Section 9.1 Notes

1

Signi%icanceTests(HypothesisTests)

Con$idenceintervalsESTIMATEanunknownparameter(μorpinourcase)bygivingusapossiblerangeofvaluesforthatparameterwithsomelevelofcon$idence.

Wecouldusethisestimatetomakestatements(withsomelevelofcon$idence)aboutthetrueparametervalue.

ForExample:Oneofyourclassmatesisrunningforstudentgovernment.Heclaimsthathehas63%oftheclasses'vote.Youcollectarandomsampleof50studentsand$indthat28ofthemwillvoteforhim.Doesyourdatasupporthisclaim?Usea95%con$idenceintervaltohelpargueyourcase.

Arethereotherwaysthatwecouldanswerquestionslikethese?

Free-ThrowActivity:Abasketballplayerclaimstomake80%ofthefreethrowsthatheattempts.Wethinkhemightbeexaggerating.Totestthisclaim,we'llaskhimtoshootsomefreethrows.

Havetheplayershoot25freethrows-recordhowmanyhemade.

Dowehaveenoughdatatodecidewhethertheplayer'sclaimisvalid?

Howmanyshotsdoweneedtomakeadecision?

Section 9.1 Notes

2

Whataresomeobservationswehaveaboutthisactivity?

Whenisiteasytodecideifaclaimiswrong?Whenisithard?

Howfarofffromtheclaimdoesthesampledataneedtobetoconvinceusitiswrong?

ThePrincetonReviewclaimsthattheirpracticecoursewillimproveSATscoresfor90%oftheirparticipants.Youaredoubtful.InordertotestPrincetonReview'sclaimyourwholeseniorclass(say,475students)decidestotakethecourseandtestwhetherPrincetonReview’sclaimiscorrect.

YourentireclasstakesthecourseandthentakestheSATasecondtime.EachstudentfromyourseniorclassanonymouslyreportswhetherornottheirscoreimprovedonthesecondSATtest.

Hereisthesampledata:475students410studentshadimprovedscores

Youwanttoknowwhattheprobabilityisofgettingthiskindofsampledata(orworse)ifinfactPrincetonReviewiscorrectand90%(andnofewer)ofparticipantshaveimprovedscores.

Whatistheprobabilitythatatmost410of475randomlyselectedstudentsimprovedtheirscorewhenthetrueproportionofstudentswhoimproveaftertakingthePrincetonReviewcourseis90%?

Section 9.1 Notes

3

Sobasedonourresultseither:

(1)PrincetonReview'sclaimiscorrect(p=0.9)and,byverybadluck,averyunlikelyoutcomehasoccurred(ourclassresults).

OR

(2)PrincetonReview'sclaimiswrongandtheproportionofimprovementisactuallylessthan0.9,sooursampleresultisnotanunlikelyoutcome.

AnoutcomethatwouldrarelyhappenifaclaimweretrueisgoodevidencethattheclaimisNOTtrue.

BasicsofaSigni%icanceTest

1.Makeastatementaboutaparameter-i.e.thatμorpareequaltosomevalue(thetestisdesignedto$indevidenceagainstthisstatement-thisisthenullhypothesisHo)

2.Determineanalternatehypothesis-i.e.thatμorparedifferentthanthevalueclaimedintheHo-couldbelessthan,greaterthan,orjustnotequalto

3.Collectsampledata

4.Determinetheprobabilityofgettingthissampledata,ormoreextreme,givenyourHoistrue

5.ConcludewhetherornotHohasbeenshowntobefalseandcanberejectedinfavorofthealternative(Ha).

Section 9.1 Notes

4

HypothesesNullHypothesis(Ho)

-theclaimthatisbeingtestedinanysigni$icancetest-testistryingto$indevidenceagainstHo-ALWAYSastatementabouttheparameter-ALWAYSstatingthatthereis"nodifference",thattheparameterisEQUALtosomething

AlternativeHypothesis(Ha)-theclaimaboutthepopulationthatwearetryingto$indevidencefor-ALWAYSstatingthatthetrueparameterislessthan,greaterthan,ornotequaltothevalueputforthinHo-canbeone-sidedortwo-sided

ALWAYSestablishyourhypothesesBEFOREyouhaveseenthedata-otherwiseitischeating!

CHECKYOURUNDERSTANDING:

Foreachofthefollowingsettings,(a)describetheparameterofinterest,and(b)stateappropriatehypothesesforasigni$icancetest.

1.AccordingtotheWebsitesleepdeprivation.com,85%ofteensaregettinglessthaneighthoursofsleepanight.Janniewondersifthisresultholdsinherlargehighschool.SheasksanSRSof100studentsattheschoolhowmuchsleeptheygetonatypicalnight.Inall,75oftheresponderssaidlessthan8hours.

2.Aspartofits2010censusmarketingcampaign,theU.S.CensusBureauadvertised"10questions,10minutes-that'sallittakes".Onthecensusformitself,weread,"TheU.S.CensusBureauestimatesthat,fortheaveragehousehold,thisformwilltakeabout10minutestocomplete,includingtimeforreviewingtheinstructionsandanswers."Wesuspectthattheactualtimeittakestocompletetheformmaybelongerthanadvertised.

Section 9.1 Notes

5

P-Value

Thesmallerthep-value,thegreatertheevidenceAGAINSTHo,providedbyourdata.

Largep-valuesdoNOTproveHotrue,theyjustfailtogiveusconvincingevidenceagainstHo.Failingto$indevidenceagainstHomeansonlythatthedataareconsistentwithHo,notthatwehaveclearevidencethatHoistrue.

The probability that the statistic (we calculated from our sample data)

would take a value as extreme or more extreme than the one actually

observed IF the null hypothesis (Ho) is true.

StatisticalSigni%icance

The$inalstepofasigni$icancetestistostateconclusions.

DeterminewhetherornotyourejectHoorfailtorejectHo.Note,onceagain,wedoNOTacceptHoastrue,weonlyfailtorejectit.

Howdowedetermine"toounlikely"?

WerejectHoifoursampleresultistoounlikelytohappenbychance.

Section 9.1 Notes

6

Somemorepractice...

Explainwhat'swrongwiththestatedhypotheses,thengivecorrecthypotheses.

1.Achangeismadethatshouldimprovestudentsatisfactionwiththeparkingsituationatyourschool.Rightnow,37%ofstudentsapproveoftheparkingthat'sprovided.Thenullhypothesisistestedagainstthealternative .

2.Inplanningastudyofthebirthweightsofbabieswhosemothersdidnotseeadoctorbeforedelivery,aresearcherstatesthehypothesesas

Section 9.1 Notes

7

3.AGallopPollreportonanationalsurveyof1028teenagersrevealedthat72%ofteenssaidtheyseldomorneverarguewiththeirfriends.Yvonnewonderswhetherthisnationalresultwouldbetrueinherlargehighschool.Soshesurveysarandomsampleof150studentsatherschoolandfoundthat96studentsinthesamplesaidtheyrarelyorneverarguewithfriends.Asigni$icancetestyieldsap-valueof0.0291.

(a)StatehypothesesforYvonne'ssigni$icancetest.Besuretode$ineanyparameters.

(b)Interpretthisresultincontext.

(c)Dothedataprovideconvincingevidenceagainstthenullhypothesis?Explain.

4.Askedtoexplainthemeaningof"statisticallysigni$icantattheα=0.05level,"astudentsays,"Thismeansthattheprobabilitythatthenullhypothesisistrueislessthan0.05."Isthisexplanationcorrect?Whyorwhynot?

The p-value is the probability that the statistic (we calculated from our

sample data) would take a value as extreme or more

extreme than the one actually observed IF the null hypothesis (Ho) is

true.

Section 9.1 Notes

8

ErrorWithHypothesisTeststhereisalwaysthepossibilitythatwewillmakeamistake.Therearetwotypeswecouldmake:

1.Werejectthenullhypothesiswheninfactit'strue.2.Wefailtorejectthenullhypothesiswheninfactit'sfalse

Section 9.1 Notes

9

SupposeIbelievethatthemeanspeedonWashingtonBlvdnearW-Lisover40mph.

Iestablishthefollowinghypotheses:

Ho:

Ha:

KeepinmindtheACTUALmeanspeedisunknown-butitiseitherexactly40mphor(basedonmyhypotheses)itishigher.

TypeIError

H0:

Section 9.1 Notes

10

AconsumeradvocacygroupclaimsthatthemeanmileagefortheCarterMotorCompany'snewsedanislessthan32milespergallon.IdentifythetypeIerrorforthetest.

a.Rejecttheclaimthatthemeanisequalto32milespergallonwhenitisactually32milespergallon.

b.Rejecttheclaimthatthemeanisequalto32milespergallonwhenitisactuallylessthan32milespergallon.

c.Failtorejecttheclaimthatthemeanisequalto32milespergallonwhenitisactuallylessthan32milespergallon.

d.Failtorejecttheclaimthatthemeanisequalto32milespergallonwhenitisactuallygreaterthan32milespergallon.

TypeIIError

H0:

Ha:

Section 9.1 Notes

11

Highwaysafetyengineerstestnewroadsigns,hopingthatincreasedre$lectivitywillmakethemmorevisibletodrivers.Volunteersdrivethroughatestcoursewithseveralofthenewandoldstylesignsandratewhichkindshowsupthebest.

WhatwouldatypeIerrorbeinthiscase?

WhatwouldatypeIIerrorbeinthiscase?

Thefeasibilityofconstructingapro$itableelectricity-producingwindmilldependsontheaveragevelocityofthewind.Foracertaintypeofwindmill,theaveragewindspeedwouldhavetoexceed20mphinorderforitsconstructiontobefeasible.Totestwhetherornotaparticularsiteisappropriateforthiswindmill,50readingsofthewindvelocityaretaken,andtheaverageiscalculated.Thetestisdesignedtoanswerthequestion,isthesitefeasible?Thatis,istheresuf$icientevidencetoconcludethattheaveragewindvelocityexceeds20mph?Wewanttotestthefollowinghypotheses.

H0:μ=20Ha:μ>20

WhatwouldatypeIIerrorinthiscasemean?

Section 9.1 Notes

12

Howcanwereduceerror?TypeI:Reducethesigni$icancelevel(α)

TypeII:IncreasePower(probabilityofthetesttoCORRECTLYrejectHo)IncreasesamplesizeIncreaseeffectsizeIncreasethesigni$icancelevel(α)

SEEHANDOUTONERRORANDPOWER!

H0:

Ha:

PowerThepowerofatestistheprobabilitythatitcorrectlyrejectsafalsenullhypothesis.

Inordertocalculatethepowerofatestweimaginethenullhypothesisisfalse.Thevalueofthepowerdependsonhowfarthetrueparameterliesfromthevalueofthenullhypothesis.Thisiscalledtheeffectsize.

Section 9.1 Notes

13

IncreasingPower

H0:

Ha:

DECREASE

TYPE II

ERROR!!!

Section9.1Homework:p.546#s1-27odd,28-30all

Section 9.1 Notes

14