columbia University; Julianne Rainbolt, Saint Louis University; TimothyA.Riggle, Baldwin-Wallace College;Mark Rizzardi, Humboldt State University; Jesse Rothstein, Princeton University; Katherine Schindler, Eastern Michigan University; Michael E. Schuckers, St. Lawrence University; Jean T. Sells, Sacred Heart University; Qin Shao, The University of Toledo; Alan Shuchat, Wellesley College; Laura J. Simon, Pennsylvania State University; Satyanand Singh, New York City College of Technology; Randall J. Swift, California State Polytechnic University, Pomona; David Sze, Monmouth University; Bruce E. Trumbo, California State University, East Bay; Harold Dean Victory, Jr., Texas Tech University; Thomas O. Vinson, Washington & Lee University; Vasant Waikar, Miami University, Ohio; Bette Warren, Eastern Michigan University; Steve White, Jacksonville State University; Shirley A. Wilson, North Central College; Lan Xue, Oregon State University; and Elaine Zanutto, The Wharton School, University of Pennsylvania. We also wish to acknowledge the contributions of Carolyn Crockett, our editor; Catie Ronquillo, assistant editor; Ashley Summers, editorial assistant; Jennifer Liang, technology project manager;Mandy Jellerichs, marketing manager; Ashley Pickering, marketing assistant; and of those involved in the production of the text: Hal Humphrey, production project manager; Betty Duncan, copyeditor; and Merrill Peterson and Sara Planck, production coordinators. Finally, we appreciate the support of our families during the writing of the various editions of this text. DENNIS D. WACKERLY WILLIAM MENDENHALL III RICHARD L. SCHEAFFER 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
NOTE TO THE STUDENTAs thetitle Mathematical Statistics with Applications implies,this text is concernedwith statistics, in both theory and application, and only deals with mathematics as anecessarytooltogiveyouafirmunderstandingofstatisticaltechniques.Thefollowingsuggestions for using the text will increase your learning and save your time.The connectivity of the book is provided by the introductions and summaries ineachchapter.These sections explain how each chapterfits into the overall picture ofstatistical inference and how each chapter relates to the preceding ones.FIGURE4alpha:3.2beta:4.6875Appletcalculationoftheprobabilitythata0.08gamma-distributedrandom variableexceeds its mean0.060.040.020.001002030405060x:15.00000Prob:0.42562xxiCopyright 2011 Cengage Lesning. All Rightspter(s)Editoria
NOTE TO THE STUDENT As the title Mathematical Statistics with Applications implies, this text is concerned with statistics, in both theory and application, and only deals with mathematics as a necessary tool to give you a firm understanding of statistical techniques. The following suggestions for using the text will increase your learning and save your time. The connectivity of the book is provided by the introductions and summaries in each chapter. These sections explain how each chapter fits into the overall picture of statistical inference and how each chapter relates to the preceding ones. FIGURE 4 Applet calculation of the probability that a gamma–distributed random variable exceeds its mean xxi 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
xxiiNotetotheStudentWithin the chapters, important concepts are set off as definitions.These should beread and reread until they are clearly understood because they form the frameworkon which everything else is built. The main theoretical results are set off as theo-rems. Although it is not necessary to understand the proof of each theorem, a clearunderstanding of the meaning and implications of the theorems is essential.It is also essential that you work manyof the exercises-for at leastfour reasons:·You can be certain that you understand what you have read only by putting yourknowledge to the test of working problems.Many of the exercises are of a practical nature and shed lighton theapplicationsof probability and statistics.·Someoftheexercisespresent newconcepts andthus extend thematerial coveredin the chapter.·Many of the applet exercises help build intuition, facilitate understanding ofconcepts, and provide answers that cannot (practically)be obtained using tablesin the appendices (see Figure 4).D.D.W.W.M.R.L.S.Copyrighs 2011 Cengage Leaning, All Rights ResesFeFLMOe0ter(s)Editorial res
xxii Note to the Student Within the chapters, important concepts are set off as definitions. These should be read and reread until they are clearly understood because they form the framework on which everything else is built. The main theoretical results are set off as theorems. Although it is not necessary to understand the proof of each theorem, a clear understanding of the meaning and implications of the theorems is essential. It is also essential that you work many of the exercises—for at least four reasons: • You can be certain that you understand what you have read only by putting your knowledge to the test of working problems. • Many of the exercises are of a practical nature and shed light on the applications of probability and statistics. • Some of the exercises present new concepts and thus extend the material covered in the chapter. • Many of the applet exercises help build intuition, facilitate understanding of concepts, and provide answers that cannot (practically) be obtained using tables in the appendices (see Figure 4). D. D. W. W. M. R. L. S. 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
CHAPTERWhat Is Statistics?1.1 Introduction1.2Characterizing a Set of Measurements:Graphical Methods1.3CharacterizingaSetofMeasurements:NumericalMethods1.4HowInferences AreMade1.5TheoryandReality1.6SummaryReferences and Further Readings1.1 IntroductionStatistical techniques are employed in almost every phase of life.Surveys are designedto collectearlyreturns on election day andforecast the outcomeof an election.Consumers are sampled toprovide informationfor predictingproductpreferencesResearchphysicians conductexperimentstodeterminetheeffectofvariousdrugsandcontrolled environmental conditions on humans in orderto infer the appropriatetreat-mentforvarious illnesses.Engineers sampleaproductqualitycharacteristicand var-ious controllable process variables to identifykeyvariables related to productqualityNewlymanufacturedelectronicdevices aresampledbeforeshippingtodecidewhetherto ship or hold individual lots.Economists observe various indices ofeconomic healthover a period oftimeandusethe information toforecastthecondition oftheeconomyinthefuture.Statistical techniquesplayan importantroleinachieving the objectiveofeachofthesepracticalsituations.Thedevelopmentofthetheoryunderlyingthesetechniques is thefocus of thistext.A prerequisite to a discussion of the theory of statistics is a definition of statis-tics and a statement of its objectives.Webster's New Collegiate Dictionary definesstatistics as“a branch of mathematics dealing with the collection, analysis, interpre-tation, and presentation of masses of numerical data." Stuart and Ord (1991) state:"Statistics is the branch of the scientific method whichdeals with the data obtainedbycountingormeasuring theproperties of populations."Rice(1995),commentingonexperimentation and statistical applications,states that statistics is“essentially con-cerned with procedures for analyzing data, especially data that in some vague sensehavea randomcharacter."FreundandWalpole(1987),among others,viewstatisticsas encompassing"the science of basing inferences on observed data and the entireCopyright 2011 CerRiahEfir
CHAPTER 1 What Is Statistics? 1.1 Introduction 1.2 Characterizing a Set of Measurements: Graphical Methods 1.3 Characterizing a Set of Measurements: Numerical Methods 1.4 How Inferences Are Made 1.5 Theory and Reality 1.6 Summary References and Further Readings 1.1 Introduction Statistical techniques are employed in almost every phase of life. Surveys are designed to collect early returns on election day and forecast the outcome of an election. Consumers are sampled to provide information for predicting product preferences. Research physicians conduct experiments to determine the effect of various drugs and controlled environmental conditions on humans in order to infer the appropriate treatment for various illnesses. Engineers sample a product quality characteristic and various controllable process variables to identify key variables related to product quality. Newlymanufactured electronic devices are sampled before shipping to decide whether to ship or hold individual lots. Economists observe various indices of economic health over a period of time and use the information to forecast the condition of the economy in the future. Statistical techniques play an important role in achieving the objective of each of these practical situations. The development of the theory underlying these techniques is the focus of this text. A prerequisite to a discussion of the theory of statistics is a definition of statistics and a statement of its objectives. Webster’s New Collegiate Dictionary defines statistics as “a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data.” Stuart and Ord (1991) state: “Statistics is the branch of the scientific method which deals with the data obtained by counting or measuring the properties of populations.” Rice (1995), commenting on experimentation and statistical applications, states that statistics is “essentially concerned with procedures for analyzing data, especially data that in some vague sense have a random character.” Freund and Walpole (1987), among others, view statistics as encompassing “the science of basing inferences on observed data and the entire 1 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