Download 8.1 Confidence Intervals: The Basics Point estimator

8.1ConfidenceIntervals:TheBasics
Pointestimator-Apointestimatorisastatisticthatprovidesanestimateofapopulationparameter.The
valueofthatstatisticfromasampleiscalledapointestimate.Ideally,apointestimateisour“bestguess”at
thevalueofanunknownparameter.
Example–FromBatteriestoSmoking
PointEstimators
PROBLEM:Ineachofthefollowingsettings,determinethepointestimatoryouwoulduseandcalculatethe
valueofthepointestimate.
(a)QualitycontrolinspectorswanttoestimatethemeanlifetimeμoftheAAbatteriesproducedinanhour
atafactory.Theyselectarandomsampleof30batteriesduringeachhourofproductionandthendrain
themunderconditionsthatmimicnormaluse.Herearethelifetimes(inhours)ofthebatteriesfromone
suchsample:
Usethesamplemeanx-barasapointestimatorforthepopulationmeanμ.Forthesedata,ourpoint
estimateis
hours.
(b)WhatproportionpofU.S.highschoolstudentssmoke?The2007YouthRiskBehavioralSurvey
questionedarandomsampleof14,041studentsingrades9to12.Ofthese,2808saidtheyhadsmoked
cigarettesatleastonedayinthepastmonth.
Usethesampleproportion
asapointestimatorforthepopulationproportionp.Forthissurvey,our
pointestimateis
.
(c)Thequalitycontrolinspectorsinpart(a)wanttoinvestigatethevariabilityinbatterylifetimesby
estimatingthepopulationvarianceσ2.
SOLUTION:
Usethesamplevariance
data,ourpointestimateis
asapointestimatorforthepopulationvarianceσ2.Forthebatterylife
.
ConfidenceInterval-Anintervalcalculatedfromthedata,whichhastheform:
estimate±marginoferror
Marginoferror-Themarginoferrortellshowclosetheestimatetendstobetotheunknownparameterin
repeatedrandomsampling.
ConfidenceLevelC–AconfidencelevelCgivestheoverallsuccessrateofthemethodforcalculatingthe
confidenceinterval.Thatis,inC%ofallpossiblesamples,themethodwouldyieldanintervalthatcapturesthe
trueparametervalue.
Aconfidenceintervalissometimesreferredtoasanintervalestimate.Thisisconsistentwithourearlieruseof
thetermpointestimate.
Weusuallychooseaconfidencelevelof90%orhigherbecausewewanttobequitesureofour
conclusions.Themostcommonconfidencelevelis95%
8.1.2InterpretingConfidenceLevelsandConfidenceIntervals
Thefigurebelowillustratestheideaofaconfidenceintervalinadifferentform.Itshowstheresultofdrawing
manySRSsfromthesamepopulationandcalculatinga95%confidenceintervalfromeachsample.Thecenter
ofeachintervalisatx-barandthereforevariesfromsampletosample.Thesamplingdistributionofxbarappearsatthetopofthefiguretoshowthelong-termpatternofthisvariation.The95%confidence
intervalsfrom25SRSsappearbelow.
Here’swhatyoushouldnotice:
• Thecenterx-barofeachintervalismarkedbyadot.
• Thedistancefromthedottoeitherendpointofthe
intervalisthemarginoferror.
• 24ofthese25intervals(that’s96%)containthetrue
valueofμ.Ifwetookallpossiblesamples,95%ofthe
resultingconfidenceintervalswouldcaptureμ.
Theconfidenceleveltellsushowlikelyitisthatthemethodweareusingwillproduceanintervalthat
capturesthepopulationparameterifweuseitmanytimes.However,inpracticewetendtocalculateonlya
singleconfidenceintervalforagivensituation.
Theconfidenceleveldoesnottellusthechancethataparticularconfidenceintervalcapturesthepopulation
parameter.Instead,theconfidenceintervalgivesusasetofplausiblevaluesfortheparameter.
Weinterpretconfidencelevelsandconfidenceintervalsinmuchthesamewaywhetherweareestimatinga
populationmean,proportion,orsomeotherparameter.
Example–DoYouUseTwitter?
Interpretingaconfidenceintervalandaconfidencelevel
Inlate2009,thePewInternetandAmericanLifeProjectaskedarandomsampleof2253U.S.adults,“Doyou
ever…useTwitteroranotherservicetoshareupdatesaboutyourselfortoseeupdatesaboutothers?”Ofthe
sample,19%said“Yes.”AccordingtoPew,theresulting95%confidenceintervalis(0.167,0.213).
Interprettheconfidenceintervalandtheconfidencelevel.
AP EXAM TIP On a given problem, you may be asked to interpret the confidence interval, the confidence level, or both. Be sure you
understand the difference: the confidence level describes the long-run capture rate of the method, and the confidence interval gives a set of
plausible values for the parameter.
Confidenceintervalsarestatementsaboutparameters.Inthepreviousexample,itwouldbewrongto
say,“Weare95%confidentthattheintervalfrom0.167to0.213containsthesampleproportionofU.S.adults
whouseTwitter.”Why?Becauseweknowthatthesampleproportion,
,isinthe
interval.Likewise,inthemysterymeanexample,itwouldbewrongtosaythat“95%ofthevaluesare
between230.79and250.79,”whetherwearereferringtothesampleorthepopulation.Allwecansay
is,“BasedonMr.Schiel’ssample,webelievethatthepopulationmeanissomewherebetween230.79and
250.79.“
CHECKYOURUNDERSTANDING
How much does the fat content of Brand X hot dogs vary? To find out, researchers measured the fat content (in
grams) of a random sample of 10 Brand X hot dogs. A 95% confidence interval for the population standard
deviation σ is 2.84 to 7.55.
1. Interpret the confidence interval.
2. Interpret the confidence level.
3. True or false: The interval from 2.84 to 7.55 has a 95% chance of containing the actual population standard
deviation σ. Justify your answer.
Amoregeneralformulaforaconfidenceinterval:
statistic±(criticalvalue)·(standarddeviationofstatistic)
ThecriticalvaluedependsonboththeconfidencelevelCandthesamplingdistributionofthestatistic.
TheconfidenceintervalforthemysterymeanμofMr.Schiel’spopulationillustratesseveralimportant
propertiesthataresharedbyallconfidenceintervalsincommonuse.Theuserchoosestheconfidence
level,andthemarginoferrorfollowsfromthischoice.Wewouldlikehighconfidenceandalsoasmallmargin
oferror.Highconfidencesaysthatourmethodalmostalwaysgivescorrectanswers.Asmallmarginoferror
saysthatwehavepinneddowntheparameterquiteprecisely.
Ourgeneralformulaforaconfidenceintervalis:
Wecanseethatthemarginoferrordependsonthecriticalvalueandthestandarddeviationofthe
statistic.Thecriticalvalueistieddirectlytotheconfidencelevel:greaterconfidencerequiresalargercritical
value.Thestandarddeviationofthestatisticdependsonthesamplesizen:largersamplesgivemoreprecise
estimates,whichmeanslessvariabilityinthestatistic.
Sothemarginoferrorgetssmallerwhen:
• Theconfidenceleveldecreases.Thereisatrade-offbetweentheconfidencelevelandthemarginof
error.Toobtainasmallermarginoferrorfromthesamedata,youmustbewillingtoacceptlower
•
confidence.Earlier,wefoundthata95%confidenceintervalforMr.Schiel’smysterymeanμis230.79
to250.79.The80%confidenceintervalforμis234.39to247.19.Thefigurebelowcomparesthesetwo
intervals.
Thesamplesizenincreases.Increasingthesamplesizenreducesthemarginoferrorforanyfixed
confidencelevel.
8.1.4UsingConfidenceIntervalsWisely
Ourgoalinthissectionhasbeentointroduceyoutothebigideasofconfidenceintervalswithoutgetting
boggeddownintechnicaldetails.Youmayhavenoticedthatweonlycalculatedintervalsinacontrived
setting:estimatinganunknownpopulationmeanμwhenwesomehowknewthepopulationstandard
deviationσ.Inpractice,whenwedon’tknowμ,wedon’tknowσeither.We’lllearntoconstructconfidence
intervalsforapopulationmeaninthismorerealisticsettinginSection8.3.First,wewillstudyconfidence
intervalsforapopulationproportionpinSection8.2.Althoughitispossibletoestimateother
parameters,confidenceintervalsformeansandproportionsarethemostcommontoolsineverydayuse.
Beforecalculatingaconfidenceintervalforμorp,therearethreeimportantconditionsthatyoushouldcheck.
1.Random:Thedatashouldcomefromawell-designedrandomsampleorrandomized
experiment.Otherwise,there’snoscopeforinferencetoapopulation(sampling)orinferenceaboutcause
andeffect(experiment).Ifwecan’tdrawconclusionsbeyondthedataathand,thenthere’snotmuchpoint
constructingaconfidenceinterval!Anotherimportantreasonforrandomselectionorrandomassignmentis
tointroducechanceintothedataproductionprocess.
Theexperimentalequivalentofasamplingdistributioniscalledarandomizationdistribution.
Wecanmodelchancebehaviorwithaprobabilitydistribution,likethesamplingdistributionsofchapter7.The
probabilitydistributionhelpsuscalculateaconfidenceinterval.
2.Normal:Themethodsweusetoconstructconfidenceintervalsforμandpdependonthefactthatthe
samplingdistributionofthestatistic(x-barorp-hat)isatleastapproximatelyNormal.
Formeans:Thesamplingdistributionofx-barisexactlyNormalifthepopulationdistributionisNormal.When
thepopulationdistributionisnotNormal,thecentrallimittheoremtellsusthatthesamplingdistributionofxbarwillbeapproximatelyNormalifnissufficientlylarge(say,atleast30).
Forproportions:WecanusetheNormalapproximationtothesamplingdistributionofp-hataslongasnp≥10
andn(1−p)≥10.
3.Independent:Theproceduresforcalculatingconfidenceintervalsassumethatindividualobservationsare
independent.Randomsamplingorrandomassignmentcanhelpensureindependent
observations.However,ourformulasforthestandarddeviationofthesamplingdistributionactasifweare
samplingwithreplacementfromapopulation.Thatisrarelythecase.Ifwe’resamplingwithoutreplacement
fromafinitepopulation,weshouldcheckthe10%condition:
Samplingmorethan10%ofapopulationcangiveusamorepreciseestimateofapopulationparameterbut
wouldrequireustouseafinitepopulationcorrection.We’llavoidsuchsituationsinthetext.
Here are two other important reminders for constructing and interpreting confidence intervals.
•
Our method of calculation assumes that the data come from an SRS of size n from the population of interest. Other types
of random samples(stratified or cluster, say) might be preferable to an SRS in a given setting, but they require more
complex calculations than the ones we’ll use.
•
The margin of error in a confidence interval covers only chance variation due to random sampling or random
assignment. The margin of error is obtained from the sampling distribution (or randomization distribution in an
experiment). It indicates how close our estimate is likely to be to the unknown parameter if we repeat the random
sampling or random assignment process many times. Practical difficulties, such as undercoverage and nonresponse in a
sample survey, can lead to additional errors that may be larger than this chance variation. Remember this unpleasant
fact when reading the results of an opinion poll or other sample survey. The way in which a survey or experiment is
conducted influences the trustworthiness of its results in ways that are not included in the announced margin of error.