Sample_Size_Calculations:样本大小的计算.pdf.pdf
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Sample_Size_Calculations:样本大小的计算.pdf.pdf
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SampleSizeCalculationsPracticalMethodsforEngineersandScientists(Softwaresolutionstoselectedexampleproblems,Version:
11June2010)SampleSizeCalculationsPracticalMethodsforEngineersandScientists(Softwaresolutionstoselectedexampleproblems,Version:
11June2010)PaulMathewsSampleSizeCalculations:
PracticalMethodsforEngineersandScientistsPaulMCopyrightc?
2010PaulMathewsAllrightsreserved.Nopartofthispublicationmaybereproducedorstoredinanyformorbyanymeanswithoutthepriorwrittenpermissionofthepublisher.Publishedby:
MathewsMalnarandBailey,Inc.217ThirdStreet,FairportHarbor,OH44077Phone:
440-350-0911Fax:
440-350-7210Web:
ISBN978-0-615-32461-6Originalrelease:
24February2010Contents1Fundamentals51.1MotivationforSampleSizeCalculations.51.2RationaleforSampleSizeandPowerCalculations.51.3RationaleforHypothesisTests.101.4PracticalConsiderations.141.5ProblemsandSolutions.201.6Software.202Means212.1Assumptions.212.2OneMean.212.3TwoIndependentMeans.322.4EquivalenceTests.442.5Contrasts.482.6MultipleComparisonsTests.493StandardDeviations553.1OneStandardDeviation.553.2TwoStandardDeviations.593.3CoefficientofVariation.634Proportions654.1OneProportion(LargePopulation).654.2OneProportion(SmallPopulation).784.3TwoProportions.814.4EquivalenceTests.914.5Chi-squareTests.95vviCONTENTS5PoissonCounts995.1OnePoissonCount.995.2TwoPoissonCounts.1035.3TestsforManyPoissonCounts.1075.4CorrectingforBackgroundCounts.1086Regression1096.1LinearRegression.1096.2LogisticRegression.1167CorrelationandAgreement1197.1PearsonsCorrelation.1197.2IntraclassCorrelation.1237.3CohensKappa.1257.4ReceiverOperatingCharacteristic(ROC)Curves.1277.5Bland-AltmanPlots.1308DesignedExperiments1338.1One-WayFixedEffectsANOVA.1338.2RandomizedBlockDesign.1408.3BalancedFullFactorialDesignwithFixedEffects.1418.4RandomandMixedModels.1468.5NestedDesigns.1488.6Two-LevelFactorialDesigns.1498.7Two-LevelFactorialDesignswithCenters.1588.8ResponseSurfaceDesigns.1599ReliabilityandSurvival1619.1ReliabilityParameterEstimation.1619.2ReliabilityDemonstrationTests.1679.3Two-SampleReliabilityTests.1759.4Interference.17810StatisticalQualityControl18110.1StatisticalProcessControl.18110.2ProcessCapability.18310.3ToleranceIntervals.18510.4AcceptanceSampling.18610.5GageR&RStudies.197CONTENTSvii11ResamplingMethods19911.1SoftwareRequirements.19911.2MonteCarlo.19911.3Bootstrap.200viiiCONTENTSCONTENTS1ThepurposeofthisdocumentistopresentsolutionstoselectedexampleproblemsfromthebookusingPASS,MINITAB,Piface,andR.(Thisversionofthedocument,compiledon11June2010,doesnotyetcontainsolutionsusingR.)Alloftheprogramshavemoresamplesizeandpowercalculationcapabilitiesthanwhatisincludedinthebook.Somepackagesareparticularlystrongincertainareas.Forexample,PASShasthebroadestscope,PifaceoffersanunmatchedcollectionofANOVAmethodsinvolvingfixed,random,mixed,andnesteddesignsandsupportscustomANOVAmodels,andMINITABhasspecialmethodsforqualityengineersincludingattributeandvariablessamplingplandesignandreliabilitystudydesign.ThefollowingfiguresshowscreencapturesofsomeofthemethodsavailableinPiface,PASS,andMINITAB.2CONTENTSCONTENTS3Mostoftheprogramsemphasizesamplesizeandpowercalculationsforhypothesistestsbuttheycanbetrickedintoperformingapproximatesamplesizecalculationsforconfidenceintervalsbysettingthehypothesistestpowerto?
=0:
50.Thistrickisexactwhenthesamplingdistributionisnormalbecausez0:
50=0anditisreasonablyaccuratewhenthesamplingdistributionisotherthannormalbutsymmetric.Bemorecarefulwhenthesamplingdistributionisasymmetric.4CONTENTSThesolutionsinthebookdontusethecontinuitycorrectionwhendiscretedistributionsareapproximatedwithcontinuousones,however,thesoftwaresolutionsoftenincludethecontinuitycorrectionsoanswerstoproblemsmaydifferslightly.Ifyouhaveaccesstosoftwarethatprovidesmoreaccuratemethods,thendefinitelyusethesoftware.Somesoftwareprovidesseveralanalysismethodsforthesameproblem.Forexample,PASSofferssixdifferentmethodsforthesignificancetestforoneproportionexpressedintermsoftheproportiondifference.Thedifferentmethodsusuallygivesimilaranswers.Thisdocumentwillberevisedoccasionally.Thecurrentversionwascompiledon11June2010.Chapter1Fundamentals1.1MotivationforSampleSizeCalculations1.2RationaleforSampleSizeandPowerCalculationsExample1.1ExpresstheconfidenceintervalP(3:
1?
StatPowerandSampleSize1-SampleZ:
56Chapter1.FundamentalsFromPASSMeansOneConfidenceIntervalofMean:
Example1.3WhatisthenewsamplesizeinExample1.2iftheprocessownerprefersa99%confidenceinterval?
Solution:
Withz?
=2=z0:
005=2:
575inEquation1.4therequiredsamplesizeisn=?
2:
575?
0:
0030:
002?
2=15.FromMINITABStatPowerandSampleSize1-SampleZ:
1.2.RationaleforSampleSizeandPowerCalculations7FromPASSMeansOneConfidenceIntervalofMean:
Example1.4WhatisthenewsamplesizeinExample1.2iftheprocessownerprefersa95%confidencelevelwith?
=0:
001mmhalf-width?
Solution:
Withz0:
025=1:
96and?
=0:
001mminEquation1.4therequiredsamplesizeisn=?
1:
96?
0:
0030:
001?
2=35.8Chapter1.FundamentalsFromMINITABStatPowerandSampleSize1-SampleZ:
FromPASSMeansOneConfidenceIntervalofMean:
Example1.5Determinethesamplesizerequiredtoestimatethemeanofapopulationwhen?
x=30isknownandthepopulationmeanmustnotexceedthesamplemeanbymorethan?
=10with95%confidence.Solution:
Aone-sidedupper95%confidenceintervalisrequiredoftheformP(?
StatPowerandSampleSize1-SampleZ:
FromPASSMeansOneConfidenceIntervalofMean:
10Chapter1.Fundamentals1.3RationaleforHypothesisTestsExample1.6AnexperimentisplannedtotestthehypothesesH0:
?
=3200versusHA:
?
6=3200.Theprocessisknowntobenormallydistributedwithstandarddeviation?
x=400.Whatsamplesizeisrequiredtodetectapracticallysignificantshiftintheprocessmeanof?
=300withpower?
=0:
90?
Solution:
With?
=1?
=0:
10andassuming?
=0:
05inEquation1.12,thesamplesizerequiredtodetectashiftfrom?
=3200to?
=2900or?
=3500with90%powerisn=?
(z0:
025+z0:
10)?
x?
2=?
(1:
96+1:
282)400300?
2=19wherethecalculatedvalueofnwasroundeduptothenearestintegervalue.FromMINITABStatPowerandSampleSize1-SampleZ:
FromPASSMeansOneInequality(Normal):
1.3.RationaleforHypothesisTests11Example1.7AnexperimentwillbeperformedtotestH0:
?
=8:
0versusHA:
?
8:
0.WhatsamplesizeisrequiredtorejectH0with90%powerwhen?
=8:
2?
Theprocessisknowntobenormallydistributedwith?
x=0:
2.Solution:
Fortheone-tailedhypothesistestwith?
=0:
05,?
=0:
2and?
=1?
=0:
10therequiredsamplesizeisn=?
(z?
+z?
)?
x?
2=?
(z0:
05+z10)?
x?
2=?
(1:
645+1:
282)0:
20:
2?
2=9.FromMINITABStatPowerandSampleSize1-SampleZ:
12Chapter1.FundamentalsFromPASSMeansOneInequality(Normal):
Example1.8CalculatethepvalueforthetestperformedundertheconditionsofExample1.6ifthesamplemeanwas?
x=3080.1.3.RationaleforHypothesisTests13Solution:
Figure1.4showsthecontributionstothepvaluefromthetwotailsofthe?
xdistributionunderH0.Thezteststatisticthatcorrespondsto?
xisz=?
x?
0?
x=?
x?
0?
x=pn=3080?
3200400=p19=?
1:
31,sothepvalueisp=1?
(?
1:
31z(?
=0:
05),theobservedsamplemeanisstatisticallyconsistentwithH0:
?
=3200,sowecannotrejectH0.FromMINITABStatPowerandSampleSize1-SampleZ:
Example1.9CalculatethepvalueforthetestperformedundertheconditionsofExample1.7ifthesamplemeanwas?
x=8:
39.Solution:
Figure1.5showsthesinglecontributiontothepvaluefromtherighttailofthe?
xdistributionunderH0.Thezteststatisticthatcorrespondsto?
xisz=8:
39?
8:
20:
2=p9=2:
85,14Chapter1.Fundamentalssothepvalueisp=?
(2:
85z1)=0:
0022.Because(p=0:
0022)StatPowerandSampleSize1-SampleZ:
1.4PracticalConsiderationsExample1.10Whatsamplesizeisrequiredforapilotstudytoestimatethestandarddeviationtobeusedinthesamplesizecalculationforaprimaryexperimentifthesamplesizefortheprimaryexperimentshouldbewithin20%ofthecorrectvaluewith90%confidence?
Solution:
With?
=0:
20and?
=0:
10inEquation1.20,therequiredsamplesizeforthepreliminaryexperimenttoestimatethestandarddeviationisn2?
1:
6450:
20?
2136.FromPifacePilotStudy:
1.4.PracticalConsiderations15FromPASSVarianceVariance:
1Group:
Example1.11Anengineermustobtainapprovalfromhismanagertotestacertainnumberofunitstodeterminethemeanresponseforavalidationstudy.Thestandarddeviationoftheresponseis?
x=600andthesmallestpracticallysignificantshiftinthemeanthattheexperimentshoulddetectisunderstoodtobe?
=400.Whatgraphshouldtheengineerusetopresenthiscase?
16Chapter1.FundamentalsSolution:
Thevalueoftheeffectsizeofinterestisfirmat?
=400.Thesamplesizeisgoingtoaffectthepowerofthetest,soanappropriategraphispowerversussamplesize.ThesamplesizerequiredtoobtainaspecifiedvalueofpowerforthetestofH0:
?
=0versusHA:
?
6=0isgivenbyEquation1.12.Figure1.6showstheresultingpowercurve.Thesamplesizerequiredtoobtain80%powerisn=18andthesamplesizerequiredfor90%powerisn=24.FromMINITABStatPowerandSampleSize1-SampleZ:
FromPASSMeansOneInequality(Normal):
1.4.PracticalConsiderations1718Chapter1.FundamentalsExample1.12SupposethatthemanagerinExample1.11approvestheuseofn=24unitsinthevalidationstudy.WhatpowerdoesthestudyhavetorejectH0whentheeffectsizeis?
=200,400,and600?
7Solution:
Thepowerisgivenby?
=?
(?
z?
zStatPowerandSampleSize1-SampleZ:
1.4.PracticalConsiderations19FromPASSMeansOneInequality(Normal):
20Chapter1.Fundamentals1.5ProblemsandSolutions1.6SoftwareChapter2Means2.1Assumptions2.2OneMeanExample2.1
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