APStatistics Name:________________________Ch.19&20GuidedNotes

Hershey’sKissesandConfidenceIntervals

Inthisactivity,wewillestimateaconfidenceintervalfortheproportionoftimesaHershey’skisslandsonitsbaseasopposedtoitsside.Todothis,wewilldropHershey’skisses,counthowmanylandontheirbase,andcalculatetheconfidenceinterval.

Totakeyoursample,gatherthreeHershey’skissesinyourcup,shakethemup,anddropthemfromaboutsixinchesaboveyourdesk.Countthenumberthatlandontheirbase.

Repeattentimestogetasampleofsize30,recordingyourresultsinthetablebelow.

Toss

NumberNumber

thatlandonbase

TossNumber

Numberthatlandonbase

1 6 2 7 3 8 4 9 5 10

Total

Whatisthepopulationofinterest?_______________________________Whatisthesample?______________________________________Yourresult(30tossescombined):𝑝=Theresultsoftheothersinyourgroup: ________ ________ ________Compareyour𝑝withtheothersinyourgroup.Didyouallgetthesameanswer?________Followthesestepstomakea95%confidenceintervalbasedonyourresult.•Calculatethestandarddeviationofyoursampleproportion(wedon’tknowp,soweuse 𝑝):

𝜎! =𝑝𝑞𝑛

•Lookupthecriticalvalue,z*,thatcorrespondsto95%confidence.Note,thisisthez-scorethatcorrespondstothemiddle95%ofthenormaldistribution.(Hint:whatpercentagewouldbeleftineachtailofthedistribution?Usethis&invNorm).Thepositivez-scoreisthecriticalvalue.

z*95%=___________

•Theconfidenceintervalis𝑝 ± 𝑧!"%∗ ×𝜎! wherethecriticalvaluemultipliedbythestandarderroristhemarginoferrorfortheinterval.Addingandsubtractingthemarginoferrorfromthesamplestatisticcreatesthelowerandupperboundsfortheconfidenceinterval.

yourlowerbound=___________ yourupperbound=____________Usethelinesbelowtoroughlydrawtheconfidenceintervalsofeachpersoninyourgroup.Indicatethelocationof𝑝asadot.(AnexampleofaCIfrom0.13to0.37with𝑝=.25isgiven.)

Aftercomparingwithyourgroup,plotyourdotstickeronyour𝑝-valueonthechartatthefrontoftheroom.Usingamarkerextendalinefromyourdottoyourlowerboundandupperbound(insimilarfashionasthediagramabove)

Interpretthe95%confidenceintervalinyourownwords.Writeyourinterpretationbelow.

Iam95%confidentthat…

Compareyourinterpretationwithyourgroupmembers’interpretationsandcometoanagreementonanappropriateinterpretation.Writeitbelow.

Ifyou’vecompletedthesetasksandquestions,eataHershey’sKisswhileyouponderthefollowing:Whichofthefollowingdoyouthinkistrue(morethanonemaybetrue)?

A. Thereisa95%probabilitythatthetrueproportionisequaltomysampleproportion.

B. Thereisa95%probabilitythatyourintervalwillincludethetrueproportion.

C. 95%ofconfidenceintervalscreatedinthiswaywillcapturethetruepopulationproportion.

D. 95%ofsampleproportionscollectedinthiswaywillfallwithinmyconfidenceinterval.

)(

Chapter19:ConfidenceIntervalsInChapter18welearnedaboutwhatthedistributionofsampleproportions(𝑝)wouldlooklike.TomakeaNormalmodelforthesamplingdistributionofproportion,weused

𝜇! = 𝜎! =

Chapter19isaboutreal-worldstuff:Intherealworld,wedon’tknowp,butwe’dliketofindarangeofreasonablevaluesforit.Ourformulasbecome:

𝜇! = 𝜎! =

Becauseintherealworld,𝑝isgenerallythebestestimatewehavefor𝑝.ThisalsomeansIneedtoaccountfor

thestandarderror(AKA___________________________________________)becausesamplesvary.

Followthesestepstomakea95%confidenceinterval:

Sayyoursamplehad𝑝 = .34 and𝑛 = 57.•Calculatethestandarddeviationofyoursampleproportion(akathestandarderror):

𝜎! =𝑝𝑞𝑛

•Findthecriticalvalue,z*,thatcorrespondsto95%confidence.Note,thisisthez-scorethatcorrespondstothemiddle95%ofthenormaldistribution.Thepositivez-scoreisthecriticalvalue.Thisnumberisalwaysthesamefor95%confidence!z*95%=___________

• Findthemarginoferror: 𝑧!"%∗ ×𝜎! =

•Theconfidenceintervalis𝑝 ± 𝑧!"%∗ ×𝜎! .Addingandsubtractingthemarginoferrorfromthesamplestatisticcreatesthelowerandupperboundsfortheconfidenceinterval.yourlowerbound=___________ yourupperbound=____________

LogicofConfidenceIntervals

__________ofsampleswillproduceasampleproportionwithin_______(butreallymorelike___________)standarddeviationsofthetruepopulationproportion.Inotherwords,95%of_________’swillbewithin~2𝜎!of_____________,thepopulationproportion.But,sincethedistancebetween𝑝 & 𝑝isthesameasthedistancebetween𝑝 & 𝑝,itfollowsthat95%ofthetime,ifIgoout2𝜎!frommyparticular________,Iwillcapture𝑝.

Createa95%confidenceintervalbyhandandoncalculator:

• Sayyoursamplehad𝑝 = .18 and𝑛 = 276.95%CI:(______________,_______________)

• Interpretyour95%confidenceintervalbywritingasentenceortwoaboutwhatitmeansaboutthepopulationproportion:95%ofintervalscalculatedwillcontainthe__________________proportion.Weare____%confidentthat

the_____________________proportionisbetween_________and_________.

Calculatorsteps:

Whatabouta90%confidenceinterval?(Careful!Youneedanewz*!)

• Sayyoursamplehad𝑝 = .34and𝑛 = 8990%CI:(______________,_______________)

• Interpretyour90%confidenceintervalbywritingasentenceortwoaboutwhatitmeansaboutthepopulationproportion:

____%ofintervalscalculatedwillcontainthe__________________proportion.Weare____%confidentthat

the_____________________proportionisbetween_________and_________.

What’swrongwiththefollowingstatements?Say𝒑 = .𝟐𝟗anda95%confidenceintervalforthepopulationproportionofvotesforcandidateAinanelectionwas(.23,.35).

• Thepopulationproportionis.29.• Wedon’tknowthepopulationproportion,butit’sdefinitelybetween.23and.35.• Thereisa95%probabilitythatthepopulationproportionis.29.

• Writeacorrectinterpretation:

ConditionsTheconditionsforconfidenceintervalsarethesameasforusingthesamplingdistributioningeneral,sinceaconfidenceintervalisbasedonthesamplingdistribution.Let’swritethem:

•

TheCIisinfluencedbychoiceofsamplesizeCIFormula:

DeterminingSizeofSampleIfIwantamarginoferrorof.03with95%confidence,howbigdoessampleneedtobe?Sayp=______,sincethiswillmakepqlargesttoallowforextremecases

Ingeneral,halvingtheMErequiresquadruplingthesamplesize(becausesamplesizeissquarerootedalongtheway)

Higheraccuracy?Lower_____________________________

Higherconfidence?__________________confidenceinterval

Chapter20:HypothesisTestingAhypothesis(forourpurposes)isaworkingmodelthatweadopttemporarily.Ourstartinghypothesisiscalledthe_________________________.Itiswhateverisgenerallyassumedtobe

true,basedonpreviousknowledgeaboutasituation.Thenullhypothesisforafaircoin?_____________

ThenullhypothesisforFDAtrialsofanewmedicine?_________________________

The___________________________containsplausiblevaluesfortheparameterifwerejectthenullhypothesis.

Itisthe__________________ofthenullhypothesis.

Writethenullandalternativehypothesisforeachexamplebelow:

Tolearnaboutmyhypotheses,performa________________________________________!

YouwouldliketoknowifthereisevidencethatWHSstudentsaresleepinglessthanthenationalaverage.Takeasampleof100studentsatWHS.Surveythemtoask“Doyousleepatleast6hoursonschoolnights?”40%respond“yes.”Ingeneral,48%ofstudentsintheUSsleepatleast6hoursonschoolnights.

1. Setuphypothesesandassumethat______istrue.(Determine_______and________,writethemclearly)

2. Gatherdatafromanexperiment,relevanttothehypothesis.(______=.4)(Usuallygiven.)

3. Modelallpossiblesamples(Samplingdistributionbasedonyournullhypothesis)

4. Determinehowunusualyoursampleis.(Findthe_____________:Theprobabilityofgettingtheobserved

results,ormoreextreme,GIVENthatH0istrue)(Wewillperforma________________________-more

testsinfuturechapters)

4.b.Alternative:Calculatorsteps

5.Makeadeterminationaboutthehypothesis,basedontheideaof"howlikelyareourresultsgiventhehypothesis?"(_____________________________________or_________________________________.)Note:Never“_____________”thenullhypothesis.Ifailedtodisprovesomething;thatdoesn’tmeanitistrue.

(Wewilluseothervalueslater,fornow___________________________isalways.05)

Statisticallysignificant≠

Statisticallysignificant=Veryunlikely,________________thatthe_________________________is______________.

P-valuelessthan.05? _____________________________

One-tailvsTwo-tailTestsDependingonmyhypotheses,my______________mightbetwiceaslarge.Youneedtoconsiderifyouareusing

aone-tailortwo-tailtestwhencalculating,butit’snottoodifficultasnormalcurvesare__________________:

Whatarethehypotheses?Wouldyouperformaone-ortwo-tailtest?

Howfaroutdoes_______needtobeinordertobestatisticallysignificant?(Inotherwords,howfarfrommynullhypothesisdoesitneedtobe?)

one-tailisbuddieswith__________

(_____wouldneedtobemorethan

________SD’sfromthehypothesized____)

two-tailisbuddieswith__________

(_____wouldneedtobemorethan

________SD’sfromthehypothesized____)

HypothesisTestingPractice

1)a)MississippiDepartmentofMotorVehicle(DMV)recordsindicatethatofallvehiclesundergoingemissionstestingduring2009,70%passedonthefirsttry.Latein2009thestateimplementedaprogramdesignedtoencouragepreventativeservicetoimproveemissionscompliance.Inlate2010,of126randomlyselectedreportsonemissions,95carspassedonthefirsttry.

Doesthisprogramwork?Writeandtestappropriatehypotheses,includinganappropriateconclusionincontext.b)Obtaina90%confidenceintervalforp.Itshouldconfirmyourdecision.

c)Whydidweusea90%confidenceintervalinsteadof95%inpart(b)?

2)51.5%ofalladultsintheU.S.arewomen.Astudyofstatelotteriesincludedarandomdigitdialing(RDD)survey.Ofthesurveyedadults,248wereclassifiedas“heavylotteryplayers.”Ofthese,96werewomen.Arewomenunderrepresentedamongheavylotteryplayers?Writeandtestappropriatehypotheses,includinganappropriateconclusionincontext.

3)AnyBostonCelticsfanorarchrivalknowsthatShaquilleO’Neal,oneoftheNBA’smostdominantcentersofthelasttwentyyears,alwayshaddifficultyshootingfreethrows.Overthecourseofhiscareer,hisoverallmadefree-throwpercentageis53.3%.Duringthisoffseason,Shaqisworkingwithanassistantcoachonhisfree-throwtechnique.Duringthefirst5gamesofthenextseason,Shagmade26ofhis39free-throwattempts.DotheseresultsprovideconvincingevidencethatShaqhassignificantlyimprovedhisfree-throwshooting?Writeandtestappropriatehypotheses,includinganappropriateconclusionincontext.

4)TwoPolishmathprofessorsandtheirstudentsspunaBelgianeurocoin250times.Itlandedonheads140times.Oneoftheprofessorsconcludedthatthecoinwasmintedasymmetrically.ArepresentativefromtheBelgianmintsaidthattheresultwasjustbychance.IsthemathprofessorortherepresentativefromtheBelgianmintcorrect?Writeandtestappropriatehypotheses,includinganappropriateconclusionincontext.

5)Adrugmanufacturerclaimsthatfewerthan10%ofpatientswhotakeitsnewdrugfortreatingAlzheimer’sdiseasewillexperiencenausea.Inarandomsampleof250patients,23experiencednausea.Isthereevidenceforthemanufacturer’sclaim?Writeandtestappropriatehypotheses,includinganappropriateconclusionincontext.

6)Challenge:Trywithmeans.

Abuscompanyadvertisedameantimeof150minutesforatripbetweentwocitieswithastandarddeviationof7.5minutes.Aconsumergrouphadreasontobelievethatthemeantimewasmorethan150minutes.Asampleof40tripsshowedameanx=̄153minutes.Testtheconsumergroup’sbelief.(Yourhypothesesneedtouse𝜇,notp!)(Assumethepopulationstandarddeviation𝜎is7.5minutes.)