Sections 8-1 and 8-2 - Gordon State College

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HYPOTHESIS TESTING
Sections 8-1 and 8-2
Review and Preview
and
Basics of Hypothesis Testing
Instatistics,ahypothesis isaclaimor
statementaboutapropertyofapopulation.
Ahypothesistest (ortestofsignificance)is
aprocedurefortestingaclaimabouta
propertyofapopulation.
Critical Value Method
RARE EVENT RULE
Thischapter,asthelastchapter,reliesonthe
RareEventRuleforInferentialStatistics.
If,underagivenassumption,the
probabilityofaparticularobservedevent
isexceptionallysmall,weconcludethatthe
assumptionisprobablynotcorrect.
OBJECTIVES FOR SECTION 8-2
Inthissection,wewilllearn:
• Howtoidentifythenullhypothesisandalternative
hypothesisfromagivenclaim,andhowtoexpressboth
insymbolicform.
• Howtocalculatethevalueoftheteststatistic,givena
claimandsampledata.
• Howtoidentifythecriticalvalue(s),givena
significancelevel.
• HowtoidentifytheP‐value,giventhevalueofthetest
statistic.
• Howtostatetheconclusionaboutaclaiminsimpleand
nontechnicalterms.
NULL HYPOTHESIS
Thenullhypothesis (denotedbyH0)isastatement
thatthevalueofapopulationparameter(suchas
proportion,mean,orstandarddeviation)isequal to
someclaimedvalue.(Thetermnull isusedtoindicate
no changeornoeffectornodifference.)Hereare
someexamples:
H0:
0.3
H0:
63.6
Wetestthenullhypothesisdirectlyinthesensethat
weassumeitistrueandreachaconclusiontoeither
rejectH0 orfailtorejectH0.
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ALTERNATIVE HYPOTHESIS
Thealternativehypothesis (denotedbyH1 orHa or
HA)isthestatementthattheparameterhasavalue
thatsomehowdiffersfromthenullhypothesis.Forthe
methodsofthischapter,thesymbolicformofthe
alternativehypothesismustuseoneofthesesymbols:
<or>or≠.Forexample:
Proportions: H1:p >0.3
Means:
H1:p <0.3
H1:p ≠0.3
H1:μ >63.6 H1:μ <63.6 H1:μ ≠63.6
NOTE ABOUT FORMING YOUR
OWN CLAIMS
Ifyouareconductingastudyandwanttousea
hypothesistesttosupport yourclaim,theclaim
mustbewordedsothatitbecomesthealternative
hypothesis.Youcanneversupportaclaimthat
someparameterisequal toaspecificvalue.
Forexample,ifyouwanttosupporttheclaimthat
yourIQimprovementcourseraisestheIQmean
above100,youmuststatetheclaimasμ > 100.So,
thenullhypothesisisH0:μ = 100andthe
alternativehypothesisisH1:μ > 100.
CRITICAL REGION
Thecriticalregion (orrejectionregion)istheset
ofallvaluesoftheteststatisticthatcauseusto
rejectthenullhypothesis.Forexample,seethered
shadedregioninthegraph.
NOTE ABOUT IDENTIFYING
H0 AND H1
Start
Identify the specific claim or hypothesis
to be tested and express it in symbolic form.
Give the symbolic form that must be true
when the original claim is false.
Of the two symbolic expressions obtained so far, let the
alternative hypothesis H1 be the one not containing
equality, so that H1 uses the symbol < or > or ≠. Let
the null hypothesis H0 be the symbolic expression that
the parameter equals a fixed value.
TEST STATISTIC
Theteststatistic isavaluecomputedfromthesample
data,anditisusedinmakingthedecisionaboutthe
rejectionofthenullhypothesis.
̂
Teststatisticfor
proportion
Teststatisticfor
mean
Teststatisticfor
standarddeviation
̅
or
̅
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SIGNIFICANCE LEVEL
Thesignificancelevel (denotedbyα)isthe
probabilitythattheteststatisticwillfallinthecritical
regionwhenthenullhypothesisisactuallytrue.The
isthesameα introducedinSection7‐2withthe
confidencelevel.
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CRITICAL VALUE
Thecriticalvalue isanyvaluethatseparatesthe
criticalregionfromthevaluesoftheteststatisticthat
donotleadtorejectionofthenullhypothesis.
TWO-TAILED TEST
Two‐tailedtest: Thecriticalregionisinthe
twoextremeregions(tails)underthecurve.
CorrespondstoH1:≠
Theareaα is
dividedequally
between
thetwotailsof
thecritical
region
LEFT-TAILED TEST
Left‐tailedtest: Thecriticalregionisinthe
extremeleftregion(tail)underthecurve.
CorrespondstoH1:<
Theareaα is
entirelyinthe
lefttail.
RIGHT-TAILED TEST
Right‐tailedtest: Thecriticalregionisinthe
extremerightregion(tail)underthecurve.
CorrespondstoH1:>
Theareaα is
entirelyinthe
righttail.
P-VALUE
TheP‐value (orp‐value orprobabilityvalue)
istheprobabilityofagettingavalueofthetest
statisticthatisatleastasextreme astheone
representingthesampledata,assumingthe
nullhypothesisistrue.Thenullhypothesisis
rejectediftheP‐valueisverysmall,suchas
0.05orless.P‐valuescanbefoundbyusingthe
procedureintheflowchartonthenextslide.
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CONCLUSIONS
Ourinitialconclusionwillalwaysbeoneofthe
following:
1.Rejectthenullhypothesis.
2.Failtorejectthenullhypothesis.
WORDING THE FINAL
CONCLUSION
“There is sufficient
evidence to warrant
rejection the claim that
. . . (original claim).”
“There is not sufficient
evidence to warrant
rejection the claim that
. . . (original claim).”
“There is sufficient
evidence to support the
claim that . . . (original
claim).”
DECISION CRITERION
P‐value
Method:
• IfP‐value≤α, rejectH0.(IfP islow,
thenullmustgo.)
• IfP‐value>α, failtorejectH0.
Critical
Value
Method:
• Iftheteststatisticfallswithinthe
criticalregion,rejectH0.
• Iftheteststatisticdoesnotfall
withinthecriticalregion,failto
rejectH0.
TYPE I ERROR
AtypeIerror isthemistakeofrejectingthe
nullhypothesiswhenitisactuallytrue.
Thesymbolα isusedtorepresentthe
probabilityofatypeIerror.
“There is not sufficient
evidence to support the
claim that . . . (original
claim).”
TYPE II ERROR
AtypeIIerror isthemistakeoffailingto
rejectthenullhypothesiswhenitisactually
false.
Thesymbolβ isusedtorepresentthe
probabilityofatypeIIerror.
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Critical Value Method
CONTROLLING TYPE I AND
TYPE II ERRORS
• Foranyfixedα,anincreaseinthesample
size willcauseadecreaseinβ.
• Foranyfixedsamplesize ,adecreaseinα
willcauseanincreaseinβ.Conversely,an
increaseinα willcauseadecreaseinβ.
• Todecreasebothα andβ,increasethe
samplesize.
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