Statistical Intervals

631 Pages·2017·6.559 MB·English

by  William Q. Meeker, Gerald J. Hahn, Luis A. Escobar

Statistical Intervals
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Statistical Intervals
A Guide for Practitioners and Researchers
SecondEdition
William Q. Meeker
DepartmentofStatistics,IowaStateUniversity
Gerald J. Hahn
GeneralElectricCompany,GlobalResearchCenter(Retired)Schenectady,NY
Luis A. Escobar
DepartmentofExperimentalStatistics,LouisianaStateUniversity
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LibraryofCongressCataloging-in-PublicationData:
Names:Meeker,WilliamQ.|Hahn,GeraldJ.|Escobar,LuisA.
Title:Statisticalintervals:aguideforpractitionersandresearchers.
Description:Secondedition/WilliamQ.Meeker,GeraldJ.Hahn,LuisA.
Escobar.|Hoboken,NewJersey:JohnWiley&Sons,Inc.,[2017]|
Includesbibliographicalreferencesandindex.
Identiiers:LCCN2016053941|ISBN9780471687177(cloth)|ISBN9781118595169(epub)
Subjects:LCSH:Mathematicalstatistics.
Classiication:LCCQA276.H222017|DDC519.5/4–dc23LCrecordavailable
athttps://lccn.loc.gov/2016053941
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Contents
PrefacetoSecondEdition xxv
PrefacetoFirstEdition xxix
Acknowledgments xxxiii
AbouttheCompanionWebsite xxxv
1 Introduction,BasicConcepts,andAssumptions 1
1.1 StatisticalInference 2
1.2 DifferentTypesofStatisticalIntervals:AnOverview 3
1.3 TheAssumptionofSampleData 3
1.4 TheCentralRoleofPracticalAssumptionsConcerningRepresentativeData 4
1.5 EnumerativeversusAnalyticStudies 5
1.5.1 DifferentiatingbetweenEnumerativeandAnalyticStudies 6
1.5.2 StatisticalInferenceforAnalyticStudies 7
1.5.3 InferentialversusPredictiveAnalyses 7
1.6 BasicAssumptionsforInferencesfromEnumerativeStudies 8
1.6.1 DeinitionoftheTargetPopulationandFrame 8
1.6.2 TheAssumptionofaRandomSample 8
1.6.3 MoreComplicatedRandomSamplingSchemes 9
1.7 ConsiderationsintheConductofAnalyticStudies 11
1.7.1 AnalyticStudies 11
1.7.2 TheConceptofStatisticalControl 11
1.7.3 OtherAnalyticStudies 12
1.7.4 HowtoProceed 12
1.7.5 PlanningandConductinganAnalyticStudy 12
1.8 ConvenienceandJudgmentSamples 13
1.9 SamplingPeople 14
1.10 IninitePopulationAssumptions 15
1.11 PracticalAssumptions:Overview 15
1.12 PracticalAssumptions:FurtherExample 17
1.13 PlanningtheStudy 18
vii
viii CONTENTS
1.14 TheRoleofStatisticalDistributions 19
1.15 TheInterpretationofStatisticalIntervals 19
1.16 StatisticalIntervalsandBigData 21
1.17 CommentConcerningSubsequentDiscussion 21
2 OverviewofDifferentTypesofStatisticalIntervals 23
2.1 ChoiceofaStatisticalInterval 23
2.1.1 PurposeoftheInterval 24
2.1.2 CharacteristicofInterest 24
2.2 ConidenceIntervals 25
2.2.1 ConidenceIntervalforaDistributionParameter 25
2.2.2 ConidenceIntervalforaDistributionQuantile 25
2.2.3 ConidenceIntervalfortheProbabilityofMeetingSpeciications 25
2.2.4 One-SidedConidenceBounds 26
2.2.5 InterpretationsofConidenceIntervalsandBounds 26
2.3 PredictionIntervals 27
2.3.1 PredictionIntervaltoContainaSingleFutureObservation 27
2.3.2 PredictionIntervaltoContainAllofmFutureObservations 27
2.3.3 PredictionIntervaltoContainatLeastkoutofmFuture
Observations 28
2.3.4 PredictionIntervaltoContaintheSampleMeanorSampleStandard
DeviationofaFutureSample 28
2.3.5 One-SidedPredictionBounds 28
2.3.6 InterpretationofPredictionIntervalsandBounds 28
2.4 StatisticalToleranceIntervals 29
2.4.1 ToleranceIntervaltoContainaProportionofaDistribution 29
2.4.2 One-SidedToleranceBounds 29
2.4.3 Interpretationofβ-ContentToleranceIntervals 30
2.4.4 β-ExpectationToleranceIntervals 30
2.5 WhichStatisticalIntervalDoIUse? 30
2.6 ChoosingaConidenceLevel 31
2.6.1 FurtherElaboration 31
2.6.2 ProblemConsiderations 31
2.6.3 SampleSizeConsiderations 32
2.6.4 APracticalConsideration 32
2.6.5 FurtherRemarks 32
2.7 Two-SidedStatisticalIntervalsversusOne-SidedStatisticalBounds 33
2.8 TheAdvantageofUsingConidenceIntervalsInsteadofSigniicanceTests 33
2.9 SimultaneousStatisticalIntervals 34
3 Constructing Statistical Intervals Assuming a Normal Distribution Using Simple
Tabulations 37
3.1 Introduction 38
3.1.1 TheNormalDistribution 38
3.1.2 UsingtheSimpleFactors 39
3.2 CircuitPackVoltageOutputExample 39
3.3 Two-SidedStatisticalIntervals 40
3.3.1 SimpleTabulationsforTwo-SidedStatisticalIntervals 40
3.3.2 Two-SidedIntervalExamples 41
3.3.3 ComparisonofTwo-SidedStatisticalIntervals 42
CONTENTS ix
3.4 One-SidedStatisticalBounds 42
3.4.1 SimpleTabulationsforOne-SidedStatisticalBounds 42
3.4.2 One-SidedStatisticalBoundExamples 44
3.4.3 ComparisonofOne-SidedStatisticalBounds 45
4 MethodsforCalculatingStatisticalIntervalsforaNormalDistribution 47
4.1 Notation 48
4.2 ConidenceIntervalfortheMeanofaNormalDistribution 49
4.3 ConidenceIntervalfortheStandardDeviationofaNormalDistribution 49
4.4 ConidenceIntervalforaNormalDistributionQuantile 50
4.5 ConidenceIntervalfortheDistributionProportionLess(Greater)thana
SpeciiedValue 52
4.6 StatisticalToleranceIntervals 53
4.6.1 Two-SidedToleranceIntervaltoControltheCenterofaDistribution 53
4.6.2 Two-SidedToleranceIntervaltoControlBothTailsofaDistribution 54
4.6.3 One-SidedToleranceBounds 55
4.7 PredictionIntervaltoContainaSingleFutureObservationortheMeanofm
FutureObservations 55
4.8 PredictionIntervaltoContainatLeastkofmFutureObservations 56
4.8.1 Two-SidedPredictionInterval 56
4.8.2 One-SidedPredictionBounds 57
4.9 PredictionIntervaltoContaintheStandardDeviationofmFuture
Observations 58
4.10 TheAssumptionofaNormalDistribution 59
4.11 AssessingDistributionNormalityandDealingwithNonnormality 60
4.11.1 ProbabilityPlotsandQ–QPlots 60
4.11.2 InterpretingProbabilityPlotsandQ–QPlots 60
4.11.3 DealingwithNonnormalData 62
4.12 DataTransformationsandInferencesfromTransformedData 63
4.12.1 PowerTransformations 63
4.12.2 ComputingStatisticalIntervalsfromTransformedData 64
4.12.3 ComparisonofInferencesUsingDifferentTransformations 65
4.12.4 Box–CoxTransformations 66
4.13 StatisticalIntervalsforLinearRegressionAnalysis 66
4.13.1 ConidenceIntervalsforLinearRegressionAnalysis 67
4.13.2 ToleranceIntervalsforLinearRegressionAnalysis 68
4.13.3 PredictionIntervalsforRegressionAnalysis 68
4.14 StatisticalIntervalsforComparingPopulationsandProcesses 68
5 Distribution-FreeStatisticalIntervals 73
5.1 Introduction 74
5.1.1 Motivation 74
5.1.2 Notation 75
5.2 Distribution-FreeConidenceIntervalsandOne-SidedConidenceBounds
foraQuantile 78
5.2.1 CoverageProbabilitiesforDistribution-FreeConidenceIntervalsor
One-SidedConidenceBoundsforaQuantile 78
5.2.2 UsingInterpolationtoObtainApproximateDistribution-Free
ConidenceBoundsorConidenceIntervalsforaQuantile 78
5.2.3 Distribution-FreeOne-SidedUpperConidenceBoundsfora
Quantile 79
x CONTENTS
5.2.4 Distribution-FreeOne-SidedLowerConidenceBoundsfora
Quantile 82
5.2.5 Distribution-FreeTwo-SidedConidenceIntervalforaQuantile 84
5.3 Distribution-FreeToleranceIntervalsandBoundstoContainaSpeciied
ProportionofaDistribution 86
5.3.1 Distribution-FreeTwo-SidedToleranceIntervals 86
5.3.2 Distribution-FreeOne-SidedToleranceBounds 88
5.3.3 MinimumSampleSizeRequiredforConstructingaDistribution-Free
Two-SidedToleranceInterval 88
5.4 PredictionIntervalsandBoundstoContainaSpeciiedOrderedObservation
inaFutureSample 89
5.4.1 CoverageProbabilitiesforDistribution-FreePredictionIntervals
andOne-SidedPredictionBoundsforaParticularOrdered
Observation 89
5.4.2 Distribution-FreeOne-SidedUpperPredictionBoundforY 90
(j)
5.4.3 Distribution-FreeOne-SidedLowerPredictionBoundforY 91
(j)
5.4.4 Distribution-FreeTwo-SidedPredictionIntervalforY 92
(j)
5.5 Distribution-FreePredictionIntervalsandBoundstoContainatLeastkof
mFutureObservations 93
5.5.1 Distribution-FreeTwo-SidedPredictionIntervalstoContainatLeast
kofmFutureObservations 93
5.5.2 Distribution-FreeOne-SidedPredictionBoundstoExceedorBe
ExceededbyatLeastkofmFutureObservations 96
6 StatisticalIntervalsforaBinomialDistribution 99
6.1 Introduction 100
6.1.1 TheBinomialDistribution 100
6.1.2 OtherDistributionsandRelatedNotation 100
6.1.3 NotationforDataandInference 101
6.1.4 BinomialDistributionStatisticalIntervalProperties 101
6.1.5 TwoExamples,Motivation,andaCaution 101
6.2 ConidenceIntervalsfortheActualProportionNonconforminginthe
SampledDistribution 102
6.2.1 Preliminaries 102
6.2.2 TheConservativeMethod 103
6.2.3 TheWald(NormalTheory)ApproximateMethod 105
6.2.4 TheAgresti–CoullAdjustedWald-ApproximationMethod 105
6.2.5 TheJeffreysApproximateMethod 107
6.2.6 ComparisonsandRecommendations 108
6.3 ConidenceIntervalfortheProportionofNonconformingUnitsinaFinite
Population 112
6.3.1 TheConservativeMethod 113
6.3.2 Large-PopulationApproximateMethod 114
6.4 ConidenceIntervalsfortheProbabilitythattheNumberofNonconforming
UnitsinaSampleisLessthanorEqualto(orGreaterthan)aSpeciied
Number 114
6.5 ConidenceIntervalsfortheQuantileoftheDistributionoftheNumberof
NonconformingUnits 116
6.5.1 Two-SidedConidenceIntervalfory 116
p
6.5.2 One-SidedConidenceBoundsfory 117
p