Download Confidence Intervals Intro - Day 1

Confidence Intervals Intro - Day 1 - Complete
SupposewehaveaNormalpopulationwithanunknownmean,μ,and
standarddeviationof20.
Ingroups,Iwouldlikeyoutoestimatethemean,μ,ofthispopulation.
Iwillgenerateasampleofsize16fromthispopulationandsincesample
mean,,isanunbiasedestimatorofpopulationmean,Iwillgiveyouthe
meanfromthissample,.
YourjobistocomeupwithanintervalofREASONABLEvaluesthatwill
containthetruevalueμ.
Confidence Intervals Intro - Day 1 - Complete
De#inition:ApointestimatorisaSTATISTICthat
providesanestimateofapopulationparameter.The
valueofthatstatisticfromasampleiscalleda point
estimate.Ideally,apointestimateisour"bestguess"at
thevalueofanunknownparameter.
Thismeansourestimatorshouldbean
_______________________andhaveasamplingdistribution
with____________________________.
Sowehaveapointestimator,,ofourunknownparameter-μ-
andwewanttoknowhowclosethisestimateistotheactualμ.Itis
VERYunlikelythatwegottheactualvalueofμwithourpoint
estimator,.
Inordertoanswerthisweneedtoknowhowthesamplemean,,
wouldvaryifwetookmanySRSsofthesamesize(16inthiscase)
fromthissamepopulation.
SAMPLINGDISTRIBUTIONSanswerthisquestion.
Confidence Intervals Intro - Day 1 - Complete
SamplingDistributions:
Shape:BecausewehaveaNormaldistributiontostartwithour
samplingdistributionofisalsoNormal.
Center:Themeanofthesamplingdistributionofisthesameas
μ(theparameterwearetryingtoestimate )
Spread:Thestandarddeviationofthesamplingdistributionof
foranSRSof n=16is
=5
Sampling Distribution of
Population Distribution
=5
σ = 20
μ
Values of X
μ
Values of
1.Toestimateμweusethesamplemean,,oftherandomsample.We
don'texpecttobeexactlyequaltoμ,sowewanttosayhowaccurate
theestimateis.
2.Inrepeatedsamples,thevaluesoffollowaNormaldistributionwith
meanμandstandarddeviation,asinthe_igureabove.
3.The95partofthe68-95-99.7ruleforNormaldistributionssaysthat
iswithin2standarddeviationsofthepopulationmeanμinabout95%of
allsamplesofsizen.
Confidence Intervals Intro - Day 1 - Complete
95% of all sample
µ-2
µ
s
µ+2
4.Wheneveriswithin2ofμ,μiswithin2of.Thishappensinabout
95%ofallpossiblesamples.Sotheintervalfromμ-2toμ+2"captures"
thepopulationmeanμinabout95%ofallsamplesofsize n.
5.Ifweestimatethatμliessomewherebetweenμ-2andμ+2,we'dbe
calculatingthisintervalusingamethodthatcapturesthetrueμinabout95%ofall
possiblesamplesofthissize.
Allcon_idenceintervalswillhaveaformsimilartothis:
estimate±marginoferror
Keepinmindthis"error"hastodowithchancevariation
duetorandomsamplingorrandomassignment-i.e.
samplingvariability-notwhetherornotwemadean
errorincalculatingorANYothertypeoferror.
Confidence Intervals Intro - Day 1 - Complete
De#initions:
Acon$idenceintervalforaparameterhastwoparts:
1)The marginoferrortellshowclosetheestimate
tendstobetotheunknownparameterinrepeated
randomsampling.
2)Acon$idencelevel,C ,whichgivestheoverallsuccess
rateoftheMETHODforcalculatingthecon_idence
interval.ThatisinC%ofALLPOSSIBLESAMPLES,the
methodwouldyieldanintervalthatcapturesthetrue
parametervalue.
InterpretingCon#idenceLevelsandCon#idenceIntervals
1.Gotowww.whfreeman.com/tps4eandlaunchtheapplet( Con$idenceInterval
applet).Thedefaultsettingforthecon_idencelevelis95%.Changethisto90%.
Wewillbeusingthisapplettoinvestigatetheideaofcon_idencelevel.
2.Click"Sample"tochooseanSRSanddisplaytheresultingcon_idenceinterval.
Didtheintervalcapturethepopulationmeanμ(whattheappletcallsa"hit")?
Dothisatotalof10times.Howmanyoftheintervalscapturedthepopulation
meanμ?
3.Resettheapplet.Click"Sample50"tochoose50SRSsanddisplaythe
con_idenceintervalsbasedonthosesamples.Howmanycapturedtheparameter
μ?
Keepclicking"Sample50"andobservethevalueof"Percenthit".Whatdoyou
notice?
Confidence Intervals Intro - Day 1 - Complete
Astheappletcon_irmed,thecon(idencelevelisthe
overall"capturerate"ifthemethodisusedanytimes.
Fora95%con_idencelevelimpliesthat inrepeated
sampling,95%oftheintervalsconstructedinthis
waywillcontaintheunknownpopulationmeanμ .
Con(idenceLevel:Tosaythatweare 95%con(identis
shorthandfor"95%ofallpossiblesamplesofagiven
sizefromthispopulationwillresultinanintervalthat
capturestheunknownparameter."
Con$idenceInterval:Tointerpreta C%con_idence
intervalforanunknownparameter,wesay,"Weare C%
con_identthattheintervalfrom_________to_______
containstheactualvalueofthe[populationparameter
INCONTEXT]."
Confidence Intervals Intro - Day 1 - Complete
What'stheprobabilitythatour95%con6idence
intervalcapturestheparameter?
NOT95%!!!
Thislevelonlytalksaboutthecon_idenceintheMETHOD,notONEsample!
BEFOREwesample,wehavea95%chanceofgettingasamplethatwill
producea95%con_idenceintervalthatcapturesthetrueparameter.
However,oncewesampleourresultingcon_idenceintervaleitherwill
containtheparameterorwon't-i.e.theprobabilitythatitcontainsthe
parameteriseither1(itdid)or0(itdidn't).
Thecon6idencelevelDOESNOTtellusthechancethatAPARTICULAR
con6idenceintervalcontainsthepopulationparameter.
Somethingtokeepinmind:
Acon_idenceintervalisastatementaboutaparameter-
notasamplestatistic.
Don'tsay"Weare95%con_identthattheintervalfrom
_______to_____containsthesampleproportion/meanof
all_______"
Weknowitcontainsthesamplestatisticsothismakes
NOsense!
Confidence Intervals Intro - Day 1 - Complete
Checkyourunderstanding:
HowmuchdoesthefatcontentofBrandXhotdogsvary?To_indout,
researchersmeasuredthefatcontent(ingrams)ofarandomsample
of10BrandXhotdogs.A95%con_idenceintervalforthepopulation
standarddeviationσis2.84to7.55.
1.Interpretthiscon_idenceinterval.
2.Interpretthecon_idencelevel.
3.Trueorfalse:Theintervalfrom2.84to7.55hasa95%chanceof
containingtheactualpopulationstandarddeviationσ.Justifyyour
answer.
Choosingacon#idencelevel
ReturntotheCon$idenceIntervalapplet.
Wearegoingtoexploretherelationshipbetweenthecon_idencelevelandthe
con_idenceinterval.
1.Setthecon_idencelevelat95%andclick"Sample50".Observethepercenthitand
thelengthofthecon_idenceinterval.
2.Changethecon_idencelevelto99%.Whathappenstothelengthofthecon_idence
interval?Tothepercenthit?
3.Nowchangethecon_idencelevelto90%.Whathappenstothelengthofthe
con_idenceinterval?Tothepercenthit?
4.Finally,changethecon_idencelevelto80%.Whathappenstothelengthofthe
con_idenceinterval?Tothepercenthit?
Confidence Intervals Intro - Day 1 - Complete
Asweincreasecon_idencelevel,weincreasethewidth
ofourinterval...
Whatelsemighteffectthewidthofourcon_idenceinterval?
Let'slookattheformofthecon_idenceintervalmoreclosely:
estimate±(criticalvalue)(standarddeviationofthestatistic)
Clearlythesmallerthemarginoferror,thenarrowerthe
con_idenceinterval-themorepreciseourestimateofthe
populationparameter.
Themarginoferrordependsontwothings:
· Thecriticalvalue -thisdependsonthesampling
distribution(moreonthis)andonthecon_idencelevel-
thegreaterthecon_idencethegreaterthecriticalvalue
·
· Thestandarddeviationofthestatistic -thisdepends
onthesamplesizen-thebiggerthesample,thesmaller
thestandarddeviationofthestatistic,andthenarrower
theinterval.
Confidence Intervals Intro - Day 1 - Complete
CONDITIONS:
1.Random:Thedatashouldcomefromawell-designedrandomsampleora
randomizedexperiment.
2.Normal:Themethodweusetoconstructcon_idenceintervalsforμdependson
thefactthatthesamplingdistributionofthestatisticisatleastapproximately
Normal
3.Independent:Theproceduresforcalculatingcon_idenceintervalsassumethat
individualobservationsareindependent.Randomsamplinghelpsensurethisso
longasourpopulationisin_inite.IfNOTandwearesamplingwithoutreplacement
weshouldcheckthe10%condition.