Lecture 12 Chapter 8 Inference for proportions12讲8章推理的比例.docx

Lecture 12 Chapter 8 Inference for proportions12讲8章推理的比例.docx

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Lecture12Chapter8Inferenceforproportions12讲8章推理的比例

Lecture12,Section8.1&8.2

Proportions

Manystatisticalstudiesproducecountsratherthanmeasurements.

Example:

Didyouvoteinthelastelection?

Theresponsewouldbeeithera“Yes”ora“No”.Thevariableiscategorical,theresponseisthevaluethevariabletakesonforeachunit/person.IfIdidasurveyofthisclass,Icouldaccumulatethecountof“Yes”responsesanddescribethiscountasaproportionofthetotal.

Example:

WhatacademicyearareyouinatPurdue.Theresponsewouldbeeither“Freshman”,“Sophomore”,“Junior”,or“Senior”.Again,Icouldaccumulatethecountofeachanddescribeeachasaproportionofthetotal.

PopulationandSampleproportions:

Instatisticalsamplingweoftenwanttoestimatetheproportion,p,of“successes”inapopulation.“Success”iswhenthecategoricalvariabletakesononeparticularvalue.

p=countofsuccessesinpopulation

sizeofpopulation

=X/N

Wetakeasampleofourpopulation;ourestimatoristhesampleproportionofsuccesses:

countofsuccessesinsample

sizeofsample

=X/n

Example:

1.Youflipacoin20timesandrecordwhetheraheadoratailistossed.Inthissample,aheadisrecorded11times.Whatisthesampleproportionofheads?

InferenceforaSingleProportion:

Sofarwehaveonlylookedatmakingstatisticalinferenceonpopulationmeans,ameasurementofsomequantitativevariableofinterest.Nowwewilllookatmakingstatisticalinferenceonacategoricalvariableusingtheproportionofsomeoutcome/successinapopulation.

Examples:

∙HowcommonisitforstudentsatPurduetofailaclass?

Outofasampleof200students,50ofthemhavefailedatleastoneclass,or25%.Basedonthesedata,whatcanwesayaboutallstudentsatPurdue?

∙WhatproportionofgolfersintheUSAhavemadeatleastonehole-in-oneintheirlifetime.FromanSRSof50golfers25ofthemhadmadeahole-in-one.WhatcanwesayaboutallgolfersintheUSA?

Inbothexamplesaboveweareinterestedinestimatingtheunknownproportionpfromapopulation.Theestimateofthatpopulationparameterpisthesampleproportion

astatistic.

SamplingDistributionofaSampleProportion:

ChooseanSRSofsizenfromalargepopulationwithpopulationproportionphavingsomecharacteristicofinterest.Wenormallycallwhatevercharacteristicwearestudyinga“success.”LetXbethecountofsuccessesinthesampleand

bethesampleproportionofsuccess,

=X/n

Also:

∙Thesamplingdistributionof

isapproximatelynormalforaSRSfromalargepopulationandisclosertoanormaldistributionwhenthesamplesizenislarge.

∙Themeanofthesamplingdistributionisp.

∙Thestandarddeviationofthesamplingdistributionis

Large-SampleConfidenceIntervalforaPopulationProportion:

ChooseanSRSofsizenfromalargepopulationwithunknownproportionpofsuccesses.Thesampleproportionis:

=X/n

Thestandarderrorof

is:

Whennislarge,anapproximatelevelCconfidenceintervalforpis:

where

isthevalueforthestandardnormaldensitycurvewithareaCbetween

and

.Themarginoferroris:

Usethisintervalwhenthenumberofsuccessesandthenumberoffailuresarebothatleast15andtheconfidencelevelis90%,95%,or99%.

Example:

1.Whentryingtohiremanagersandexecutives,companiessometimesverifytheacademiccredentialsdescribedbytheapplicants.Onecompanythatperformsthesecheckssummarizeditsfindingsforasix-monthperiod.Ofthe84applicantswhosecredentialswerechecked,15liedabouthavingadegree.

a.Givetheestimateoftheproportionofapplicantswholiedabouthavingadegreeandgivetheestimateforthestandarderrorof

.

b.Considerthesedatatobearandomsampleofcredentialsfromalargecollectionofsimilarapplicants.Givea95%confidenceintervalforthetrueproportionofapplicantswholieabouthavingadegree.

Large-SampleSignificanceTestforaPopulationProportion:

1.StatetheNullandAlternativehypothesis.

or

or

2.Findtheteststatistic:

DrawaSRSofsizenfromalargepopulationwithunknownproportionpofsuccesses.Totestthehypothesis,computethezstatistic:

3.Calculatethep-value.

IntermsofastandardnormalrandomvariableZ,theapproximateP-valueforatestof

against

is

is

is

4.Statetheconclusionsintermsoftheproblem.Chooseasignificancelevelsuchasα=0.05,thencomparetheP-valuetotheαlevel.

IfP-value

α,thenreject

IfP-value>α,thenfailtoreject

UsetheLarge-SampleSignificanceTestforaPopulationProportioniftheexpectednumberofsuccesses

andtheexpectednumberoffailures

arebothgreaterthan10.Ifthisisnotmet,orifthepopulationislessthan10timesaslargeasthesample,otherproceduresshouldbeused.

Example:

1.Shereka,astartingplayerforamajorcollegebasketballteam,made60%ofherfreethrowsinherlastthreeseasons.Duringthesummersheworkedhardondevelopingasoftershotinthehopeofimprovingherfree-throwaccuracy.InthefirstninegamesofthisseasonSherekamade48freethrowsin67attempts.Letpbeherprobabilityofsuccess,makingeachfreethrowsheshoots.

a.Statethenullhypothesis

thatShereka’sfree-throwprobabilityhasremainedthesameasthelastthreeseasonsandthealternative

thatherworkinthesummerresultedinahigherprobabilityofsuccess.

b.Calculatethezstatisticfortesting

versus

.

c.Doyouacceptorreject

forα=0.05?

FindtheP-value.

d.Givea90%confidenceintervalforShereka’sfree-throwsuccessprobabilityforthenewseason.Areyouconvincedthatsheisnowabetterfree-throwershooter?

e.Whatassumptionsareneededforthevalidityofthetestandconfidenceintervalcalculationsthatyouperformed?

SampleSizeforDesiredMarginofError:

ThelevelCconfidenceintervalforaproportionpwillhaveamarginoferrorapproximatelyequaltoaspecifiedvaluemwhenthesamplesizesatisfies

Herez*isthecriticalvalueforconfidenceC,andp*isanestimatedorguessedvaluefortheproportionofsuccessesinthefuturesample.Theestimatedvaluecanbeeitherbasedonapreviouspilotstudyoritcanbeassumedtobe.5,thevalueofp*thatgeneratesthelargestsamplesize.Themarginoferrorwillbelessthanorequaltomifp*ischosentobe0.5.Thesamplesizerequiredisthengivenby

Example:

1.Youwanttoestimatetheproportionofstudentsatyourcollegeoruniversitywhoareemployedfor10ormorehoursperweekwhileclassesareinsession.Youplantopresentyourresultsbya95%confidenceinterval.Usingtheestimatedvalue

findthesamplesizerequirediftheintervalistohaveanapproximatemarginoferrorof

.

ComparingTwoProportions

AssumptionsforcomparingtwoProportions:

∙ThedataconsistofthetwoindependentSRS’s

∙ThetwoSRS’sarelarge.

Typicallywewanttocomparetwoproportionsbygivingaconfidenceintervalforthedifference,

orbytestingthehypothesisofnodifference,

.

ConfidenceIntervalsforComparingTwoProportions

ChooseanSRSofsize

fromalargepopulationhavingproportion

ofsuccesses;chooseaSRSofsize

fromanotherlargepopulationhavingproportion

ofsuccess.

AnapproximatelevelCconfidenceintervalfor

is

Where

and

aretheestimatesofthepopulationproportions;thestandarderrorofthedifferenceis

andz*isthevalueforthestandardNormaldensitycurvewithareaCbetween–z*andz*.Themarginoferroris:

Usethismethodwhenthenumberofsuccessesandthenumberoffailuresinbothsamplesizesareatleast10andtheconfidencelevelis90%,95%,or99%.

Example

1.Islyingaboutcredentialsbyjobapplicantschanging?

Fromapreviousexample,onecompanythatperformsthesecheckssummarizeditsfindingsforthefirstsix-monthperiod:

ofthe84applicantswhosecredentialswerechecked,15liedabouthavingadegree.Thecompanyperformedthesamechecksforasecondperiod,theresultsforbothareshownbelow.

Period

n

X（lied）

1

84

15

2

106

35

a）Findthe95%confidenceintervalforthedifferenceoftheproportionofapplicantswholiedabouthavingadegreeforthetwoperiods.

b）Basedonthisconfidenceinterval,whatcanyousayaboutthechangeovertimeoftheproportionofapplicantsthatliedontheirapplicationforthetwoperiods?

SignificanceTestsforComparingTwoProportions

1.StatetheNullandAlternativehypothesis.

or

or

2.Findtheteststatistic:

DrawaSRSofsizenfromalargepopulationwithunknownproportionpofsuccesses.Totestthehypothesis,computethezstatistic:

where

and

andwherethepooledstandarderroris

andwhere

3.Calculatethep-value.

IntermsofastandardnormalrandomvariableZ,theapproximateP-valueforatestof

against

is

is

is

4.Statetheconclusionsintermsoftheproblem.Chooseasignificancelevelsuchasα=0.05,thencomparetheP-valuetotheαlevel.

IfP-value

α,thenreject

IfP-value>α,thenfailtoreject

Examples

1.Dataontheproportionofapplicantswholiedabouthavingadegreeintwoconsecutivesix-monthperiodsaregiveninthepreviousexampleas:

Period

n

X（lied）

1

84

15

2

106

35

a.Formulateappropriatenullandalternativehypothesesthatcanbeaddressedwiththesedata,carryoutthesignificancetest,andsummarizetheresults.