6西格玛绿带培训Materials(PPT 277页)

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,Revision: 1.00,Date: June 2001,6,西格玛绿带培训,Materials,TWO,-6,-4,-2,0,2,4,6,标准偏差,第二天,: Testsof Hypotheses,Week1 recapofStatistics Terminology,IntroductiontoStudentT distribution,Examplein usingStudentT distribution,Summaryof formula for Confidence Limits,IntroductiontoHypothesis Testing,TheelementsofHypothesis Testing,-Break-,Large sampleTest ofHypothesisabout apopulation mean,p-Values, the observed significancelevels,Small sampleTest ofHypothesisabout apopulation mean,Measuring the powerof hypothesis testing,CalculatingTypeIIError probabilities,Hypothesis ExerciseI,-Lunch-,Hypothesis ExerciseI Presentation,Comparing 2population Means: Independent Sampling,Comparing 2population Means: PairedDifferenceExperiments,Comparing 2population Proportions:F-Test,-Break-,Hypothesis Testing ExerciseII (paper clip),Hypothesis Testing Presentation,第一天,wrapup,第二天,: Analysis of variance,和,simple linear regression,Chi-square :A testof independence,Chi-square :Inferencesabout apopulation variance,Chi-square exercise,ANOVA -Analysisofvariance,ANOVA Analysisofvariancecase study,-Break-,Testingthefittnessofa probability distribution,Chi-square:a goodness of fit test,TheKolmogorov-Smirnov Test,Goodnessoffitexerciseusing dice,Result,和,discussion on exercise,-Lunch-,Probabilistic,关,系,系,hipofaregressionmodel,Fittingmodelwithleastsquareapproach,Assumptions,和,varianceestimator,Makinginferenceabouttheslope,CoefficientofCorrelation,和,Determination,Exampleofsimplelinearregression,Simplelinearregressionexercise(usingstatapult),-Break-,Simplelinearregressionexercise(con,t),Presentationofresults,第,二,二,天,天,wrapup,Day3:Multipleregression,和,modelbuilding,Introductiontomultipleregressionmodel,Buildinga model,Fitting themodelwith least squaresapproach,Assumptionsformodel,Usefulness of amodel,Analysisofvariance,Usingthemodelfor estimation,和,prediction,Pitfallsinprediction model,-Break-,Multipleregression exercise (statapult),Presentationfor multiple regressionexercise,-Lunch-,- Qualitative data,和,dummyvariables,Modelswith2 or morequantitative independent variables,Testing themodel,Modelswithonequalitativeindependentvariable,Comparingslopes,和,responsecurve,-Break-,Modelbuildingexample,Stepwiseregression anapproachtoscreen outfactors,Day3wrap up,Day4:,设计,ofExperiment,OverviewofExperimentalDesign,What is adesignedexperiment,Objectiveofexperimental,设计和,itscapabilityinidentifyingtheeffect of factors,Onefactor at atime(OFAT)versus,设计,ofexperiment (DOE) formodelling,Orthogonality,和,itsimportancetoDOE,H,和,calculationforbuildingsimple linearmodel,Type,和,uses of DOE,(i.e.linear screening,linear modelling,和,non-linear modelling),OFAT versusDOE,和,itsimpact in ascreening experiment,TypesofscreeningDOEs,-Break-,Pointstonotewhen conductingDOE,ScreeningDOE exercise using statapult,Interpretatingthescreening DOEs result,-Lunch-,ModellingDOE (Full factoria withinteractions),Interpretinginteractionoffactors,Paretooffactorssignificance,GraphicalinterpretationofDOE results,某些,rulesofthumbinDOE,实例,ofModellingDOE,和,itsanalysis,-Break-,ModellingDOEexercisewithstatapult,Targetpractice,和,confirmationrun,Day4wrapup,Day5:Statistical,流程,Control,WhatisStatistical,流程,Control,Controlchart,thevoiceofthe,流程,流程,controlversus,流程,capability,Typesofcontrolchartavailable,和,itsapplication,Observingtrendsforcontrolchart,OutofControlreaction,IntroductiontoXbarRChart,XbarRChartexample,Assignable,和,ChancecausesinSPC,RuleofthumbforSPCruntest,-Break-,XbarRChartexercise(usingDice),IntroductiontoXbarSChart,ImplementingXbarSChart,为什,么,么,XbarSChart?,IntroductiontoIndividualMovingRangeChart,ImplementingIndividualMovingRangeChart,为什,么,么,XbarSChart?,-Lunch-,Choosingthesub-group,Choosingthecorrectsamplesize,Samplingfrequency,Introductiontocontrolchartsforattributedata,npCharts,pCharts,cCharts,uCharts,-Break-,Attribute control chartexercise(paperclip),Outof control not necessarily is bad,Day5 wrap up,Recap ofStatisticalTerminology,Distributions differs in location,Distributions differs in spread,Distributions differs in shape,Normal Distribution,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,- 99.9999998% -,- 99.73% -,- 95.45% -,-68.27%-, 3,variation is called natural tolerance,Areaunder aNormalDistribution,流程,capability potential, Cp,Based ontheassumptionsthat :,流程,is normal,Normal Distribution,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,Lower Spec Limit,LSL,Upper Spec Limit,USL,Specification Center,It is a2-sidedspecification,流程,meaniscenteredtothedevice specification,Spread in specification,Naturaltolerance,CP =,USL - LSL,6,8,6,= 1.33,流程,Capability Index, Cpk,Based ontheassumptionthatthe,流程,is normal,和,in control,2.Anindex thatcomparethe,流程,center withspecification center,Normal Distribution,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,Lower Spec Limit,LSL,Upper Spec Limit,USL,Specification Center,Therefore when ,Cpk 20),Estimated,标准偏差, R/d,2,Population,标准偏差,(when sample size, n,20),ProbabilityTheory,Probabilityis the chance for anevent to occur.,Statisticaldependence /independence,Posterior probability,Relativefrequency,Makedecision through probability distributions,(i.e. Binomial,Poisson,Normal),Central Limit Theorem,Regardless the actual distribution of the population, the distribution of the mean for sub-groups of sample from that distribution, will be normally distributed with sample mean approximately equal to the population mean.,Set confidence interval for sample based on normal distribution.,A basis to compare samples using normal distribution, hence making statistical comparison of the actual populations.,It does not implies that the population is always normally distributed.,(Cp, Cpk must always based on the assumption that,流程,is normal),InferentialStatistics,The,流程,of interpretingthesample datato drawconclusionsabout the populationfrom whichthesample was taken.,Confidence Interval,(Determine confidence levelfora sampling meanto fluctuate),T-Test,和,F-Test,(Determine if the underlyingpopulationsissignificantly differentin termsofthemeans,和,variations),Chi-Square Testof Independence,(Test ifthesampleproportionsaresignificantly different),Correlation,和,Regression,(Determine if,关系,hipbetweenvariables exists,和,generatemodel equationto predict the outcome of asingle output variable),CentralLimit Theorem,The mean x of the sampling distribution will approximately equal to the population mean regardless of the sample size. The,larger the sample size, the closer the sample mean is towards the population mean.,2. The sampling distribution of the mean will approach normality regardless of the actual population distribution.,3.It assures us that the sampling,distribution of the mean approaches normal,as the,sample size increases,.,m,= 150,Population distribution,x = 150,Sampling distribution,(n = 5),x = 150,Sampling distribution,(n = 20),x = 150,Sampling distribution,(n = 30),m,= 150,Population distribution,x = 150,Sampling distribution,(n = 5),某些,takeaways for sample size,和,samplingdistribution,For large sample size (i.e. n,30), the sampling distribution of x will approach normality regardless the actual distribution of the sampled population.,For small sample size (i.e. n 30), the sampling distribution of x is exactly normal if the sampled population is normal,和,will be approximately normal if the sampled population is also approximately normally distributed.,The point estimate of population,标准偏差,using,S,equation may,提供,a poor estimation if the sample size is small.,Introduction to Student t Distrbution,Discovered in 1908 by W.S. Gosset from Guinness Brewery in Ireland.,To compensate for,标准偏差,dependence on small sample size.,Contain two random quantities (x,和,S), whereas normal distribution contains only one random quantity (x only),As sample size increases, the t distribution will become closer to that of standard normal distribution (or z distribution).,Percentilesof the tDistribution,Whereby,df= Degree offreedom,= n(samplesize) 1,Shaded area=one-tailed probability of occurence,a,= 1,Shadedarea,Applicable when:,Sample size450).,Designated as H,1,ElementsofHypothesis Testing(cont),Example if we wanttotestwhether apopulationmean is equal to 500,wewouldtranslate it to:,Null Hypothesis, H,0,:,m,p,= 500,和,consideralternatehypothesisas:,AlternateHypothesis,H,1,:,m,p,500;(2tailstest),Rememberconfidence interval,at95%confidencelevelstatesthat:,95%ofthe timethe meanvaluewillfluctuatewithin theconfidence interval (limit),5%chancethatthemeanisnatural fluctuation,butwethinkitisnot alpha(,a,) probability,-,Confidence limit -,m,H,0,= 500,0.025,of area,0.025,of area,(,a,/2),reject area,(,a,/2),reject area,1.96,std error,1.96,std error,Type II Error,Acceptinganull hypothesis(H,0,),when it is false.Probabilityofthis error equals,b,Type IError,Rejectingthe nullhypothesis(H,0,),when it is true. Probability of thiserrorequals,a,If,m,p is within confidence limit, accept the null hypothesis H,0,.,If,m,p is in reject area, reject the null hypothesis H,0,.,Usethe stderrorobservedfrom thesampletoset confidencelimiton500 (,m,H,0,).Theassumptionis,m,H,0,hasthe samevarianceas,m,p.,ElementsofHypothesis Testing(cont),Otherpossiblealternatehypothesisare:,AlternateHypothesis,H,1,:,m,p, 500; (1 tailtest),AlternateHypothesis,H,1,:,m,p, 500,For95% confidencelevel,a,= 0.05.,SinceH,1,isonetailtest,rejectareadoes notneed to be dividedby2.,From standard normal distribution table:,Z-value of 1.645 will give 0.95 area, leaving,a,to be 0.05.,Therefore if,m,p,is more than 500 by,1.645 std error, it will be in the reject area,和,we will reject the null hypothesis H,0, concluding on alternate hypothesis H,1,that,m,p,is 500.,某些,hypothesis testings thatare applicabletoengineers:,The impacton response measurementwithnew,和,old,流程,parameters.,Comparisonofa newvendorsparts(which are slightlymoreexpensive) tothe present vendor,whenvariationis amajorissue.,Is the yield onTester ECTZ21the same as theyield onTester ECTZ33 ?,流程,Situations,Comparisonof one population from asingle,流程,to adesirablestandard,Comparisonof two populationsfromtwo different,流程,es,or,Single sided: comparisonconsidersa difference only ifit is greateror only ifit is less, but notboth.,Two sided:comparison considers anydifference ofine,质量,important,Inferencesbased ona single sample,“,Largesample test ofhypothesis about a,populationmean,”,Example:,An automotive manufacturer wants toevaluateif their new throttle,设计,on all thelatest car model isableto give an adequateresponsetime,resultingin an predictable pick-up ofthe vehicle speed when the fuel pedal isbeing depressed. Based on finite element modelling, the,设计,teamcommittedthatthe throttle response time is1.2 msec,和,thisis the recommended valuethatwillgivethe driverthebestcontrol over the vehicleacceleration.,The test engineer ofthisproject has testedon 100 vehicleswiththenew throttle,设计和,obtain anaverage throttle responsetimeof 1.05 msec with a,标准偏差,S of0.5 msec.Basedon 99% confidence level,canhe concluded that the newthrottle,设计,willgivean averageresponsetimeof 1.2 msec ?,“,Largesample test ofhypothesis about a,populationmean,” (cont),Solution:,Since the sample size is relatively large (i.e. 30), we should use z statistic.,m ,X = 1.05 msec ;,s ,S = 0.5 msec ;n = 100 ;,Null hypothesis,H,0,:,m,=,m,H,0,(1.2 msec),Alternate hypothesis,H,1,:,m,m,H,0,(1.2 msec),-,Acceptance Area -,m,H,0,= 1.2,0.005,of area,0.005,of area,(,a,/2),Reject Area,(,a,/2),Reject Area,2.58,std error,2.58,std error,Fromstandard normaldistribution table,The Zvalue correspondingto 0.005tailareais,2.58.,a,= 0.01 (2tails), since2 tails test, therefore tail area =,a,/2 =0.005;,How many std error is X away from 1.2 msec ?,X = 1.02,Therefore X is 3 std errors away from 1.2 msec.,-,Acceptance Area -,m,H,0,= 1.2,0.005,of area,0.005,of area,(,a,/2),Reject Area,(,a,/2),Reject Area,2.58,std error,2.58,std error,X = 1.02,“,Largesample test ofhypothesis about a,populationmean,” (cont),Baed on 99% confidence level, since X is at the negative reject area, we will reject the null hypothesis,和,conclude on the alternate hypo