DecidingtoRejecttheNullHypothesis决定拒绝零假设.docx
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DecidingtoRejecttheNullHypothesis决定拒绝零假设.docx
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DecidingtoRejecttheNullHypothesis决定拒绝零假设
DecidingtoRejecttheNullHypothesis:
决定拒绝零假设
Chapter7–HypothesisTestingwithOneSample
7.1StatisticalHypotheses
Astatisticalhypothesisisamathematicalclaimaboutapopulationparameter.
Examples:
,65,Themeanheightofwomenislessthan65inchestall.,
p,.57,ThepercentageofFloridiansfavoringabullettrainis57%.
TheaveragedistancedrivenperyearbyAmericansismorethan10,000miles.
,10,000
p,.05,Atleast5%ofAmericansearnmorethan$100,000peryear.
HypothesisTesting-BasicProcedure
Ifwewantedtoknowwhetheranyoftheabovehypothesesaretrue,wewouldconductahypothesistest.Whenwetestastatisticalhypothesis,wefollowthefollowingbasicprocedure:
1.Drawarandomsamplefortherandomvariableinquestion.
2.Determineiftheresultsfromthesampledataareconsistentorinconsistent
withthehypothesis.
3.Ifthesampledataissignificantlydifferentfromthenull
hypothesis(H),wewouldrejectthenullhypothesis(aso
beingfalse),andsupportthealternativehypothesis(H).Ifa
thedataisnotsignificantlydifferent,wewouldnotrejectthenullhypothesis–
therewouldnotbeanyevidencetosupportthealternativehypothesis.
Wecanonly“support”aclaimifthenullisrejectedandwesupportthealternative.Thegoal,then,istorejecttheHoandsayHaiscorrect.TheHowillalwayscontaintheconditionofequality.
1
FormalHypothesisTests
Inaformalhypothesistest,theoppositeclaimsaregiventhenamesnullhypothesis
Handalternativehypothesis.Thenullhypothesisisdenotedbyand0
Hthealternativehypothesisisdenotedby.Thenullandalternativea
hypothesesneedtobeassignedasfollows(seealsop367):
Thenullhypothesisisthehypothesisbeingtested.Hmust:
0
bethehypothesiswewanttoreject
(,,),,,,containtheconditionofequality
Thealternativehypothesisisalwaystheoppositeofthenullhypothesis.
Hmust:
a
bethehypothesiswewanttosupport
(,,),,,,notcontaintheconditionofequality
HAformalhypothesistestwillalwaysconcludewithadecisiontoreject0
basedonsampledata,orthedecisionthatthereisnotstrongenoughevidencetorejectH.0
Example:
Abatterymanufacturerclaimsthattheaveragelifeofitsbatteriesisatleast300minutes.Totesttheclaim,asampleof100batteriesisdrawn.Thesampleistestedandthemeanbatterylifeofthesampleis294minuteswithasamplestandarddeviationof20minutes.
a)Writethenullandalternativehypotheses.
b)Ifthenullhypothesiswererejected,whatwouldyouconclude?
c)Ifthenullhypothesiswerenotrejected,whatwouldyouconclude?
2
Example:
Atiremanufacturerclaimsitstireslastonaveragemorethan35,000
H,miles.Ifwethinktheclaimisfalse,thenwewouldwritetheclaimas0
rememberingtoincludetheconditionofequality.Ourhypothesisforthistestwouldbe:
H:
o
H:
a
Wewouldthenhopethatoursampledatawouldallowtherejectionofthenullhypothesis,refutingthecompany’sclaim.
Example:
Ontheotherhand,ifweworkedforthetirecompanyandwantedto
Hgatherevidencetosupporttheirclaim,thenwewouldmakethecompany’sclaim,a
andrememberthatequalityshouldnotbeincludedintheclaim.Ourhypothesistestwouldthenusethehypotheses:
H:
o
H:
a
HIfthesampledatawereabletosupporttherejectionof,thiswouldbestrong0
Hevidencetosupporttheclaimwhichiswhatthecompanybelievestobetrue.a
Note:
Forsimplicity,inyourtextandonallquizzesandtests,ittheconditionofequalityisinferredbythewording,thatwillautomaticallybethenullhypothesis.
(Lookattheexampleaboveanddecidewhichwouldbethebest“asworded”.)
3
TypesofError
Wheneversampledataisusedtomakeanestimateofapopulationparameter,thereisalwaysaprobabilityoferrorduetodrawinganunusualsample.There
aretwomaintypesoferrorthatoccurinhypothesistests.
TypeIError–Asampleischosenwhosesampledataleadstothe
Hrejectionofthenullhypothesiswhen,infact,istrue.0
TypeIIError–Asampleischosenwhosesampledataleadstonot
Hrejectingthenullhypothesiswhen,infact,isfalse.0
Summarizing:
TrueFalseHH00
RejectedTypeIErrorCorrectDecisionH0
NotRejectedCorrectDecisionTypeIIErrorH0
Inlayman’sterms,TypeIerroristothinkyouareright(becauseyou
haverejectedthenullhypothesis)whenyouarewrong,andTypeIIerroristothinkyoumaybewrong(becauseyoudidnotrejectthenullhypothesis)whenyouareactuallyright.(Alsoseep368.)WewouldrathermakeaTypeIIthanaTypeIerror.
Exercise:
GobackanddeterminewhataTypeIErrorandaTypeIIErrorwouldbeforourtireexamples.
4
LevelofSignificance
Inhypothesistests,aconservativeapproachisusuallytakentowardtherejectionofthenullhypothesis.Thatis,wewanttheprobabilityofmakingaTypeIErrortobesmall.(We’dratheractuallyberightandthinkwe’rewrongthanthinkwe’rerightandactuallybewrong.)
Themaximumacceptableprobability(ofmakingaTypeIerror)isusuallychosenat
thebeginningofthehypothesistest,andiscalledthelevelofsignificanceforthe
test.Thelevelofsignificanceisdenotedby,andthemostcommonlyusedvalues
,.10,,.01,,.05are,,and.Eachoftheserepresentsthe________________
____________________________________________.
TheprobabilityofmakingaTypeIIErrorinahypothesistestisdenotedby.Once
isdetermined,thevalueofisalsofixed,butthecalculationofthisvalueis,
beyondthescopeofthiscourse.
5
TypesofTests
Therearethreebasictypesofhypothesistests(seep338text,oldbookp326):
Thelocationofthe“tail”isbasedonwhatweactuallywanttoprove(H).Thea
HYPOTHESISEDparameter(meanorproportion)willbe“inthemiddle.”
Left-tailedTest–usedwhenthenullhypothesisbeingtestedisaclaimthatthe
(),populationparameteratleastagivenvalue.Notethatthealternative
hypothesisthenclaimsthattheparameterislessthan(<)the
value.
Right-tailedTest–usedwhenthenullhypothesisbeingtestedisaclaimthatthe
(),populationparameterisatmostagivenvalue.Notethatthealternative
hypothesisthenclaimsthattheparameterisgreaterthan(>)the
value.
Two-tailedTest–usedwhenthenullhypothesisbeingtestedisaclaimthatthepopulationparameterisequalto(=)agivenvalue.Notethatthealternative
(),hypothesisthenclaimsthattheparameterisnotequaltothe
value.
6
Giventhethreeexamplesbelow,decidewhattypeoftestwouldbeused:
aLEFT-
TAILED,RIGHT-TAILEDorTWO-TAILEDtest.
Example:
TheCensusBureauclaimsthatthepercentageofTampaArearesidentswithabachelor’sdegreeorhigheris24.4%.Wewouldwritethenullandalternative
hypothesesforthisclaimas:
Hp:
.244,0
Hp:
.244,a
HInthiscase,wewouldhavetorejectifoursamplepercentagewaseither0
significantlymorethan24.4%,orsignificantlylessthan24.4%.Thatis,ifour
sampleproportionwasin________________________ofthedistributionofallsampleproportions.
Example:
Fromourfirsttiremanufacturerexample:
H:
35,000,,0
H:
35,000,,a
WewouldrejectinthecaseaboveifoursamplemeanweresignificantlylessthanH0
35,000.Thatis,ifoursamplemeanwasin_____________________ofthe
distributionofallsamplemeans.
Example:
Fromoursecondtiremanufacturerexample:
H:
35,000,,0
H:
35,000,,a
WewouldrejectinthiscaseifoursamplemeanweresignificantlymorethanH0
35,000.Thatis,ifoursamplemeanwasin_______________________ofthe
distributionofallsamplemeans.
7
7.2TestingaClaimabouttheMeanUsingaLargeSample
Whenahypothesistestinvolvesaclaimaboutapopulationmean,thenwewilldrawasampleandlookatthesamplemeantotesttheclaim.Ifthesampledrawnislarge
n,30enough(),thentheCentralLimitTheoremapplies,andthedistributionofsample
,,meansisapproximatelynormal.Asusual,wealsohavethat:
andx
s,,,.xnn
Note:
Sincesandnareknownfromthesampledata,wehaveagoodestimateof,,,butwedonotknowsincethisistheparameterwearetestingaclaimabout.Inx
ordertohaveavaluefor,wewillalwaysassumethatthenull
hypothesisistrueinanyhypothesistest.
Sincethenullhypothesismustbeofoneofthefollowingtypes:
,,,,,,,,,,or000
whereisaconstant,wewillalwaysassumeforthepurposeofourtestthat0
,,.0
Thatis,thehypothesizedparameter(meanorproportion)willbeinthe“middle”ofournormaldistributionsketch.
8
TheStandardizedTestStatistic
Therearethreemethodsofdeterminingifyoushouldrejectornotrejectthenullhypothesis:
Confidenceintervals
(isthehypothesizedstatisticwithintheconfidenceintervalforthesample)
P-values
(howprobableitisthatthesamplestatisticwillbeat/below/abovethe
hypothesizedparameter-thatis,theprobabilityofrejectingH)Thismeans0
thatastheP-valuegetssmaller,the________likelywearetomakeaType
Ierror(rejectthenullhypothesiswhenitisactuallytrue.)
CriticalValues
(howmanystandarddeviationsthesampleisawayfromthehypothesized
parameter)Themorestandarddeviationsfromthehypothesizedparameter,
the____________likelywearetorejectH0.
Note:
Thesameconclusionshouldbemaderegardlessofthemethodchosen.
Let’sgobacktoourpreviousexampletoseehowthismightwork...
9
Example:
Abatterymanufacturerclaimsthattheaveragelifeofitsbatteriesisat
,300least300minutes().Totesttheclaimasampleofn=100batteriesisd
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