sigma统计概念培训.ppt

Hence P 4 5 x 8 5 0 9864 0 3745 0 6119 approxequalto0 6123bybinomialdistribution 天马行空官方博客 IntroductiontoSampling Whatispopulationinstatistic Apopulationinstatisticreferstoallitemsthathavebeenchosenforstudy Whatisasampleinstatistic Asampleinstatisticreferstoaportionchosenfromapopulation bywhichthedataobtaincanbeusedtoinferontheactualperformanceofthepopulation Population Sample2 Sample6 Sample8 Sample1 Sample3 Sample7 Sample4 Sample5 Samplingdistribution adistributionofsamplemeansIfyoutake10samplesoutofthesamepopulations youwillmostlikelyendupwith10differentsamplemeans和sample标准偏差s ASamplingdistributiondescribestheprobabilityofallpossiblemeansofthesamplestakenfromthesamepopulation Whensamplesizeincreases thestandarderror orthestddeviationofsamplingdistribution willgetsmaller Samplingdistribution con t CentralLimitTheorem Exampleofsamplingdistribution Thepopulationdistributionofannualincomeofengineersisskewednegatively Thisdistributionhasameanof 48 000 和a标准偏差of 5000 Ifwedrawasampleof100engineers whatistheprobabilitythattheiraverageannualincomeis 48700和more Exampleofsamplingdistribution con t Therefore mean 48000sigma 500X 50000Z 48700 48000 500 700 500 1 4 Fromthestandardizednormaldistributiontable P X 48700 0 9192Therefore P X 48700 1 0 9192 0 0808Thus wehavedeterminedthatithasonly8 08 chancefortheaverageannualincomeof100engineerstobemorethan 48700 CentralLimitTheoremExercise Breakinto4groupsasbelow Group1 Thepopulationgroup Group2to4 Thesamplesub groupThepopulationgroupwillhave3oftheirmembersthrowingasingledices60timeseach Atotalof180throwswillberecorded和thisdatawillbethepopulationdata Eachsamplesub groupwillhave3oftheirmembersthrowing5dicesatonetime 和collectthesum和averagevalueoftheparticularthrow Eachmemberistoconduct20throws和obtainthesamplemeanofeachthrow Attheendoftheexercise atotalof180samplemeanswillbecollectedfromthe3sub groups Fromthearriveddata plotthehistogram和commentonthedistributionofboththepopulation和thesamplings Thefinitepopulationmultiplier Previouslywesaythat Finitepopulationmultiplierwithrespecttopopulation和samplesize RuleofthumbsThefinitepopulationmultiplierneedonlybeincludedifpopulationsizetosamplesizeratioislessthan25 ConfidenceInterval Pointestimates Apointestimateisasinglenumberthatisusedtoestimateanunknownpopulationparameter Whatdoes95 confidenceintervalsmeans Itdefines95 ofthetime theaveragevalueofarandomsamplingwillfallwithinavaluerangewhichis 1 96standarderrorfromthesamplemean 为什么1 96standarderror Referringtothestandardizednormaldistributiontable whenz 1 96 theassociatedprobabilityis0 975 or97 5 asbelow Butthisisa2tailsinterval i e 1 96 henceweneedtominusoffanother2 5 fromtheotherend givingatotalcoverageof95 EquationforcomputingconfidenceintervalsForcontinuousdata Fordiscretedata ConfidenceintervalforcontinuousdataAlargediskdrivemanufacturerneedsanestimateofthemeanlifeitcanexpectfromthemagneticmediabyreciprocallyswitchingitsdigitalstateat1MHzfrequency Thedevelopmentteamhasdeterminedpreviouslythatthevarianceofthepopulationlifeis36months 和hadconductedareliabilitytestingfor100samples collectingdataofitsusefullifeasbelow Fromtheabovedata whatisthe95 confidenceintervalfortheusefulmeanlifeofthemagneticmediainadiskdrive Whatdoesthismean Confidenceintervalforcontinuousdata con t Applyingtheconfidenceintervalsequation Upperconfidencelimit 27 75 1 96 0 6 28 926Lowerconfidencelimit 27 75 1 96 0 6 26 574 Assuch thereis95 confidencelevelthattheaverageusefullifeofthemagneticmediatofallbetween26 574和28 926months ConfidenceintervalfordiscretedataBreakinto4or5teams combinedalltheM M sinoneteam和calculatetheconfidenceintervalsforeachcolortype usingbelowtable Combineallthedatafromthe4teams whatarethechangestotheconfidenceintervals Determinethesamplesizeforconfidenceintervals continuousdata GivenSamplemean 21S 6Whatisthesamplesizerequired suchthatthereisa95 confidencelevelthattheaveragevaluewillfallwithin 1 176fromitsmean Ans Applyingtheequation n 1 96x6 1 176 2 10 2 100Thesamplesizemustbe100 Determinethesamplesizeforconfidenceintervals discretedata Given p 0 4 Proportionagree q 0 6 Proportionnotagree Whatisthesamplesizerequired suchthatthereisa99 confidencelevelthattheproportionagreewillfallwithin 0 146fromp Ans Applyingtheequation n 2 58 2 0 4 0 6 0 146 2 74 95 75Thesamplesizemustbe75 Revision 1 00Date June2001 第二天 TestsofHypothesesWeek1recapofStatisticsTerminologyIntroductiontoStudentTdistributionExampleinusingStudentTdistributionSummaryofformulaforConfidenceLimitsIntroductiontoHypothesisTestingTheelementsofHypothesisTesting Break LargesampleTestofHypothesisaboutapopulationmeanp Values theobservedsignificancelevelsSmallsampleTestofHypothesisaboutapopulationmeanMeasuringthepowerofhypothesistestingCalculatingTypeIIErrorprobabilitiesHypothesisExerciseI Lunch HypothesisExerciseIPresentationComparing2populationMeans IndependentSamplingComparing2populationMeans PairedDifferenceExperimentsComparing2populationProportions F Test Break HypothesisTestingExerciseII paperclip HypothesisTestingPresentation第一天wrapup 第二天 Analysisofvariance和simplelinearregressionChi square AtestofindependenceChi square InferencesaboutapopulationvarianceChi squareexerciseANOVA AnalysisofvarianceANOVA Analysisofvariancecasestudy Break TestingthefittnessofaprobabilitydistributionChi square agoodnessoffittestTheKolmogorov SmirnovTestGoodnessoffitexerciseusingdiceResult和discussiononexercise Lunch Probabilistic关系hipofaregressionmodelFittingmodelwithleastsquareapproachAssumptions和varianceestimatorMakinginferenceabouttheslopeCoefficientofCorrelation和DeterminationExampleofsimplelinearregressionSimplelinearregressionexercise usingstatapult Break Simplelinearregressionexercise con t Presentationofresults第二天wrapup Day3 Multipleregression和modelbuildingIntroductiontomultipleregressionmodelBuildingamodelFittingthemodelwithleastsquaresapproachAssumptionsformodelUsefulnessofamodelAnalysisofvarianceUsingthemodelforestimation和predictionPitfallsinpredictionmodel Break Multipleregressionexercise statapult Presentationformultipleregressionexercise Lunch Qualitativedata和dummyvariablesModelswith2ormorequantitativeindependentvariablesTestingthemodelModelswithonequalitativeindependentvariableComparingslopes和responsecurve Break ModelbuildingexampleStepwiseregression anapproachtoscreenoutfactorsDay3wrapup Day4 设计ofExperimentOverviewofExperimentalDesignWhatisadesignedexperimentObjectiveofexperimental设计和itscapabilityinidentifyingtheeffectoffactorsOnefactoratatime OFAT versus设计ofexperiment DOE formodellingOrthogonality和itsimportancetoDOEH和calculationforbuildingsimplelinearmodelType和usesofDOE i e linearscreening linearmodelling 和non linearmodelling OFATversusDOE和itsimpactinascreeningexperimentTypesofscreeningDOEs Break PointstonotewhenconductingDOEScreeningDOEexerciseusingstatapultInterpretatingthescreeningDOE sresult Lunch ModellingDOE Fullfactoriawithinteractions InterpretinginteractionoffactorsParetooffactorssignificanceGraphicalinterpretationofDOEresults某些rulesofthumbinDOE实例ofModellingDOE和itsanalysis Break ModellingDOEexercisewithstatapultTargetpractice和confirmationrunDay4wrapup Day5 Statistical流程ControlWhatisStatistical流程ControlControlchart thevoiceofthe流程流程controlversus流程capabilityTypesofcontrolchartavailable和itsapplicationObservingtrendsforcontrolchartOutofControlreactionIntroductiontoXbarRChartXbarRChartexampleAssignable和ChancecausesinSPCRuleofthumbforSPCruntest Break XbarRChartexercise usingDice IntroductiontoXbarSChartImplementingXbarSChart为什么XbarSChart IntroductiontoIndividualMovingRangeChartImplementingIndividualMovingRangeChart为什么XbarSChart Lunch Choosingthesub groupChoosingthecorrectsamplesizeSamplingfrequencyIntroductiontocontrolchartsforattributedatanpCharts pCharts cCharts uCharts Break Attributecontrolchartexercise paperclip OutofcontrolnotnecessarilyisbadDay5wrapup RecapofStatisticalTerminology AreaunderaNormalDistribution 流程capabilitypotential CpBasedontheassumptionsthat 流程isnormal Itisa2 sidedspecification 流程meaniscenteredtothedevicespecification Spreadinspecification Naturaltolerance 流程CapabilityIndex Cpk Basedontheassumptionthatthe流程isnormal和incontrol2 Anindexthatcomparethe流程centerwithspecificationcenter Thereforewhen Cpk Cp then流程isnotcentered Cpk Cp then流程iscentered The流程ofcollecting presenting和describingsampledata usinggraphical工具和numbers ParetoChartPopulationmeanHistogramPopulation标准偏差 DescriptiveStatistics ProbabilityTheory Probabilityisthechanceforaneventtooccur Statisticaldependence independencePosteriorprobabilityRelativefrequencyMakedecisionthroughprobabilitydistributions i e Binomial Poisson Normal InferentialStatistics The流程ofinterpretingthesampledatatodrawconclusionsaboutthepopulationfromwhichthesamplewastaken ConfidenceInterval Determineconfidencelevelforasamplingmeantofluctuate T Test和F Test Determineiftheunderlyingpopulationsissignificantlydifferentintermsofthemeans和variations Chi SquareTestofIndependence Testifthesampleproportionsaresignificantlydifferent Correlation和Regression Determineif关系hipbetweenvariablesexists 和generatemodelequationtopredicttheoutcomeofasingleoutputvariable CentralLimitTheorem 某些takeawaysforsamplesize和samplingdistribution PercentilesofthetDistribution Whereby df Degreeoffreedom n samplesize 1Shadedarea one tailedprobabilityofoccurencea 1 ShadedareaApplicablewhen Samplesize 30标准偏差isunknownPopulationdistributionisatleastapproximatelynormallydistributed PercentilesoftheNormalDistribution ZDistribution Whereby Shadedarea one tailedprobabilityofoccurencea 1 Shadedarea StudenttDistrbutionexample FDArequirespharmaceuticalcompaniestoperformextensivetestsonallnewdrugsbeforetheycanbemarketedtothepublic Thefirstphaseoftestingwillbeonanimals whilethesecondphasewillbeonhumanonalimitedbasis PWDisapharmaceuticalcompanycurrentlyinthesecondphaseoftestingonanewantibioticproject Thechemistsareinterestedtoknowtheeffectofthenewantibioticonthehumanbloodpressure 和theyareonlyallowedtoteston6patients Theresultoftheincreaseinbloodpressureofthe6testedpatientsareasbelow 1 7 3 0 0 8 3 4 2 7 2 1 Constructa95 confidenceintervalfortheaverageincreaseinbloodpressureforpatientstakingthenewantibiotic usingbothnormal和tdistributions StudenttDistrbutionexample con t Usingnormalorzdistribution Usingstudenttdistribution Althoughtheconfidencelevelisthesame usingtdistributionwillresultinalargerintervalvalue because 标准偏差 Sforsmallsamplesizeisprobablynotaccurate标准偏差 SforsmallsamplesizeisprobablytoooptimisticWiderintervalisthereforenecessarytoachievetherequiredconfidencelevel Summaryofformulaforconfidencelimit 6Sigma流程和1 5SigmaShiftinMean Statistically a流程thatis6Sigmawithrespecttoitsspecificationsis ButMotoroladefines6Sigmawithascenarioof1 5Sigmashiftinmean DPM 3 4Cp 2Cpk 1 5 某些Explanationson1 5SigmaMeanShift Motorlahasconductedalotofexperiments 和foundthatinlongterm the流程meanwillshiftwithin1 5sigmaifthe流程isundercontrol 1 5sigmameanshiftina3Sigma流程controlplanwillbetranslatedtoapproximately14 ofthetimeadatapointwillbeoutofcontrol 和thisisdeemacceptableinstatistical流程control SPC practices OurExplanation MostfrequentlyusedsamplesizeforSPCinindustryis3to5unitspersampling Takethemiddlevalueof4asanaveragesamplesizeusedinthesampling Assumingthe流程isof6sigmacapability isincontrol 和isnormallydistributed Undertheconfidenceintervalforsamplingdistribution weexpecttheaveragevalueofthesamplestofluctuatewithin 3standarderrors i e naturaltolerance givingconfidenceintervalof IntroductiontoHypothesisTesting Whatishypothesistestinginstatistic Ahypothesisis atentativeassumptionmadeinordertodrawoutortestitslogicalorempiricalconsequences Astatisticalhypothesisisastatementaboutthevalueofoneofthecharacteristicsforoneormorepopulations Thepurposeofthehypothesisistoestablishabasis sothatonecangatherevidencetoeitherdisprovethestatementoracceptitastrue ExampleofstatisticalhypothesisTheaveragecommutetimeusingHighway92isshorterthanusingFranceAvenue This流程changewillnotcauseanyeffectonthedownstream流程es ThevariationofVendorB spartsare40 widerthanthoseofVendorA ElementsofHypothesisTesting Possibleoutcomesforhypothesistestingontwotestedpopulations 为什么HypothesisTesting Manyproblemsrequireadecisiontoacceptorrejectastatementaboutaparameter ThatstatementisaHypothesis Itrepresentsthetranslationofapracticalquestionintoastatisticalquestion Statisticaltesting提供sanobjectivesolution withknownrisks toquestionswhicharetraditionallyansweredsubjectively Itisasteppingstoneto设计ofExperiment DOE HypothesisTestingDescriptions HypothesisTestinganswersthepracticalquestion IstherearealdifferencebetweenA和B Inhypothesistesting relativelysmallsamplesareusedtoanswerquestionsaboutpopulationparameters Thereisalwaysachancethatasamplethatisnotrepresentativeofthepopulationbeingselected和resultsindrawingawrongconclusion ElementsofHypothesisTesting con t TheNullHypothesisStatementgenerallyassumedtobetrueunlesssufficientevidenceisfoundtobecontraryOftenassumedtobethestatusquo orthepreferredoutcome However itsometimesrepresentsastateyoustronglywanttodisprove DesignatedasH0Inhypothesistesting wealwaysbiastowardnullhypothesis TheAlternativeHypothesis orResearchHypothesis Statementthatwillbeacceptedonlyifdata提供convincingevidenceofitstruth i e byrejectingthenullhypothesis Insteadofcomparingtwopopulations itcanalsobebasedonaspecificengineeringdifferenceinacharacteristicvaluethatonedesirestodetect i e insteadofaskingism1 m2 weaskism1 450 DesignatedasH1 ElementsofHypothesisTesting con t Exampleifwewanttotestwhetherapopulationmeanisequalto500 wewouldtranslateitto NullHypothesis H0 mp 500和consideralternatehypothesisas AlternateHypothesis H1 mp500 2tailstest Rememberconfidenceinterval at95 confidencelevelstatesthat 95 ofthetimethemeanvaluewillfluctuatewithintheconfidenceinterval limit 5 chancethatthemeanisnaturalfluctuation butwethinkitisnot alpha a probability TypeIIErrorAcceptinganullhypothesis H0 whenitisfalse Probabilityofthiserrorequalsb TypeIErrorRejectingthenullhypothesis H0 whenitistrue Probabilityofthiserrorequalsa Usethestderrorobservedfromthesampletosetconfidencelimiton500 mH0 TheassumptionismH0hasthesamevarianceasmp ElementsofHypothesisTesting con t Otherpossiblealternatehypothesisare AlternateHypothesis H1 mp 500 1tailtest AlternateHypothesis H1 mp 500 1tailtest Takingexampleforalternatehypothesis H1 mp 500For95 confidencelevel a 0 05 SinceH1isonetailtest rejectareadoesnotneedtobedividedby2 某些hypothesistestingsthatareapplicabletoengineers Theimpactonresponsemeasurementwithnew和old流程parameters Comparisonofanewvendors parts whichareslightlymoreexpensive tothepresentvendor whenvariationisamajorissue IstheyieldonTesterECTZ21thesameastheyieldonTesterECTZ33 流程SituationsComparisonofonepopulationfromasingle流程toadesirablestandardComparisonoftwopopulationsfromtwodifferent流程esorSinglesided comparisonconsidersadifferenceonlyifitisgreateroronlyifitisless butnotboth Twosided comparisonconsidersanydifferenceofine质量important Inferencesbasedonasinglesample Largesampletestofhypothesisaboutapopulationmean Example Anautomotivemanufacturerwantstoevaluateiftheirnewthrottle设计onallthelatestcarmodelisabletogiveanadequateresponsetime resultinginanpredictablepick upofthevehiclespeedwhenthefuelpedalisbeingdepressed Basedonfiniteelementmodelling the设计teamcommittedthatthethrottleresponsetimeis1 2msec 和thisistherecommendedvaluethatwillgivethedriverthebestcontroloverthevehicleacceleration Thetestengineerofthisprojecthastestedon100vehicleswiththenewthrottle设计和obtainanaveragethrottleresponsetimeof1 05msecwitha标准偏差Sof0 5msec Basedon99 confidencelevel canheconcludedthatthenewthrottle设计willgiveanaverageresponsetimeof1 2msec Largesampletestofhypothesisaboutapopulationmean con t Fromstandardnormaldistributiontable TheZvaluecorrespondingto0 005tailareais 2 58 a 0 01 2tails since2tailstest thereforetailarea a 2 0 005 Largesampletestofhypothesisaboutapopulationmean con t Whatdoes99 confidencelevelmeansintheaboveexample Itdefinesthelimitswhereby99 oftheaveragesamplingvalueshouldfallwithin giventhedesirable hypothesised meanasmH0 AnyvaluefalloutsidethisconfidencelimitindicatesthesamplemeanissignificantlydifferentfrommH0 Inotherwords wewillonlyconcludethealternatehypothesisH1 thatthemeansaredifferent ifwearemorethan99 sure TheObservedSignificancelevel p value p valueistheprobabilityforconcludingthenullhypothesisH0thatbothpopulationmeansareequalwiththeobservedsampledata Hence1 pvaluewillbetheconfidencelevelwehaveonthealternatehypothesis Usingthethrottlequestionasanexample Weknowthatthemeanresponsetimeis3standarderrorawayfrom1 2msec mH0 therefore Z 3 Sincethisisa2tailstest p value P Z3 2P Z 3 Fromstandardnormaldistributiontable P Z 3 0 9987P Z 3 1 0 9987 0 0013p value 2P Z 3 0 0026 Instatisticalterm itmeansthereisonly0 0026probabilitythattheaveragethrottleresponsetimetobe1 02mseciftheactualpopulationmeanis1 2msecassuggestedbyfiniteelementanalysis Smallsampletestofhypothesisaboutapopulationmean Example AmyisthePersonnelOfficerofamulti nationalcompanywhoisinchargeofrecruitingalargenumberofemployeesforanoverseasassignment Astheseoverseasassignmentsareverycrucialforthecompanysuccessinmeetingtheirbusinessplan anaptitudetestwasformulatedtotestthe质量ofallpotentialcandidateshead huntedbythe招聘Agency Themanagementwantstoknowtheeffectivenessofthe招聘Agency asitwasbelievedthattheaveragetestscoreforalltheidentifiedcandidatesshouldbeequalormorethan90inordertoreducetheriskofassigningthewrongcandidatesforthetask WhenAmyreviewsthetestsresultofaparticularbatchof20candidates shefindsthatthemeanscoreis84和the标准偏差is11 Asthisisaverycritical招聘project Amywantstobemorereservewithheranalysis 和decidedtobemorebiastowardsprovingthatthepopulationmeanislesserthan90 Asaresult aconfidencelevelof90 willbeusedinheranalysis Smallsampletestofhypothesisaboutapopulationmean con t Fromstudent t distributiontable Thetvaluewith19dfcorrespondingto0 1tailareais 1 3277 Largesampletestofhypothesisaboutapopulationproportion Amethodcurrentlyusedbydoctorstoscreenforpossiblestomachulcerfailstodetecttheulcerin20 ofthepatientswhoactuallyhavethedisease Supposeanewmethodhasbeendevelopedthatresearchershopewilldetectstomachulcermoreaccurately Thisnewmethodwasusedtoscreenara