6西格玛绿带培训Materials(PPT 277页)

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1、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,Example

2、in usingStudentT distribution,Summaryof formula for Confidence Limits,IntroductiontoHypothesis Testing,TheelementsofHypothesis Testing,-----------------------------------------------------Break--------------------------------------------------------------------,Large sampleTest ofHypothesisabout apo

3、pulation mean,p-Values, the observed significancelevels,Small sampleTest ofHypothesisabout apopulation mean,Measuring the powerof hypothesis testing,CalculatingTypeIIError probabilities,Hypothesis ExerciseI,-----------------------------------------------------Lunch-----------------------------------

4、---------------------------------,Hypothesis ExerciseI Presentation,Comparing 2population Means: Independent Sampling,Comparing 2population Means: PairedDifferenceExperiments,Comparing 2population Proportions:F-Test,-----------------------------------------------------Break--------------------------

5、------------------------------------------,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 -Analys

6、isofvariance,ANOVA –Analysisofvariancecase study,-----------------------------------------------------Break--------------------------------------------------------------------,Testingthefittnessofa probability distribution,Chi-square:a goodness of fit test,TheKolmogorov-Smirnov Test,Goodnessoffitexe

7、rciseusing dice,Result,和,discussion on exercise,------------------------------------------------------Lunch-------------------------------------------------------------------,Probabilistic,关,系,系,hipofaregressionmodel,Fittingmodelwithleastsquareapproach,Assumptions,和,varianceestimator,Makinginference

8、abouttheslope,CoefficientofCorrelation,和,Determination,Exampleofsimplelinearregression,Simplelinearregressionexercise(usingstatapult),------------------------------------------------------Break-------------------------------------------------------------------,Simplelinearregressionexercise(con,’,’t

9、),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

10、 model,--------------------------------------------------------Break-----------------------------------------------------------------,Multipleregression exercise (statapult),Presentationfor multiple regressionexercise,--------------------------------------------------------Lunch---------------------

11、--------------------------------------------,- Qualitative data,和,dummyvariables,Modelswith2 or morequantitative independent variables,Testing themodel,Modelswithonequalitativeindependentvariable,Comparingslopes,和,responsecurve,--------------------------------------------------------Break-----------

12、------------------------------------------------------,Modelbuildingexample,Stepwiseregression –anapproachtoscreen outfactors,Day3wrap up,,Day4:,设计,ofExperiment,,OverviewofExperimentalDesign,What is adesignedexperiment,Objectiveofexperimental,设计和,itscapabilityinidentifyingtheeffect of factors,Onefac

13、tor 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,----

14、-----------------------------------------------Break----------------------------------------------------------------------,Pointstonotewhen conductingDOE,ScreeningDOE exercise using statapult,Interpretatingthescreening DOE’s result,---------------------------------------------------Lunch------------

15、----------------------------------------------------------,ModellingDOE (Full factoria withinteractions),Interpretinginteractionoffactors,Paretooffactorssignificance,GraphicalinterpretationofDOE results,某些,rulesofthumbinDOE,实例,ofModellingDOE,和,itsanalysis,--------------------------------------------

16、------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,Typesofcontrolch

17、artavailable,和,itsapplication,Observingtrendsforcontrolchart,OutofControlreaction,IntroductiontoXbarRChart,XbarRChartexample,Assignable,和,ChancecausesinSPC,RuleofthumbforSPCruntest,-------------------------------------------------------------Break-----------------------------------------------------

18、-------,XbarRChartexercise(usingDice),IntroductiontoXbarSChart,ImplementingXbarSChart,为什,么,么,XbarSChart?,IntroductiontoIndividualMovingRangeChart,ImplementingIndividualMovingRangeChart,为什,么,么,XbarSChart?,-------------------------------------------------------------Lunch------------------------------

19、------------------------------,Choosingthesub-group,Choosingthecorrectsamplesize,Samplingfrequency,Introductiontocontrolchartsforattributedata,npCharts,pCharts,cCharts,uCharts,-------------------------------------------------------------Break----------------------------------------------------------

20、--,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,,

21、------------------------------ 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 nor

22、mal,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,Ca

23、pability 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< Cp

24、; then,流程,is not centered,,Cpk= Cp; then,流程,is centered,USL - Y,3,,Y - LSL,3,,Cpk = min ,,The,流程,of collecting, presenting,和,describing sample data,using graphical,工具和,numbers.,Pareto Chart,Population mean,Histogram,Population,标准偏差,DescriptiveStatistics,Estimates for

25、Descriptive Statistics,The,流程,of estimating the population parameters from sample(s) that was taken from the population.,Sample mean, X,,Population mean,,m,Sample,标准偏差,, S,,Population,标准偏差,,,,(when sample size, n > 20),Estimated,标准偏差,, R/d,2,,,Population,标准偏差,,,,(when sample size, n,,20),Proba

26、bilityTheory,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 distri

27、bution 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 statistica

28、l 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 take

29、n.,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,Regr

30、ession,(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

31、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

32、 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), th

33、e 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 a

34、lso 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

35、 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= Deg

36、ree offreedom,= n(samplesize) –1,Shaded area=one-tailed probability of occurence,a,= 1,–,–Shadedarea,,Applicable when:,Sample size< 30,标准偏差,is unknown,Population distributionis at leastapproximately normally distributed,t (,a,,,u,),a,Area under the curve,Percentilesof the Normal Distribution /Z Di

37、stribution,Z,a,Area under the curve,Whereby,,,Shaded area=one-tailed probability of occurence,a,= 1,–,–Shadedarea,Studentt Distrbution example,FDArequirespharmaceuticalcompanies toperformextensive testsonallnewdrugs beforethey can bemarketed tothepublic.Thefirst phaseoftestingwillbeon animals,while

38、 the secondphase willbe on humanon alimitedbasis.PWDis apharmaceuticalcompanycurrently inthesecondphase oftestingona new antibioticproject. The chemists are interested toknow the effectofthenewantibiotic on the humanblood pressure,,和,theyareonly allowed totest on6 patients.Theresultof the increasein

39、 bloodpressure ofthe6 tested patients are as below:,,( 1.7, 3.0, 0.8, 3.4, 2.7, 2.1),,,Construct a95%confidence intervalfortheaverageincreaseinblood pressure for patientstaking the new antibiotic, usingboth normal,和,t distributions.,Studentt Distrbution example (con’t),Using normal or z distri

40、bution,Using student t distribution,Althoughtheconfidencelevel isthesame, usingt distributionwillresultin alargerintervalvalue,because:,标准偏差,, Sforsmall samplesize isprobably not accurate,标准偏差,, Sforsmall samplesize isprobably too optimistic,Widerintervalisthereforenecessary to achievethe required c

41、onfidencelevel,Summary of formulafor confidencelimit,6,Sigma,流程和,1.5SigmaShiftinMean,Statistically,a,流程,that is 6Sigmawithrespect to itsspecificationsis:,Normal Distribution,-6,,-5,,-4,,-3,,-2,,-1,,0,1,,2,,3,,4,,5,,6,,------------------------------ 99.9999999998% -------------------

42、---------,LSL,USL,DPM = 0.002,,Cp = 2,,Cpk = 2,ButMotoroladefines6 Sigma withascenarioof1.5Sigmashiftinmean,DPM=3.4,,Cp= 2,,Cpk=1.5,1.5,,某些,Explanationson1.5 Sigma MeanShift,Motorla hasconductedalotofexperiments,,和,foundthat in longterm,the,流程,mean willshiftwithin 1.5sigmaifthe,流程,isundercontrol.,,

43、1.5sigmameanshiftina 3Sigma,流程,control planwillbetranslated to approximately14% of thetime adatapointwill be outofcontrol,,和,this is deemacceptableinstatistical,流程,control (SPC) practices.,Normal Distribution,-3,,-2,,-1,,0,1,,2,,3,,------------------ 99.74% -----------------,LCL,UCL,Distribu

44、tion with 1.5 Sigma Shift,-3,,-2,,-1,,0,1,,2,,3,,----------------- 86.64% ----------------,LCL,UCL,Out of control data points,OurExplanation,Most frequentlyusedsamplesizeforSPC in industry is 3to5unitspersampling.,,Take themiddlevalueof4asanaverage samplesize usedinthe sampling.,,Assumingthe

45、,流程,isof6 sigma capability, is in control,,和,isnormallydistributed.,,Undertheconfidenceintervalforsamplingdistribution,weexpectthe averagevalueofthe samplestofluctuate within,,3 standard errors(i.e.natural tolerance), givingconfidence interval of:,IntroductiontoHypothesisTesting ?,What is hypothesi

46、stestinginstatistic?,A hypothesisis,“,“atentative assumptionmadeinordertodraw outortest itslogical or empiricalconsequences.”,A statistical hypothesisisastatementaboutthe value of oneofthecharacteristics foroneormorepopulations.,Thepurposeofthehypothesisistoestablishabasis,sothatonecan gatherevidenc

47、etoeitherdisprovethe statementoracceptitastrue.,Example of statistical hypothesis,,Theaveragecommute timeusingHighway92isshorter thanusingFrance Avenue.,,This,流程,changewillnotcauseany effectonthedownstream,流程,es.,,Thevariation of VendorB’spartsare 40%widerthan those of VendorA.,ElementsofHypothesis

48、Testing,Possibleoutcomesforhypothesistesting on twotestedpopulations:,No Significant Difference,Significant Difference in Variation,Significant Difference in Mean,Significant Difference in both Mean,和,Variation,m,1 ,m,2,,1 =,,2,m,1 ,m,2,,1 ,,2,m,1 =,m,2,,1 ,,2,m,1 =,m,2,,1 =,,2,为什么,Hypothesi

49、s Testing?,Many problems requireadecisiontoacceptorreject a,statement,abouta parameter.,,That statementisa,Hypothesis,.Itrepresents thetranslationofa practicalquestioninto astatisticalquestion.,,Statisticaltesting,提供,s an,objective,solution,withknown,risks,, to questionswhicharetraditionallyanswered

50、subjectively.,,Itisa stepping stone to,设计,ofExperiment,DOE.,Hypothesis TestingDescriptions,Hypothesis Testinganswersthepractical question:,“,“Isthereareal differencebetweenA,和,B ?,”,”,Inhypothesis testing, relativelysmallsamplesareusedtoanswerquestions about populationparameters.,,Thereisalwaysachan

51、cethata samplethat is notrepresentativeofthepopulationbeingselected,和,results in drawingawrongconclusion.,,,ElementsofHypothesis Testing(con’t),TheNullHypothesis,Statement,generallyassumedtobetrue,unlesssufficientevidenceisfoundtobecontrary,Oftenassumed to be thestatusquo,orthepreferred outcome. How

52、ever, it sometimes,represents astateyou strongly wanttodisprove,.,Designated as H,0,Inhypothesis testing, we alwaysbias towardnull hypothesis,TheAlternativeHypothesis(orResearchHypothesis),Statementthatwill be accepted onlyifdata,提供,convincing evidence of itstruth(i.e.byrejectingthe nullhypothesis).

53、,Instead of comparingtwopopulations, it canalso be based on aspecificengineeringdifferenceina characteristic value thatone desirestodetect(i.e.instead of askingis,m,1 =,m,2,weaskis,m,1 >450).,Designated as H,1,ElementsofHypothesis Testing(con’t),Example if we wanttotestwhether apopulationmean is equ

54、al 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%chancethatthe

55、meanisnatural 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,

56、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,).Th

57、eassumptionis,m,H,0,hasthe samevarianceas,m,p.,ElementsofHypothesis Testing(con’t),Otherpossiblealternatehypothesisare:,AlternateHypothesis,H,1,:,m,p,> 500; (1 tailtest),AlternateHypothesis,H,1,:,m,p,< 500; (1 tailtest),1.645,std error,Acceptance area,m,H,0,= 500,0.05,of area,(,a,),reject area,Takin

58、gexampleforalternate hypothesis,,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 err

59、or, 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 newvendors’parts(which are sligh

60、tlymoreexpensive) 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: comparisonc

61、onsidersa 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 th

62、rottle,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 valuetha

63、twillgivethe 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,willgivea

64、n averageresponsetimeof 1.2 msec ?,“,Largesample test ofhypothesis about a,populationmean,” (con’t),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),Al

65、ternate 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

66、 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,” (con’t),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