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1.Engineering—Statisticalmethods.2.Probabilities.I.Walpole,RonaldE. TA340.P7382011
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1.4MeasuresofVariability..
1.7GeneralTypesofStatisticalStudies:DesignedExperiment, ObservationalStudy,andRetrospectiveStudy..
2.8PotentialMisconceptionsandHazards;RelationshiptoMaterial inOtherChapters ...............................................79
3RandomVariablesandProbabilityDistributions ......
3.1ConceptofaRandomVariable..................................81
3.2DiscreteProbabilityDistributions..
3.3ContinuousProbabilityDistributions
3.4JointProbabilityDistributions.
3.5PotentialMisconceptionsandHazards;RelationshiptoMaterial inOtherChapters
4MathematicalExpectation ................................
4.1MeanofaRandomVariable.....................................111
4.2VarianceandCovarianceofRandomVariables...................119
4.3MeansandVariancesofLinearCombinationsofRandomVariables128
4.4Chebyshev’sTheorem...........................................135
4.5PotentialMisconceptionsandHazards;RelationshiptoMaterial inOtherChapters ...............................................142
5SomeDiscreteProbabilityDistributions ................
5.1IntroductionandMotivation....................................143
5.2BinomialandMultinomialDistributions..
5.3HypergeometricDistribution...
5.4NegativeBinomialandGeometricDistributions...
5.5PoissonDistributionandthePoissonProcess.
5.6PotentialMisconceptionsandHazards;RelationshiptoMaterial inOtherChapters ...............................................169
6SomeContinuousProbabilityDistributions
6.1ContinuousUniformDistribution
6.2NormalDistribution...
6.3AreasundertheNormalCurve..
6.4ApplicationsoftheNormalDistribution...
6.5NormalApproximationtotheBinomial...
6.6GammaandExponentialDistributions.
6.7Chi-SquaredDistribution
6.8BetaDistribution
6.9LognormalDistribution
6.10WeibullDistribution(Optional).
6.11PotentialMisconceptionsandHazards;RelationshiptoMaterial
7FunctionsofRandomVariables(Optional) ..............
7.1Introduction....................................................211
7.2TransformationsofVariables....................................211
7.3MomentsandMoment-GeneratingFunctions
8FundamentalSamplingDistributionsand DataDescriptions
8.1RandomSampling..
8.2SomeImportantStatistics.......................................227
8.3SamplingDistributions.
8.4SamplingDistributionofMeansandtheCentralLimitTheorem.233
8.5SamplingDistributionof
8.6
8.7 F
8.8QuantileandProbabilityPlots...
8.9PotentialMisconceptionsandHazards;RelationshiptoMaterial
9One-andTwo-SampleEstimationProblems
9.1Introduction....................................................265
9.2StatisticalInference.............................................265
9.3ClassicalMethodsofEstimation... ..............................266
9.4SingleSample:EstimatingtheMean ............................269
9.5StandardErrorofaPointEstimate .............................276
9.6PredictionIntervals.............................................277
9.7ToleranceLimits................................................280
9.8TwoSamples:EstimatingtheDifferencebetweenTwoMeans...285
9.9PairedObservations.............................................291
9.10SingleSample:EstimatingaProportion. ........................296
9.11TwoSamples:EstimatingtheDifferencebetweenTwoProportions300 Exercises...................................................302
9.12SingleSample:EstimatingtheVariance. ........................303
9.13TwoSamples:EstimatingtheRatioofTwoVariances..
9.14MaximumLikelihoodEstimation(Optional).
9.15PotentialMisconceptionsandHazards;RelationshiptoMaterial inOtherChapters ...............................................316
10One-andTwo-SampleTestsofHypotheses
10.1StatisticalHypotheses:GeneralConcepts..
10.3TheUseof P -ValuesforDecisionMakinginTestingHypotheses.331 Exercises...................................................334
10.4SingleSample:TestsConcerningaSingleMean...
10.5TwoSamples:TestsonTwoMeans.............................342
10.6ChoiceofSampleSizeforTestingMeans
10.7GraphicalMethodsforComparingMeans.
10.8OneSample:TestonaSingleProportion..
10.9TwoSamples:TestsonTwoProportions...
10.10One-andTwo-SampleTestsConcerningVariances.
10.11Goodness-of-FitTest...
10.12TestforIndependence(CategoricalData).......................373
10.13TestforHomogeneity
11.1IntroductiontoLinearRegression...
11.7ChoiceofaRegressionModel.
11.8Analysis-of-VarianceApproach.
11.9TestforLinearityofRegression:DatawithRepeatedObservations416
11.10DataPlotsandTransformations.................................424
12.2EstimatingtheCoefficients..
12.4PropertiesoftheLeastSquaresEstimators
12.6ChoiceofaFittedModelthroughHypothesisTesting.
12.7SpecialCaseofOrthogonality(Optional)
12.8CategoricalorIndicatorVariables...............................472
12.9SequentialMethodsforModelSelection.........................476
12.10StudyofResidualsandViolationofAssumptions(ModelChecking).............................................................482
12.11CrossValidation, Cp ,andOtherCriteriaforModelSelection ....487
12.12SpecialNonlinearModelsforNonidealConditions...
12.13PotentialMisconceptionsandHazards;RelationshiptoMaterial inOtherChapters
13One-FactorExperiments:General
13.1Analysis-of-VarianceTechnique..
13.2TheStrategyofExperimentalDesign ............................508
13.3One-WayAnalysisofVariance:CompletelyRandomizedDesign (One-WayANOVA).............................................509
13.4TestsfortheEqualityofSeveralVariances.
13.5Single-Degree-of-FreedomComparisons..
13.6MultipleComparisons
13.7ComparingaSetofTreatmentsinBlocks
13.8RandomizedCompleteBlockDesigns............................533
13.9GraphicalMethodsandModelChecking... .....................540
13.10DataTransformationsinAnalysisofVariance...................543
13.11RandomEffectsModels...
13.12CaseStudy... ..................................................551
13.13PotentialMisconceptionsandHazards;RelationshiptoMaterial inOtherChapters
14FactorialExperiments(TwoorMoreFactors) ..........
14.1Introduction....................................................561
14.2InteractionintheTwo-FactorExperiment.......................562
14.3Two-FactorAnalysisofVariance................................565
14.4Three-FactorExperiments
14.5FactorialExperimentsforRandomEffectsandMixedModels
14.6PotentialMisconceptionsandHazards;RelationshiptoMaterial
15.1Introduction....................................................597
15.2The2k Factorial:CalculationofEffectsandAnalysisofVariance598
15.3Nonreplicated2k FactorialExperiment..........................604
15.4FactorialExperimentsinaRegressionSetting
15.6FractionalFactorialExperiments................................626
15.7AnalysisofFractionalFactorialExperiments....................632
15.8HigherFractionsandScreeningDesigns
15.9ConstructionofResolutionIIIandIVDesignswith8,16,and32 DesignPoints...................................................637
15.10OtherTwo-LevelResolutionIIIDesigns;ThePlackett-Burman
15.11IntroductiontoResponseSurfaceMethodology ..................639
15.12RobustParameterDesign.. .....................................643 Exercises...................................................652
15.13PotentialMisconceptionsandHazards;RelationshiptoMaterial
16.1NonparametricTests
16.3WilcoxonRank-SumTest..
16.5RunsTest.......................................................671
16.6ToleranceLimits................................................674
16.7RankCorrelationCoefficient
Preface
GeneralApproachandMathematicalLevel
Ouremphasisincreatingthenintheditionislessonaddingnewmaterialandmore onprovidingclarityanddeeperunderstanding.Thisobjectivewasaccomplishedin partbyincludingnewend-of-chaptermaterialthataddsconnectivetissuebetween chapters.Weaffectionatelycallthesecommentsattheendofthechapter“Pot Holes.”Theyareveryusefultoremindstudentsofthebigpictureandhoweach chapterfitsintothatpicture,andtheyaidthestudentinlearningaboutlimitations andpitfallsthatmayresultifproceduresaremisused.Adeeperunderstanding ofreal-worlduseofstatisticsismadeavailablethroughclassprojects,whichwere addedinseveralchapters.Theseprojectsprovidetheopportunityforstudents alone,oringroups,togathertheirownexperimentaldataanddrawinferences.In somecases,theworkinvolvesaproblemwhosesolutionwillillustratethemeaning ofaconceptorprovideanempiricalunderstandingofanimportantstatistical result.Someexistingexampleswereexpandedandnewoneswereintroducedto create“casestudies,”inwhichcommentaryisprovidedtogivethestudentaclear understandingofastatisticalconceptinthecontextofapracticalsituation.
Inthisedition,wecontinuetoemphasizeabalancebetweentheoryandapplications.Calculusandothertypesofmathematicalsupport(e.g.,linearalgebra) areusedataboutthesamelevelasinpreviouseditions.Thecoverageofanalyticaltoolsinstatisticsisenhancedwiththeuseofcalculuswhendiscussion centersonrulesandconceptsinprobability.ProbabilitydistributionsandstatisticalinferencearehighlightedinChapters2through10.Linearalgebraand matricesareverylightlyappliedinChapters11through15,wherelinearregressionandanalysisofvariancearecovered.Studentsusingthistextshouldhave hadtheequivalentofonesemesterofdifferentialandintegralcalculus.Linear algebraishelpfulbutnotnecessarysolongasthesectioninChapter12onmultiplelinearregressionusingmatrixalgebraisnotcoveredbytheinstructor.As inpreviouseditions,alargenumberofexercisesthatdealwithreal-lifescientific andengineeringapplicationsareavailabletochallengethestudent.Themany datasetsassociatedwiththeexercisesareavailablefordownloadfromthewebsite http://www.pearsonhighered.com/datasets.
• Classprojectswereaddedinseveralchapterstoprovideadeeperunderstandingofthereal-worlduseofstatistics.Studentsareaskedtoproduceorgather theirownexperimentaldataanddrawinferencesfromthesedata.
• Morecasestudieswereaddedandothersexpandedtohelpstudentsunderstandthestatisticalmethodsbeingpresentedinthecontextofareal-lifesituation.Forexample,theinterpretationofconfidencelimits,predictionlimits, andtolerancelimitsisgivenusingareal-lifesituation.
• “PotHoles”wereaddedattheendofsomechaptersandexpandedinothers. Thesecommentsareintendedtopresenteachchapterinthecontextofthe bigpictureanddiscusshowthechaptersrelatetooneanother.Theyalso providecautionsaboutthepossiblemisuseofstatisticaltechniquespresented inthechapter.
• Chapter1hasbeenenhancedtoincludemoreonsingle-numberstatisticsas wellasgraphicaltechniques.Newfundamentalmaterialonsamplingand experimentaldesignispresented.
• ExamplesaddedtoChapter8onsamplingdistributionsareintendedtomotivate P -valuesandhypothesistesting.Thispreparesthestudentforthemore challengingmaterialonthesetopicsthatwillbepresentedinChapter10.
• Chapter12containsadditionaldevelopmentregardingtheeffectofasingle regressionvariableinamodelinwhichcollinearitywithothervariablesis severe.
• Chapter15nowintroducesmaterialontheimportanttopicofresponsesurface methodology(RSM).TheuseofnoisevariablesinRSMallowstheillustration ofmeanandvariance(dualresponsesurface)modeling.
• Thecentralcompositedesign(CCD)isintroducedinChapter15.
• MoreexamplesaregiveninChapter18,andthediscussionofusingBayesian methodsforstatisticaldecisionmakinghasbeenenhanced.
ContentandCoursePlanning
Thistextisdesignedforeitheraone-oratwo-semestercourse.Areasonable planforaone-semestercoursemightincludeChapters1through10.Thiswould resultinacurriculumthatconcludedwiththefundamentalsofbothestimation andhypothesistesting.Instructorswhodesirethatstudentsbeexposedtosimple linearregressionmaywishtoincludeaportionofChapter11.Forinstructors whodesiretohaveanalysisofvarianceincludedratherthanregression,theonesemestercoursemayincludeChapter13ratherthanChapters11and12.Chapter 13featuresone-factoranalysisofvariance.Anotheroptionistoeliminateportions ofChapters5and/or6aswellasChapter7.Withthisoption,oneormoreof thediscreteorcontinuousdistributionsinChapters5and6maybeeliminated. Thesedistributionsincludethenegativebinomial,geometric,gamma,Weibull, beta,andlognormaldistributions.Otherfeaturesthatonemightconsiderremovingfromaone-semestercurriculumincludemaximumlikelihoodestimation,
prediction,and/ortolerancelimitsinChapter9.Aone-semestercurriculumhas built-inflexibility,dependingontherelativeinterestoftheinstructorinregression, analysisofvariance,experimentaldesign,andresponsesurfacemethods(Chapter 15).Thereareseveraldiscreteandcontinuousdistributions(Chapters5and6) thathaveapplicationsinavarietyofengineeringandscientificareas.
Chapters11through18containsubstantialmaterialthatcanbeaddedforthe secondsemesterofatwo-semestercourse.Thematerialonsimpleandmultiple linearregressionisinChapters11and12,respectively.Chapter12aloneoffersa substantialamountofflexibility.Multiplelinearregressionincludessuch“special topics”ascategoricalorindicatorvariables,sequentialmethodsofmodelselection suchasstepwiseregression,thestudyofresidualsforthedetectionofviolations ofassumptions,crossvalidationandtheuseofthePRESSstatisticaswellas Cp ,andlogisticregression.Theuseoforthogonalregressors,aprecursortothe experimentaldesigninChapter15,ishighlighted.Chapters13and14offera relativelylargeamountofmaterialonanalysisofvariance(ANOVA)withfixed, random,andmixedmodels.Chapter15highlightstheapplicationoftwo-level designsinthecontextoffullandfractionalfactorialexperiments(2k ).Special screeningdesignsareillustrated.Chapter15alsofeaturesanewsectiononresponse surfacemethodology(RSM)toillustratetheuseofexperimentaldesignforfinding optimalprocessconditions.Thefittingofasecondordermodelthroughtheuseof acentralcompositedesignisdiscussed.RSMisexpandedtocovertheanalysisof robustparameterdesigntypeproblems.Noisevariablesareusedtoaccommodate dualresponsesurfacemodels.Chapters16,17,and18containamoderateamount ofmaterialonnonparametricstatistics,qualitycontrol,andBayesianinference. Chapter1isanoverviewofstatisticalinferencepresentedonamathematically simplelevel.Ithasbeenexpandedfromtheeightheditiontomorethoroughly coversingle-numberstatisticsandgraphicaltechniques.Itisdesignedtogive studentsapreliminarypresentationofelementaryconceptsthatwillallowthemto understandmoreinvolveddetailsthatfollow.Elementaryconceptsinsampling, datacollection,andexperimentaldesignarepresented,andrudimentaryaspects ofgraphicaltoolsareintroduced,aswellasasenseofwhatisgarneredfroma dataset.Stem-and-leafplotsandbox-and-whiskerplotshavebeenadded.Graphs arebetterorganizedandlabeled.Thediscussionofuncertaintyandvariationin asystemisthoroughandwellillustrated.Thereareexamplesofhowtosort outtheimportantcharacteristicsofascientificprocessorsystem,andtheseideas areillustratedinpracticalsettingssuchasmanufacturingprocesses,biomedical studies,andstudiesofbiologicalandotherscientificsystems.Acontrastismade betweentheuseofdiscreteandcontinuousdata.Emphasisisplacedontheuse ofmodelsandtheinformationconcerningstatisticalmodelsthatcanbeobtained fromgraphicaltools.
Chapters2,3,and4dealwithbasicprobabilityaswellasdiscreteandcontinuousrandomvariables.Chapters5and6focusonspecificdiscreteandcontinuous distributionsaswellasrelationshipsamongthem.Thesechaptersalsohighlight examplesofapplicationsofthedistributionsinreal-lifescientificandengineering studies.Examples,casestudies,andalargenumberofexercisesedifythestudent concerningtheuseofthesedistributions.Projectsbringthepracticaluseofthese distributionstolifethroughgroupwork.Chapter7isthemosttheoreticalchapter
inthetext.Itdealswithtransformationofrandomvariablesandwilllikelynotbe usedunlesstheinstructorwishestoteacharelativelytheoreticalcourse.Chapter 8containsgraphicalmaterial,expandingonthemoreelementarysetofgraphicaltoolspresentedandillustratedinChapter1.Probabilityplottingisdiscussed andillustratedwithexamples.Theveryimportantconceptofsamplingdistributionsispresentedthoroughly,andillustrationsaregiventhatinvolvethecentral limittheoremandthedistributionofasamplevarianceundernormal,independent (i.i.d.)sampling.The t and F distributionsareintroducedtomotivatetheiruse inchapterstofollow.NewmaterialinChapter8helpsthestudenttovisualizethe importanceofhypothesistesting,motivatingtheconceptofa P -value.
Chapter9containsmaterialonone-andtwo-samplepointandintervalestimation.Athoroughdiscussionwithexamplespointsoutthecontrastbetweenthe differenttypesofintervals—confidenceintervals,predictionintervals,andtoleranceintervals.Acasestudyillustratesthethreetypesofstatisticalintervalsinthe contextofamanufacturingsituation.Thiscasestudyhighlightsthedifferences amongtheintervals,theirsources,andtheassumptionsmadeintheirdevelopment,aswellaswhattypeofscientificstudyorquestionrequirestheuseofeach one.Anewapproximationmethodhasbeenaddedfortheinferenceconcerninga proportion.Chapter10beginswithabasicpresentationonthepragmaticmeaningofhypothesistesting,withemphasisonsuchfundamentalconceptsasnulland alternativehypotheses,theroleofprobabilityandthe P -value,andthepowerof atest.Followingthis,illustrationsaregivenoftestsconcerningoneandtwosamplesunderstandardconditions.Thetwo-sample t-testwithpairedobservations isalsodescribed.Acasestudyhelpsthestudenttodevelopaclearpictureof whatinteractionamongfactorsreallymeansaswellasthedangersthatcanarise wheninteractionbetweentreatmentsandexperimentalunitsexists.Attheendof Chapter10isaveryimportantsectionthatrelatesChapters9and10(estimation andhypothesistesting)toChapters11through16,wherestatisticalmodelingis prominent.Itisimportantthatthestudentbeawareofthestrongconnection. Chapters11and12containmaterialonsimpleandmultiplelinearregression, respectively.Considerablymoreattentionisgiveninthiseditiontotheeffectthat collinearityamongtheregressionvariablesplays.Asituationispresentedthat showshowtheroleofasingleregressionvariablecandependinlargepartonwhat regressorsareinthemodelwithit.Thesequentialmodelselectionprocedures(forward,backward,stepwise,etc.)arethenrevisitedinregardtothisconcept,and therationaleforusingcertain P -valueswiththeseproceduresisprovided.Chapter12offersmaterialonnonlinearmodelingwithaspecialpresentationoflogistic regression,whichhasapplicationsinengineeringandthebiologicalsciences.The materialonmultipleregressionisquiteextensiveandthusprovidesconsiderable flexibilityfortheinstructor,asindicatedearlier.AttheendofChapter12iscommentaryrelatingthatchaptertoChapters14and15.Severalfeatureswereadded thatprovideabetterunderstandingofthematerialingeneral.Forexample,the end-of-chaptermaterialdealswithcautionsanddifficultiesonemightencounter. Itispointedoutthattherearetypesofresponsesthatoccurnaturallyinpractice (e.g.proportionresponses,countresponses,andseveralothers)withwhichstandardleastsquaresregressionshouldnotbeusedbecausestandardassumptionsdo notholdandviolationofassumptionsmayinduceseriouserrors.Thesuggestionis
madethatdatatransformationontheresponsemayalleviatetheprobleminsome cases.FlexibilityisagainavailableinChapters13and14,onthetopicofanalysis ofvariance.Chapter13coversone-factorANOVAinthecontextofacompletely randomizeddesign.Complementarytopicsincludetestsonvariancesandmultiple comparisons.Comparisonsoftreatmentsinblocksarehighlighted,alongwiththe topicofrandomizedcompleteblocks.GraphicalmethodsareextendedtoANOVA toaidthestudentinsupplementingtheformalinferencewithapictorialtypeofinferencethatcanaidscientistsandengineersinpresentingmaterial.Anewproject isgiveninwhichstudentsincorporatetheappropriaterandomizationintoeach planandusegraphicaltechniquesand P -valuesinreportingtheresults.Chapter 14extendsthematerialinChapter13toaccommodatetwoormorefactorsthat areinafactorialstructure.TheANOVApresentationinChapter14includeswork inbothrandomandfixedeffectsmodels.Chapter15offersmaterialassociated with2k factorialdesigns;examplesandcasestudiespresenttheuseofscreening designsandspecialhigherfractionsofthe2k .Twonewandspecialfeaturesare thepresentationsofresponsesurfacemethodology(RSM)androbustparameter design.Thesetopicsarelinkedinacasestudythatdescribesandillustratesa dualresponsesurfacedesignandanalysisfeaturingtheuseofprocessmeanand varianceresponsesurfaces.
ComputerSoftware
Casestudies,beginninginChapter8,featurecomputerprintoutandgraphical materialgeneratedusingbothSASandMINITAB.Theinclusionofthecomputer reflectsourbeliefthatstudentsshouldhavetheexperienceofreadingandinterpretingcomputerprintoutandgraphics,evenifthesoftwareinthetextisnotthat whichisusedbytheinstructor.Exposuretomorethanonetypeofsoftwarecan broadentheexperiencebaseforthestudent.Thereisnoreasontobelievethat thesoftwareusedinthecoursewillbethatwhichthestudentwillbecalledupon touseinpracticefollowinggraduation.Examplesandcasestudiesinthetextare supplemented,whereappropriate,byvarioustypesofresidualplots,quantileplots, normalprobabilityplots,andotherplots.Suchplotsareparticularlyprevalentin Chapters11through15.
Supplements
Instructor’sSolutionsManual.Thisresourcecontainsworked-outsolutionstoall textexercisesandisavailablefordownloadfromPearsonEducation’sInstructor ResourceCenter.
StudentSolutionsManual ISBN-10:0-321-64013-6;ISBN-13:978-0-321-64013-0. Featuringcompletesolutionstoselectedexercises,thisisagreattoolforstudents astheystudyandworkthroughtheproblemmaterial.
PowerPoint R LectureSlides ISBN-10:0-321-73731-8;ISBN-13:978-0-321-737311.Theseslidesincludemostofthefiguresandtablesfromthetext.Slidesare availabletodownloadfromPearsonEducation’sInstructorResourceCenter.
StatCruncheText.Thisinteractive,onlinetextbookincludesStatCrunch,apowerful,web-basedstatisticalsoftware.EmbeddedStatCrunchbuttonsallowusers toopenalldatasetsandtablesfromthebookwiththeclickofabuttonand immediatelyperformananalysisusingStatCrunch.
StatCrunch TM .StatCrunchisweb-basedstatisticalsoftwarethatallowsusersto performcomplexanalyses,sharedatasets,andgeneratecompellingreportsof theirdata.UserscanuploadtheirowndatatoStatCrunchorsearchthelibrary ofovertwelvethousandpubliclyshareddatasets,coveringalmostanytopicof interest.Interactivegraphicaloutputshelpusersunderstandstatisticalconcepts andareavailableforexporttoenrichreportswithvisualrepresentationsofdata. Additionalfeaturesinclude
• Afullrangeofnumericalandgraphicalmethodsthatallowuserstoanalyze andgaininsightsfromanydataset.
• Reportingoptionsthathelpuserscreateawidevarietyofvisuallyappealing representationsoftheirdata.
• Anonlinesurveytoolthatallowsuserstoquicklybuildandadministersurveys viaawebform.
StatCrunchisavailabletoqualifiedadopters.Formoreinformation,visitour websiteatwww.statcrunch.comorcontactyourPearsonrepresentative.
Acknowledgments
Weareindebtedtothosecolleagueswhoreviewedthepreviouseditionsofthisbook andprovidedmanyhelpfulsuggestionsforthisedition.TheyareDavidGroggel, MiamiUniversity;LanceHemlow, RaritanValleyCommunityCollege;YingJi, UniversityofTexasatSanAntonio ;ThomasKline, UniversityofNorthernIowa; SheilaLawrence, RutgersUniversity;LuisMoreno, BroomeCountyCommunity College;DonaldWaldman, UniversityofColorado—Boulder;andMarleneWill, SpaldingUniversity.WewouldalsoliketothankDelraySchulz, MillersvilleUniversity;RoxaneBurrows, HockingCollege;andFrankChmelyforensuringthe accuracyofthistext.
WewouldliketothanktheeditorialandproductionservicesprovidedbynumerouspeoplefromPearson/PrenticeHall,especiallytheeditorinchiefDeirdre Lynch,acquisitionseditorChristopherCummings,executivecontenteditorChristineO’Brien,productioneditorTracyPatruno,andcopyeditorSallyLifland.Many usefulcommentsandsuggestionsbyproofreaderGailMaginaregreatlyappreciated.WethanktheVirginiaTechStatisticalConsultingCenter,whichwasthe sourceofmanyreal-lifedatasets.
Chapter1 IntroductiontoStatistics andDataAnalysis
1.1 Overview:StatisticalInference,Samples,Populations,
andtheRoleofProbability
Beginninginthe1980sandcontinuingintothe21stcentury,aninordinateamount ofattentionhasbeenfocusedon improvementofquality inAmericanindustry. MuchhasbeensaidandwrittenabouttheJapanese“industrialmiracle,”which beganinthemiddleofthe20thcentury.TheJapanesewereabletosucceedwhere weandothercountrieshadfailed–namely,tocreateanatmospherethatallows theproductionofhigh-qualityproducts.MuchofthesuccessoftheJapanesehas beenattributedtotheuseof statisticalmethods andstatisticalthinkingamong managementpersonnel.
Theuseofstatisticalmethodsinmanufacturing,developmentoffoodproducts, computersoftware,energysources,pharmaceuticals,andmanyotherareasinvolves thegatheringofinformationor scientificdata.Ofcourse,thegatheringofdata isnothingnew.Ithasbeendoneforwelloverathousandyears.Datahave beencollected,summarized,reported,andstoredforperusal.However,thereisa profounddistinctionbetweencollectionofscientificinformationand inferential statistics.Itisthelatterthathasreceivedrightfulattentioninrecentdecades.
Theoffspringofinferentialstatisticshasbeenalarge“toolbox”ofstatistical methodsemployedbystatisticalpractitioners.Thesestatisticalmethodsaredesignedtocontributetotheprocessofmakingscientificjudgmentsinthefaceof uncertainty and variation.Theproductdensityofaparticularmaterialfroma manufacturingprocesswillnotalwaysbethesame.Indeed,iftheprocessinvolved isabatchprocessratherthancontinuous,therewillbenotonlyvariationinmaterialdensityamongthebatchesthatcomeofftheline(batch-to-batchvariation), butalsowithin-batchvariation.Statisticalmethodsareusedtoanalyzedatafrom aprocesssuchasthisoneinordertogainmoresenseofwhereintheprocess changesmaybemadetoimprovethe quality oftheprocess.Inthisprocess,qual-
itymaywellbedefinedinrelationtoclosenesstoatargetdensityvalueinharmony with whatportionofthetime thisclosenesscriterionismet.Anengineermaybe concernedwithaspecificinstrumentthatisusedtomeasuresulfurmonoxidein theairduringpollutionstudies.Iftheengineerhasdoubtsabouttheeffectiveness oftheinstrument,therearetwo sourcesofvariation thatmustbedealtwith. Thefirstisthevariationinsulfurmonoxidevaluesthatarefoundatthesame localeonthesameday.Thesecondisthevariationbetweenvaluesobservedand the true amountofsulfurmonoxidethatisintheairatthetime.Ifeitherofthese twosourcesofvariationisexceedinglylarge(accordingtosomestandardsetby theengineer),theinstrumentmayneedtobereplaced.Inabiomedicalstudyofa newdrugthatreduceshypertension,85%ofpatientsexperiencedrelief,whileitis generallyrecognizedthatthecurrentdrug,or“old”drug,bringsreliefto80%ofpatientsthathavechronichypertension.However,thenewdrugismoreexpensiveto makeandmayresultincertainsideeffects.Shouldthenewdrugbeadopted?This isaproblemthatisencountered(oftenwithmuchmorecomplexity)frequentlyby pharmaceuticalfirmsinconjunctionwiththeFDA(FederalDrugAdministration). Again,theconsiderationofvariationneedstobetakenintoaccount.The“85%” valueisbasedonacertainnumberofpatientschosenforthestudy.Perhapsifthe studywererepeatedwithnewpatientstheobservednumberof“successes”would be75%!Itisthenaturalvariationfromstudytostudythatmustbetakeninto accountinthedecisionprocess.Clearlythisvariationisimportant,sincevariation frompatienttopatientisendemictotheproblem.
VariabilityinScientificData
Intheproblemsdiscussedabovethestatisticalmethodsusedinvolvedealingwith variability,andineachcasethevariabilitytobestudiedisthatencounteredin scientificdata.Iftheobservedproductdensityintheprocesswerealwaysthe sameandwerealwaysontarget,therewouldbenoneedforstatisticalmethods. Ifthedeviceformeasuringsulfurmonoxidealwaysgivesthesamevalueandthe valueisaccurate(i.e.,itiscorrect),nostatisticalanalysisisneeded.Ifthere werenopatient-to-patientvariabilityinherentintheresponsetothedrug(i.e., iteitheralwaysbringsreliefornot),lifewouldbesimpleforscientistsinthe pharmaceuticalfirmsandFDAandnostatisticianwouldbeneededinthedecision process.Statisticsresearchershaveproducedanenormousnumberofanalytical methodsthatallowforanalysisofdatafromsystemslikethosedescribedabove. Thisreflectsthetruenatureofthesciencethatwecallinferentialstatistics,namely, usingtechniquesthatallowustogobeyondmerelyreportingdatatodrawing conclusions(orinferences)aboutthescientificsystem.Statisticiansmakeuseof fundamentallawsofprobabilityandstatisticalinferencetodrawconclusionsabout scientificsystems.Informationisgatheredintheformof samples,orcollections of observations.TheprocessofsamplingisintroducedinChapter2,andthe discussioncontinuesthroughouttheentirebook.
Samplesarecollectedfrom populations,whicharecollectionsofallindividualsorindividualitemsofaparticulartype.Attimesapopulationsignifiesa scientificsystem.Forexample,amanufacturerofcomputerboardsmaywishto eliminatedefects.Asamplingprocessmayinvolvecollectinginformationon50 computerboardssampledrandomlyfromtheprocess.Here,thepopulationisall
computerboardsmanufacturedbythefirmoveraspecificperiodoftime.Ifan improvementismadeinthecomputerboardprocessandasecondsampleofboards iscollected,anyconclusionsdrawnregardingtheeffectivenessofthechangeinprocessshouldextendtotheentirepopulationofcomputerboardsproducedunder the“improvedprocess.”Inadrugexperiment,asampleofpatientsistakenand eachisgivenaspecificdrugtoreducebloodpressure.Theinterestisfocusedon drawingconclusionsaboutthepopulationofthosewhosufferfromhypertension. Often,itisveryimportanttocollectscientificdatainasystematicway,with planningbeinghighontheagenda.Attimestheplanningis,bynecessity,quite limited.Weoftenfocusonlyoncertainpropertiesorcharacteristicsoftheitemsor objectsinthepopulation.Eachcharacteristichasparticularengineeringor,say, biologicalimportancetothe“customer,”thescientistorengineerwhoseekstolearn aboutthepopulation.Forexample,inoneoftheillustrationsabovethequality oftheprocesshadtodowiththeproductdensityoftheoutputofaprocess.An engineermayneedtostudytheeffectofprocessconditions,temperature,humidity, amountofaparticularingredient,andsoon.Heorshecansystematicallymove these factors towhateverlevelsaresuggestedaccordingtowhateverprescription or experimentaldesign isdesired.However,aforestscientistwhoisinterested inastudyoffactorsthatinfluencewooddensityinacertainkindoftreecannot necessarilydesignanexperiment.Thiscasemayrequirean observationalstudy inwhichdataarecollectedinthefieldbut factorlevels cannotbepreselected. Bothofthesetypesofstudieslendthemselvestomethodsofstatisticalinference. Intheformer,thequalityoftheinferenceswilldependonproperplanningofthe experiment.Inthelatter,thescientistisatthemercyofwhatcanbegathered. Forexample,itissadifanagronomistisinterestedinstudyingtheeffectofrainfall onplantyieldandthedataaregatheredduringadrought.
Theimportanceofstatisticalthinkingbymanagersandtheuseofstatistical inferencebyscientificpersonneliswidelyacknowledged.Researchscientistsgain muchfromscientificdata.Dataprovideunderstandingofscientificphenomena. Productandprocessengineerslearnagreatdealintheiroff-lineeffortstoimprove theprocess.Theyalsogainvaluableinsightbygatheringproductiondata(onlinemonitoring)onaregularbasis.Thisallowsthemtodeterminenecessary modificationsinordertokeeptheprocessatadesiredlevelofquality.
Therearetimeswhenascientificpractitionerwishesonlytogainsomesortof summaryofasetofdatarepresentedinthesample.Inotherwords,inferential statisticsisnotrequired.Rather,asetofsingle-numberstatisticsor descriptive statistics ishelpful.Thesenumbersgiveasenseofcenterofthelocationof thedata,variabilityinthedata,andthegeneralnatureofthedistributionof observationsinthesample.Thoughnospecificstatisticalmethodsleadingto statisticalinference areincorporated,muchcanbelearned.Attimes,descriptive statisticsareaccompaniedbygraphics.Modernstatisticalsoftwarepackagesallow forcomputationof means, medians, standarddeviations,andothersinglenumberstatisticsaswellasproductionofgraphsthatshowa“footprint”ofthe natureofthesample.Definitionsandillustrationsofthesingle-numberstatistics andgraphs,includinghistograms,stem-and-leafplots,scatterplots,dotplots,and boxplots,willbegiveninsectionsthatfollow.
TheRoleofProbability
Inthisbook,Chapters2to6dealwithfundamentalnotionsofprobability.A thoroughgroundingintheseconceptsallowsthereadertohaveabetterunderstandingofstatisticalinference.Withoutsomeformalismofprobabilitytheory, thestudentcannotappreciatethetrueinterpretationfromdataanalysisthrough modernstatisticalmethods.Itisquitenaturaltostudyprobabilitypriortostudyingstatisticalinference.Elementsofprobabilityallowustoquantifythestrength or“confidence”inourconclusions.Inthissense,conceptsinprobabilityforma majorcomponentthatsupplementsstatisticalmethodsandhelpsusgaugethe strengthofthestatisticalinference.Thedisciplineofprobability,then,provides thetransitionbetweendescriptivestatisticsandinferentialmethods.Elementsof probabilityallowtheconclusiontobeputintothelanguagethatthescienceor engineeringpractitionersrequire.Anexamplefollowsthatwillenablethereader tounderstandthenotionofa P -value,whichoftenprovidesthe“bottomline”in theinterpretationofresultsfromtheuseofstatisticalmethods.
Example1.1: Supposethatanengineerencountersdatafromamanufacturingprocessinwhich 100itemsaresampledand10arefoundtobedefective.Itisexpectedandanticipatedthatoccasionallytherewillbedefectiveitems.Obviouslythese100items representthesample.However,ithasbeendeterminedthatinthelongrun,the companycanonlytolerate5%defectiveintheprocess.Now,theelementsofprobabilityallowtheengineertodeterminehowconclusivethesampleinformationis regardingthenatureoftheprocess.Inthiscase,the population conceptually representsallpossibleitemsfromtheprocess.Supposewelearnthat iftheprocess isacceptable,thatis,ifitdoesproduceitemsnomorethan5%ofwhicharedefective,thereisaprobabilityof0.0282ofobtaining10ormoredefectiveitemsin arandomsampleof100itemsfromtheprocess.Thissmallprobabilitysuggests thattheprocessdoes,indeed,havealong-runrateofdefectiveitemsthatexceeds 5%.Inotherwords,undertheconditionofanacceptableprocess,thesampleinformationobtainedwouldrarelyoccur.However,itdidoccur!Clearly,though,it wouldoccurwithamuchhigherprobabilityiftheprocessdefectiverateexceeded 5%byasignificantamount.
Fromthisexampleitbecomesclearthattheelementsofprobabilityaidinthe translationofsampleinformationintosomethingconclusiveorinconclusiveabout thescientificsystem.Infact,whatwaslearnedlikelyisalarminginformationto theengineerormanager.Statisticalmethods,whichwewillactuallydetailin Chapter10,produceda P -valueof0.0282.Theresultsuggeststhattheprocess verylikelyisnotacceptable.Theconceptofa P-value isdealtwithatlength insucceedingchapters.Theexamplethatfollowsprovidesasecondillustration.
Example1.2: Oftenthenatureofthescientificstudywilldictatetherolethatprobabilityand deductivereasoningplayinstatisticalinference.Exercise9.40onpage294provides dataassociatedwithastudyconductedattheVirginiaPolytechnicInstituteand StateUniversityonthedevelopmentofarelationshipbetweentherootsoftreesand theactionofafungus.Mineralsaretransferredfromthefungustothetreesand sugarsfromthetreestothefungus.Twosamplesof10northernredoakseedlings wereplantedinagreenhouse,onecontainingseedlingstreatedwithnitrogenand
theothercontainingseedlingswithnonitrogen.Allotherenvironmentalconditions wereheldconstant.Allseedlingscontainedthefungus Pisolithustinctorus.More detailsaresuppliedinChapter9.Thestemweightsingramswererecordedafter theendof140days.ThedataaregiveninTable1.1.
Table1.1:DataSetforExample1.2
Figure1.1:Adotplotofstemweightdata.
Inthisexampletherearetwosamplesfromtwo separatepopulations.The purposeoftheexperimentistodetermineiftheuseofnitrogenhasaninfluence onthegrowthoftheroots.Thestudyisacomparativestudy(i.e.,weseekto comparethetwopopulationswithregardtoacertainimportantcharacteristic).It isinstructivetoplotthedataasshowninthedotplotofFigure1.1.The ◦ values representthe“nitrogen”dataandthe × valuesrepresentthe“no-nitrogen”data. Noticethatthegeneralappearanceofthedatamightsuggesttothereader that,onaverage,theuseofnitrogenincreasesthestemweight.Fournitrogenobservationsareconsiderablylargerthananyoftheno-nitrogenobservations.Most oftheno-nitrogenobservationsappeartobebelowthecenterofthedata.The appearanceofthedatasetwouldseemtoindicatethatnitrogeniseffective.But howcanthisbequantified?Howcanalloftheapparentvisualevidencebesummarizedinsomesense?Asintheprecedingexample,thefundamentalsofprobability canbeused.Theconclusionsmaybesummarizedinaprobabilitystatementor P-value.Wewillnotshowherethestatisticalinferencethatproducesthesummary probability.AsinExample1.1,thesemethodswillbediscussedinChapter10. Theissuerevolvesaroundthe“probabilitythatdatalikethesecouldbeobserved” giventhatnitrogenhasnoeffect,inotherwords,giventhatbothsampleswere generatedfromthesamepopulation.Supposethatthisprobabilityissmall,say 0.03.Thatwouldcertainlybestrongevidencethattheuseofnitrogendoesindeed influence(apparentlyincreases)averagestemweightoftheredoakseedlings.
