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Chapter6:ConfidenceIntervalsandHypothesisTesting

Whenanalyzingdata,wecantjustacceptthesamplemeanorsampleproportionastheofficial

meanorproportion.Whenweestimatethestatistics ,x p (samplemeanandsampleproportion),wegetdifferentanswersduetovariability.Sowehavetoperformstatisticalinference:ConfidenceInterval:whenyouwanttoestimateapopulationparameterSignificanceTesting:whenwewanttoassesstheevidenceprovidedbythedatainfavorofsomeclaimaboutthepopulation.

Section6.1:ConfidenceIntervalsallowustoestimatearangeofvaluesforthepopulationmeanorproportion.Thetruemeanorproportionforthepopulationexistsandisafixednumber,butwedontknowit!Usingsamplestatisticswegetanestimateofwhereweexpectthepopulationparametertobe.

Ifwetakeasinglesample,oursingleconfidenceintervalnetmayormaynotincludethepopulationparameter.

Howeverifwetakemanysamplesofthesamesizeandcreateaconfidenceintervalfromeachsamplestatistic,overthelongrun95%ofourconfidenceintervalswillcontainthetruepopulationparameter(ifweareusinga95%confidencelevel).

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Ifyouincreaseyoursamplesize(n),youdecreaseyourmarginoferror

Ifyouincreaseyourconfidencelevel(C),thenyouincreaseyourmarginoferror

Asmallermarginoferrorisgoodbecausewegetasmallerrangeofwheretoexpectthetruepopulationparameter.

Confidenceintervalformulaslooklikeestimatemarginoferror.

Wewritetheintervalsas(lowerbound,upperbound).

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ConfidenceIntervalforaPopulationMean,:

* xx zn

wherez*isthevalueonthestandardnormalcurvewithareCbetweenz*andz*.

z* 1.645 1.960 2.576

C 90% 95% 99%

(TableDinthebackofthebookcontainsmorevalues,butthesearethemostcommon)

SampleSize,n,forDesiredMarginofError,m:

2* xznm

Notethatitisthesamplesize,n,thatinfluencesthemarginoferror.Thepopulationsizehasnothingtodowithit.

Waystoreduceyourmarginoferror:1.) Increasesamplesize2.) Usealowerlevelofconfidence(smallerC)3.) Reduce x

Becareful!!!!Youcanonlyusetheformula* xx z

n

undercertaincircumstances: DatamustbeanSRSfromthepopulation.

DonotuseifthesamplingisanythingmorecomplicatedthananSRS.

Datamustbecollectedcorrectly(nobias).Themarginoferrorcoversonlyrandom

samplingerrors.Undercoverageandnonresponsearenotcovered.

Outlierscanhaveabigeffectontheconfidenceinterval.(ThismakessensebecauseweusethemeanandstandarddeviationtogetaCI.)

Youmustknowthestandarddeviationofthepopulation, x .

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EXAMPLE1:Aquestionnaireofspendinghabitswasgiventoarandomsampleofcollegestudents.Eachstudentwasaskedtorecordandreporttheamountofmoneytheyspentontextbooksinasemester.Thesampleof130studentsresultedinanaverageof$422withstandarddeviationof$57.

a) Givea90%confidenceintervalforthemeanamountofmoneyspentbycollegestudentsontextbooks.

b) Isittruethat90%ofthestudentsspenttheamountofmoneyfoundintheintervalfrompart(a)?Explainyouranswer.

c) Whatisthemarginoferrorforthe90%confidenceinterval?

d) Howmanystudentsshouldyousampleifyouwantamarginoferrorof$5fora90%confidenceinterval?

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EXAMPLE2:Asampleof12STAT301studentsyieldsthefollowingExam1scores:

78 62 99 85 94 5388 90 86 92 75 92

Assumethatthepopulationstandarddeviationis10.ThesamplemeancanbecalculatedusingSPSSorcalculatortobe82.83.(Note:DoNOTuseanySPSSconfidenceintervalstheyaregoodonlyforChapter7,notthistypeofCI.YoumustgettheseZconfidenceintervalsbyhand.)

a) Findthe90%confidenceintervalforthemeanscoreforSTAT301students.

b) Findthe95%confidenceinterval.

c) Findthe99%confidenceinterval.

d) Howdothemarginsoferrorin(b),(c),and(d)changeastheconfidencelevelincreases?Why?

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Section6.2:HypothesisTesting

The4stepscommontoalltestsofsignificance:1. StatethenullhypothesisH0andthealternativehypothesisHa.2. Calculatethevalueoftheteststatistic.3. DrawapictureofwhatHalookslike,andfindthePvalue.4. Stateyourconclusionaboutthedatainasentence,usingthePvalueand/orcomparing

thePvaluetoasignificancelevelforyourevidence.STEP1:StatethenullhypothesisH0andthealternativehypothesisHa.Todoasignificancetest,youneed2hypotheses:

H0,NullHypothesis:thestatementbeingtested,usuallyphrasedasnoeffectornodifference.

Ha,AlternativeHypothesis:thestatementwehopeorsuspectistrueinsteadofH0.Hypothesesalwaysrefertosomepopulationormodel.Nottoaparticularoutcome.Hypothesescanbeonesidedortwosided.

Onesidedhypothesis:coversjustpartoftherangeforyourparameter

H0:=10 OR H0:=10Ha:>10 Ha:

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Example(Exercise6.37,p.418):Eachofthefollowingsituationsrequiresasignificancetestaboutapopulationmean.StatetheappropriatenullhypothesisH0andalternativehypothesisHaineachcase:

a. CensusBureaudatashowsthatthemeanhouseholdincomeintheareaservedbyashoppingmallis$72,500peryear.Amarketresearchfirmquestionsshoppersatthemalltofindoutwhetherthemeanhouseholdincomeofmallshoppersishigherthanthatofthegeneralpopulation.

b. Lastyear,yourcompanysservicetechnicianstookanaverageof1.8hourstorespondtotroublecallsfrombusinesscustomerswhohadpurchasedservicecontracts.Dothisyearsdatashowadifferentaverageresponsetime?

STEP2:Calculatethevalueoftheteststatistic.AteststatisticmeasurescompatibilitybetweentheH0andthedata.Theformulafortheteststatisticwillvarybetweendifferenttypesofproblems.InproblemslikethosewestudiedinChapter6,theteststatisticwillbetheZscore.STEP3:DrawapictureofwhatHalookslike,andfindthePvalue.Pvalue:theprobability,computedassumingthatH0istrue,thattheteststatisticwouldtakeavalueasextremeormoreextremethanthatactuallyobservedduetorandomfluctuation.Itisameasureofhowunusualyoursampleresultsare.

ThesmallerthePvalue,thestrongertheevidenceagainstH0providedbythedata.

CalculatethePvaluebyusingthesamplingdistributionoftheteststatistic(onlythenormaldistributionforChapter6).

STEP4:CompareyourPvaluetoasignificancelevel.Stateyourconclusionaboutthedatainasentence.

ComparePvaluetoasignificancelevel,.

IfthePvalue,wecanrejectH0.

IfyoucanrejectH0,yourresultsaresignificant.

IfyoudonotrejectH0,yourresultsarenotsignificant.

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ZTestforaPopulationMeanTotestthehypothesisH0:=0basedonanSRSofsizenfromapopulationwithunknownmeanandknownstandarddeviation,

computetheteststatistic:

00 /

xZn

thePvaluesforatestofH0against:

Ha:>0isP(ZZ0)

Ha:

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EXAMPLES

1. LastyearthegovernmentmadeaclaimthattheaverageincomeoftheAmericanpeoplewas$33,950.However,asampleof50peopletakenrecentlyshowedanaverageincomeof$34,076withapopulationstandarddeviationof$324.Isthegovernmentsestimatetoolow?Conductasignificancetesttoseeifthetruemeanismorethanthereportedaverage.Usean=0.01.

2. Anenvironmentalistcollectsaliterofwaterfrom45differentlocationsalongthebanksofastream.Hemeasurestheamountofdissolvedoxygenineachspecimen.Themeanoxygenlevelis4.62mg,withtheoverallstandarddeviationof0.92.Awaterpurifyingcompanyclaimsthatthemeanlevelofoxygeninthewateris5mg.Conductahypothesistestwith=0.001todeterminewhetherthemeanoxygenlevelislessthan5mg.

3. Anagroeconomistexaminesthecellulosecontentofavarietyofalfalfahay.Supposethatthecellulosecontentinthepopulationhasastandarddeviationof8mg.Asampleof15cuttingshasameancellulosecontentof145mg.Apreviousstudyclaimedthatthemeancellulosecontentwas140mg.Performahypothesistesttodetermineifthemeancellulosecontentisdifferentfrom140mgif=0.05.

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Howdoesrelatetoconfidenceintervals?

Ifyouhavea2sidedtest,andiftheandconfidenceleveladdto100%,youcanrejectH0if0(thenumberyouwerechecking)isnotintheconfidenceinterval.

a) Finda95%confidenceintervalforthemeancellulosecontentfromtheaboveexample.

b) Nowtrythetestfrompartnumber3againusingtheconfidenceintervalfrompartbtodothehypothesistest.(Theresultshouldbethesame.)

AnnualDrinkingWaterQualityReport,2004,TownofBrookston,IN

Impleasedtoreportthatourdrinkingwaterissafeandmeetsfederalandstaterequirements.

TestResults(MCListhemaximumcontaminantlevel,thehighestlevelofacontaminantthatisallowedindrinkingwater.)

Contaminant ViolationY/N

LevelDetected

Unitmeasurement

MCL

Beta/photonemitters N 2.1 3.2 mrem/yr 4

Alphaemitters N 0 1.6 pCi/l 15

Barium N 0.216 ppm 2

Copper N 0.039to0.453 ppm 1.3

Fluoride N 0.01 ppm 4

Sodium N 0.0 ppm N/A

Oneoftheseviolationreportsshouldactuallybeayesinsteadofano.Whichoneisitandwhy?Whathypothesesgoalongwiththeseconfidenceintervals?

Note:WhenIcalledthetownofBrookstonofficetoaskthemaboutthis,thewatermanagercalledthestateEPAofficetogetmoreinformation.Whattheytoldhimwasthat,yes,technicallyIwascorrect,butthattheydontusetheconfidenceintervalsthatarereported.ApparentlythesearetheFEDERALEPArules.Theyonlyusethemean.Itriedtogetsamplesizeorotherinformation,butIwasntabletolearnanythingmore.

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Pvaluescanbemoreinformativethanareject/donotrejectH0basedon.AsPvaluegetssmallertheevidenceforrejectingH0getsstronger.

Justbecauseweuse=0.05alotdoesntmeanthatsthelevelyouhavetouseitsjustthemostcommon.Theresnothingparticularlyspecialaboutthatlevel.

Inalargesample,eventinydeviationsfromthenullhypothesiscanbeimportant.

IfwefailtorejectH0,itmaybebecauseH0istrueorbecauseoursamplesizeisinsufficienttodetectthealternative.

PlotyourdataandlookatyourPvaluetodetermineyourconclusions.Couldoutliersbepartoftheproblem?

Aconfidenceintervalactuallyestimatesthesizeofaneffectratherthansimplyaskingifitistoolargetoreasonablyoccurbychancealone.

Youmusthaveawelldesignedexperimentinorderforstatisticalinferencetowork.Randomizationisimportant.