One-SampleTestsofHypothesisGOALSWhenyouhavecompletedthischapter,youwillbeableto:ONE

Defineahypothesisandhypothesistesting.TWO

Describethefivestephypothesistestingprocedure.THREE

Distinguishbetweenaone-tailedandatwo-tailedtestofhypothesis.FOUR

Conductatestofhypothesisaboutapopulationmean.GOALSWhenyouhavecompletedthischapter,youwillbeableto:FIVE

Conductatestofhypothesisaboutapopulationproportion.SIX

DefineTypeIandTypeIIerrors.SEVEN

ComputetheprobabilityofaTypeIIerror.One-SampleTestsofHypothesisWhatisaHypothesis?AHypothesisisastatementaboutthevalueofapopulationparameterdevelopedforthepurposeoftesting.Examplesofhypothesesmadeaboutapopulationparameterare:Themeanmonthlyincomeforsystemsanalystsis$3,625.TwentypercentofallcustomersatBovine’sChopHousereturnforanothermealwithinamonth.WhatisHypothesisTesting?Hypothesistestingisaprocedure,basedonsampleevidenceandprobabilitytheory,usedtodeterminewhetherthehypothesisisareasonablestatementandshouldnotberejected,orisunreasonableandshouldberejected.HypothesisTestingDefinitionsNullHypothesisH0:Astatementaboutthevalueofapopulationparameter.AlternativeHypothesisH1:Astatementthatisacceptedifthesampledataprovideevidencethatthenullhypothesisisfalse.LevelofSignificance:Theprobabilityofrejectingthenullhypothesiswhenitisactuallytrue.TypeIError:Rejectingthenullhypothesiswhenitisactuallytrue.DefinitionsTypeIIError:Acceptingthenullhypothesiswhenitisactuallyfalse.Teststatistic:Avalue,determinedfromsampleinformation,usedtodeterminewhetherornottorejectthenullhypothesis.Criticalvalue:Thedividingpointbetweentheregionwherethenullhypothesisisrejectedandtheregionwhereitisnotrejected.One-TailedTestsofSignificanceAtestisone-tailedwhenthealternatehypothesis,H1,statesadirection,suchas:H1:Themeanyearlycommissionsearnedbyfull-timerealtorsismorethan$35,000.(μ>$35,000)H1:ThemeanspeedoftruckstravelingonI-95inGeorgiaislessthan60milesperhour.(μ<60)H1:Lessthan20percentofthecustomerspaycashfortheirgasolinepurchase.(p<.20)Two-TailedTestsofSignificanceAtestistwo-tailedwhennodirectionisspecifiedinthealternatehypothesisH1,suchas:H1:ThemeanamountspentbycustomersattheWal-MartinGeorgetownisnotequalto$25.(μ≠$25).H1:Themeanpriceforagallonofgasolineisnotequalto$1.54.(μ≠$1.54).TestingforthePopulationMean:LargeSample,PopulationStandardDeviationKnownWhentestingforthepopulationmeanfromalargesampleandthepopulationstandarddeviationisknown,theteststatisticisgivenby:EXAMPLE1TheprocessorsofFries’Catsupindicateonthelabelthatthebottlecontains16ouncesofcatsup.Thestandarddeviationoftheprocessis0.5ounces.Asampleof36bottlesfromlasthour’sproductionrevealedameanweightof16.12ouncesperbottle.Atthe.05significancelevelistheprocessoutofcontrol?Thatis,canweconcludethatthemeanamountperbottleisdifferentfrom16ounces?EXAMPLE1continuedStep1:Statethenullandthealternativehypotheses:

H0:μ=16; H1:μ≠16Step3:Identifytheteststatistic. Becauseweknowthepopulationstandarddeviation,theteststatisticisz.

Step2:Selectthelevelofsignificance. Inthiscaseweselectedthe.05significancelevel.

EXAMPLE1continuedStep4:Statethedecisionrule: RejectH0ifz>1.96

orz<-1.96Step5:Computethevalueoftheteststatisticandarriveatadecision.

Donotrejectthenullhypothesis.Wecannotconcludethemeanisdifferentfrom16ounces.p-ValueinHypothesisTestingAp-Valueistheprobability,assumingthatthenullhypothesisistrue,offindingavalueoftheteststatisticatleastasextremeasthecomputedvalueforthetest.Ifthep-Valueissmallerthanthesignificancelevel,H0isrejected.Ifthep-Valueislargerthanthesignificancelevel,H0isnotrejected.Computationofthep-ValueOne-TailedTest:p-Value=P{zabsolutevalueofthecomputedteststatisticvalue}Two-TailedTest:p-Value=2P{zabsolutevalueofthecomputedteststatisticvalue}FromEXAMPLE1,z=1.44,andbecauseitwasatwo-tailedtest,thep-Value=2P{z1.44}=2(.5-.4251)=.1498.Because.1498>.05,donotrejectH0.TestingforthePopulationMean:LargeSample,PopulationStandardDeviationUnknownHereσisunknown,soweestimateitwiththesamplestandarddeviations.Aslongasthesamplesizen≥30,zcanbeapproximatedwith:EXAMPLE2Roder’sDiscountStorechainissuesitsowncreditcard.Lisa,thecreditmanager,wantstofindoutifthemeanmonthlyunpaidbalanceismorethan$400.Thelevelofsignificanceissetat.05.Arandomcheckof172unpaidbalancesrevealedthesamplemeantobe$407andthesamplestandarddeviationtobe$38.ShouldLisaconcludethatthepopulationmeanisgreaterthan$400,orisitreasonabletoassumethatthedifferenceof$7($407-$400)isduetochance?EXAMPLE2continuedStep1:

H0:≤$400,H1:μ>$400Step2:Thesignificancelevelis.05Step3:

Becausethesampleislargewecanusethez

distributionastheteststatistic.Step4:

H0isrejectedifz>1.65Step5:Performthecalculationsandmakeadecision.

H0isrejected.Lisacanconcludethatthemeanunpaidbalanceisgreaterthan$400.TestingforaPopulationMean:SmallSample,PopulationStandardDeviationUnknownTheteststatisticisthetdistribution.Theteststatisticfortheonesamplecaseisgivenby:

Example3

Thecurrentrateforproducing5lampfusesatNearyElectricCo.is250perhour.Anewmachinehasbeenpurchasedandinstalledthat,accordingtothesupplier,willincreasetheproductionrate.Asampleof10randomlyselectedhoursfromlastmonthrevealedthemeanhourlyproductiononthenewmachinewas256units,withasamplestandarddeviationof6perhour.Atthe.05significancelevelcanNearyconcludethatthenewmachineisfaster?Example3continuedStep1:Statethenullandthealternatehypothesis.

H0:μ≤250;H1:μ>250Step2:Selectthelevelofsignificance.

Itis.05.Step3:Findateststatistic.Itisthetdistributionbecausethepopulationstandarddeviationisnotknownandthesamplesizeislessthan30.Example3continuedStep4:Statethedecisionrule.

Thereare10–1=9degreesoffreedom.Thenullhypothesisisrejectedift>1.833.Step5:Makeadecisionandinterprettheresults.

Thenullhypothesisisrejected.Themeannumberproducedismorethan250perhour.TestsConcerningProportionAProportionisthefractionorpercentagethatindicatesthepartofthepopulationorsamplehavingaparticulartraitofinterest.Thesampleproportionisdenotedbypandisfoundby:TestStatisticforTestingaSinglePopulationProportionThesampleproportionispandπisthepopulationproportion.EXAMPLE4Inthepast,15%ofthemailordersolicitationsforacertain