Business Statistics

BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-1DepartmentofQuantitativeMethods&InformationSystemsChapter3MultipleRegressionCompleteExampleECON504Dr.MohammadZainalSpring2013ReviewGoalsAftercompletingthislecture,youshouldbeableto:FormulatenullandalternativehypothesesforapplicationsinvolvingasinglepopulationmeanorproportionFormulateadecisionrulefortestingahypothesisKnowhowtousetheteststatistic,criticalvalue,andp-valueapproachestotestthenullhypothesisKnowwhatTypeIandTypeIIerrorsare2BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-2ReviewGoalsexplainmodelbuildingusingmultipleregressionanalysisapplymultipleregressionanalysistobusinessdecision-makingsituationsanalyzeandinterpretthecomputeroutputforamultipleregressionmodeltestthesignificanceoftheindependentvariablesinamultipleregressionmodel(continued)3ReviewGoalsrecognizepotentialproblemsinmultipleregressionanalysisandtakestepstocorrecttheproblemsincorporatequalitativevariablesintotheregressionmodelbyusingdummyvariablesusevariabletransformationstomodelnonlinearrelationships(continued)4BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-3WhatisaHypothesis?Ahypothesisisaclaim(assumption)aboutapopulationparameter:populationmeanpopulationproportionExample:Themeanmonthlycellphonebillofthiscityisµ=$42Example:TheproportionofadultsinthiscitywithcellphonesisP=.685TheNullHypothesis,H0Statestheassumption(numerical)tobetestedExample:TheaveragenumberofTVsetsinU.S.Homesisatleastthree()Isalwaysaboutapopulationparameter,notaboutasamplestatistic3μ:H03μ:H03x:H06BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-4TheNullHypothesis,H0BeginwiththeassumptionthatthenullhypothesisistrueSimilartothenotionofinnocentuntilprovenguiltyReferstothestatusquoAlwayscontains“=”,“≤”or“”signMayormaynotberejected(continued)7TheAlternativeHypothesis,HAIstheoppositeofthenullhypothesise.g.:TheaveragenumberofTVsetsinU.S.homesislessthan3(HA:µ<3)ChallengesthestatusquoNevercontainsthe“=”,“≤”or“”signMayormaynotbeacceptedIsgenerallythehypothesisthatisbelieved(orneedstobesupported)bytheresearcher–aresearchhypothesis8BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-5FormulatingHypothesesExample1:FordmotorcompanyhasworkedtoreduceroadnoiseinsidethecaboftheredesignedF150pickuptruck.Itwouldliketoreportinitsadvertisingthatthetruckisquieter.Theaverageofthepriordesignwas68decibelsat60mph.Whatistheappropriatehypothesistest?9FormulatingHypothesesExample1:FordmotorcompanyhasworkedtoreduceroadnoiseinsidethecaboftheredesignedF150pickuptruck.Itwouldliketoreportinitsadvertisingthatthetruckisquieter.Theaverageofthepriordesignwas68decibelsat60mph.Whatistheappropriatetest?H0:µ≥68(thetruckisnotquieter)statusquoHA:µ<68(thetruckisquieter)wantstosupportIfthenullhypothesisisrejected,Fordhassufficientevidencetosupportthatthetruckisnowquieter.10BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-6FormulatingHypothesesExample2:TheaverageannualincomeofbuyersofFordF150pickuptrucksisclaimedtobe$65,000peryear.Anindustryanalystwouldliketotestthisclaim.Whatistheappropriatehypothesistest?11FormulatingHypothesesExample1:TheaverageannualincomeofbuyersofFordF150pickuptrucksisclaimedtobe$65,000peryear.Anindustryanalystwouldliketotestthisclaim.Whatistheappropriatetest?H0:µ=65,000(incomeisasclaimed)statusquoHA:µ≠65,000(incomeisdifferentthanclaimed)Theanalystwillbelievetheclaimunlesssufficientevidenceisfoundtodiscreditit.12BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-7PopulationClaim:thepopulationmeanageis50.NullHypothesis:REJECTSupposethesamplemeanageis20:x=20SampleNullHypothesisIsx=20likelyifµ=50?HypothesisTestingProcessIfnotlikely,Nowselectarandomsample:H0:µ=5013SamplingDistributionofxμ=50IfH0istrueIfitisunlikelythatwewouldgetasamplemeanofthisvalue......thenwerejectthenullhypothesisthatμ=50.ReasonforRejectingH020...ifinfactthiswerethepopulationmean…x14BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-8ErrorsinMakingDecisionsTypeIErrorRejectatruenullhypothesisConsideredaserioustypeoferrorTheprobabilityofTypeIErrorisCalledlevelofsignificanceofthetestSetbyresearcherinadvance15ErrorsinMakingDecisionsTypeIIErrorFailtorejectafalsenullhypothesisTheprobabilityofTypeIIErrorisββisacalculatedvalue,theformulaisdiscussedlaterinthechapter(continued)16BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-9OutcomesandProbabilitiesStateofNatureDecisionDoNotRejectH0Noerror(1-)TypeIIError(β)RejectH0TypeIError()PossibleHypothesisTestOutcomesH0FalseH0TrueKey:Outcome(Probability)NoError(1-β)17TypeI&IIErrorRelationshipTypeIandTypeIIerrorscannothappenatthesametimeTypeIerrorcanonlyoccurifH0istrueTypeIIerrorcanonlyoccurifH0isfalseIfTypeIerrorprobability(),thenTypeIIerrorprobability(β)18BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-10FactorsAffectingTypeIIErrorAllelseequal,βwhenthedifferencebetweenhypothesizedparameteranditstruevalueβwhenβwhenσβwhennTheformulausedtocomputethevalueofβisdiscussedlaterinthechapter19LevelofSignificance,DefinesunlikelyvaluesofsamplestatisticifnullhypothesisistrueDefinesrejectionregionofthesamplingdistributionIsdesignatedby,(levelofsignificance)Typicalvaluesare.01,.05,or.10IsselectedbytheresearcheratthebeginningProvidesthecriticalvalue(s)ofthetest20BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-11HypothesisTestsfortheMeanσKnownσUnknownHypothesisTestsforAssumefirstthatthepopulationstandarddeviationσisknown211.Specifypopulationparameterofinterest2.Formulatethenullandalternativehypotheses3.Specifythedesiredsignificancelevel,α4.Definetherejectionregion5.Takearandomsampleanddeterminewhetherornotthesampleresultisintherejectionregion6.ReachadecisionanddrawaconclusionProcessofHypothesisTesting22BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-12LevelofSignificanceandtheRejectionRegionH0:μ≥3HA:μ<30H0:μ≤3HA:μ>3H0:μ=3HA:μ≠3/2LowertailtestLevelofsignificance=0/2UppertailtestTwotailedtest0-zαzα-zα/2zα/2RejectH0RejectH0RejectH0RejectH0DonotrejectH0DonotrejectH0DonotrejectH0Example:Example:Example:23RejectH0DonotrejectH0Thecutoffvalue,or,iscalledacriticalvalue-zαxα-zαxα0µ=3H0:μ≥3HA:μ<3nσzμxCriticalValueforLowerTailTest24BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-13RejectH0DonotrejectH0CriticalValueforUpperTailTestzαxα0H0:μ≤3HA:μ>3nσzμxµ=3Thecutoffvalue,or,iscalledacriticalvaluezαxα25DonotrejectH0RejectH0RejectH0Therearetwocutoffvalues(criticalvalues):orCriticalValuesforTwoTailedTests/2-zα/2xα/2±zα/2xα/20H0:μ=3HA:μ3zα/2xα/2nσzμx/2/2LowerUpperxα/2LowerUpper/2µ=326BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-14TheRejectionRegionH0:μ≥3HA:μ<30H0:μ≤3HA:μ>3H0:μ=3HA:μ≠3/2Lowertailtest0/2UppertailtestTwotailedtest0-zαzα-zα/2zα/2RejectH0RejectH0RejectH0RejectH0DonotrejectH0DonotrejectH0DonotrejectH0Example:Example:Example:RejectH0ifz<-zαi.e.,ifx<xααxαxα/2(L)xα/2(U)xRejectH0ifz>zαi.e.,ifx>xαRejectH0ifz<-zα/2orz>zα/2i.e.,ifx<xα/2(L)orx>xα/2(U)27z-units:Forgiven,findthecriticalzvalue(s):-zα,zα,or±zα/2Convertthesamplemeanxtoazteststatistic:RejectH0ifzisintherejectionregion,otherwisedonotrejectH0xunits:Given,calculatethecriticalvalue(s)xα,orxα/2(L)andxα/2(U)Thesamplemeanistheteststatistic.RejectH0ifxisintherejectionregion,otherwisedonotrejectH0TwoEquivalentApproachestoHypothesisTestingnσμxz28BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-15HypothesisTestingExampleTesttheclaimthatthetruemean#ofTVsetsinUShomesisatleast3.(Assumeσ=0.8)1.SpecifythepopulationvalueofinterestThemeannumberofTVsinUShomes2.FormulatetheappropriatenullandalternativehypothesesH0:μ3HA:μ<3(Thisisalowertailtest)3.SpecifythedesiredlevelofsignificanceSupposethat=.05ischosenforthistest29RejectH0DonotrejectH04.Determinetherejectionregion=.05-zα=-1.6450Thisisaone-tailedtestwith=.05.Sinceσisknown,thecutoffvalueisazvalue:RejectH0ifz<z=-1.645;otherwisedonotrejectH0HypothesisTestingExample(continued)30BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-165.ObtainsampleevidenceandcomputetheteststatisticSupposeasampleistakenwiththefollowingresults:n=100,x=2.84(=0.8isassumedknown)Thentheteststatisticis:2.0.08.161000.832.84nσμxzHypothesisTestingExample31RejectH0DonotrejectH0=.05-1.64506.Reachadecisionandinterprettheresult-2.0Sincez=-2.0<-1.645,werejectthenullhypothesisthatthemeannumberofTVsinUShomesisatleast3.Thereissufficientevidencethatthemeanislessthan3.HypothesisTestingExample(continued)z32BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-17RejectH0=.052.8684DonotrejectH03Analternatewayofconstructingrejectionregion:2.84Sincex=2.84<2.8684,werejectthenullhypothesisHypothesisTestingExample(continued)xNowexpressedinx,notzunits2.86841000.81.6453nσzμxαα33p-ValueApproachtoTestingConvertSampleStatistic()toTestStatistic(azvalue,ifσisknown)Determinethep-valuefromatableorcomputerComparethep-valuewithIfp-value<,rejectH0Ifp-value,donotrejectH0x34BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-18p-ValueApproachtoTestingp-value:Probabilityofobtainingateststatisticmoreextreme(≤or)thantheobservedsamplevaluegivenH0istrueAlsocalledobservedlevelofsignificanceSmallestvalueofforwhichH0canberejected(continued)35p-value=.0228=.05p-valueexampleExample:Howlikelyisittoseeasamplemeanof2.84(orsomethingfurtherbelowthemean)ifthetruemeanis=3.0?2.868432.84x.02282.0)P(z1000.83.02.84zP3.0)μ|2.84xP(0-1.645-2.0z36BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-19Comparethep-valuewithIfp-value<,rejectH0Ifp-value,donotrejectH0Here:p-value=.0228=.05Since.0228<.05,werejectthenullhypothesis(continued)p-valueexamplep-value=.0228=.052.868432.8437Example:UpperTailzTestforMean(Known)Aphoneindustrymanagerthinksthatcustomermonthlycellphonebillhaveincreased,andnowaverageover$52permonth.Thecompanywishestotestthisclaim.(Assume=10isknown)H0:μ≤52theaverageisnotover$52permonthHA:μ>52theaverageisgreaterthan$52permonth(i.e.,sufficientevidenceexiststosupportthemanager’sclaim)Formhypothesistest:38BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-20RejectH0DonotrejectH0Supposethat=.10ischosenforthistestFindtherejectionregion:=.10zα=1.280RejectH0RejectH0ifz>1.28Example:FindRejectionRegion(continued)39Review:FindingCriticalValue-OneTailZ.07.091.1.3790.3810.38301.2.3980.40151.3.4147.4162.4177z01.28.08StandardNormalDistributionTable(Portion)Whatiszgiven=0.10?=.10CriticalValue=1.28.90.3997.10.40.5040BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-21ObtainsampleevidenceandcomputetheteststatisticSupposeasampleistakenwiththefollowingresults:n=64,x=53.1(=10wasassumedknown)Thentheteststatisticis:0.8864105253.1nσμxzExample:TestStatistic(continued)41RejectH0DonotrejectH0Example:Decision=.101.280RejectH0DonotrejectH0sincez=0.88≤1.28i.e.:thereisnotsufficientevidencethatthemeanbillisover$52z=.88Reachadecisionandinterprettheresult:(continued)42BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-22RejectH0=.10DonotrejectH01.280RejectH0z=.88Calculatethep-valueandcompareto(continued).1894.3106.50.88)P(z641052.053.1zP52.0)μ|53.1xP(p-value=.1894p-ValueSolutionDonotrejectH0sincep-value=.1894>=.1043CriticalValueApproachtoTestingWhenσisknown,convertsamplestatistic()toazteststatisticxKnownUnknownHypothesisTestsforTheteststatisticis:nσμxz44BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-23CriticalValueApproachtoTestingWhenσisunknown,convertsamplestatistic()toatteststatisticxKnownUnknownHypothesisTestsforTheteststatisticis:nsμxt1n(Thepopulationmustbeapproximatelynormal)45HypothesisTestsforμ,σUnknown1.Specifythepopulationvalueofinterest2.Formulatetheappropriatenullandalternativehypotheses3.Specifythedesiredlevelofsignificance4.Determinetherejectionregion(criticalvaluesarefromthet-distributionwithn-1d.f.)5.Obtainsampleevidenceandcomputethetteststatistic6.Reachadecisionandinterprettheresult46BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-24Example:Two-TailTest(Unknown)TheaveragecostofahotelroominNewYorkissaidtobe$168pernight.Arandomsampleof25hotelsresultedinx=$172.50ands=$15.40.Testatthe=0.05level.(Assumethepopulationdistributionisnormal)H0:μ=168HA:μ16847=0.05n=25CriticalValues:t24=±2.0639isunknown,souseatstatisticExampleSolution:Two-TailTestDonotrejectH0:notsufficientevidencethattruemeancostisdifferentthan$168RejectH0RejectH0/2=.025-tα/2DonotrejectH00tα/2/2=.025-2.06392.06391.462515.40168172.50nsμxt1n1.46H0:μ=168HA:μ16848BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-25RejectH0:μ52DonotrejectH0:μ52TypeIIErrorTypeIIerroristheprobabilityoffailingtorejectafalseH05250SupposewefailtorejectH0:μ52wheninfactthetruemeanisμ=5049RejectH0:52DonotrejectH0:52TypeIIErrorSupposewedonotrejectH0:52wheninfactthetruemeanis=505250Thisisthetruedistributionofxif=50ThisistherangeofxwhereH0isnotrejected(continued)50BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-26RejectH0:μ52DonotrejectH0:μ52TypeIIErrorSupposewedonotrejectH0:μ52wheninfactthetruemeanisμ=505250βHere,β=P(xcutoff)ifμ=50(continued)51RejectH0:μ52DonotrejectH0:μ52Supposen=64,σ=6,and=.055250Soβ=P(x50.766)ifμ=50Calculatingβ50.7666461.64552nσzμxcutoff(forH0:μ52)50.76652BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-27RejectH0:μ52DonotrejectH0:μ52.1539.3461.51.02)P(z6465050.766zP50)μ|50.766xP(Supposen=64,σ=6,and=.055250Calculatingβ(continued)ProbabilityoftypeIIerror:β=.153953HypothesisTestsinMinitab54BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-28HypothesisTestsinMinitab55SampleMinitabOutput56BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-29HypothesisTestsSummaryAddressedhypothesistestingmethodologyPerformedzTestforthemean(σknown)Discussedp–valueapproachtohypothesistestingPerformedone-tailandtwo-tailtests...57HypothesisTestsSummaryPerformedttestforthemean(σunknown)PerformedztestfortheproportionDiscussedTypeIIerrorandcomputeditsprobability(continued)58BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-30MultipleRegressionAssumptionsThemodelerrorsareindependentandrandomTheerrorsarenormallydistributedThemeanoftheerrorsiszeroErrorshaveaconstantvariancee=(y–y)<Errors(residuals)fromtheregressionmodel:59ModelSpecificationDecidewhatyouwanttodoandselectthedependentvariableDeterminethepotentialindependentvariablesforyourmodelGathersampledata(observations)forallvariables60BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-31TheCorrelationMatrixCorrelationbetweenthedependentvariableandselectedindependentvariablescanbefoundusingExcel:FormulaTab:DataAnalysis/CorrelationCancheckforstatisticalsignificanceofcorrelationwithattest61ExampleAdistributoroffrozendesertpieswantstoevaluatefactorsthoughttoinfluencedemandDependentvariable:Piesales(unitsperweek)Independentvariables:Price(in$)Advertising($100’s)Dataarecollectedfor15weeks62BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-32PieSalesModelSales=b0+b1(Price)+b2(Advertising)WeekPieSalesPrice($)Advertising($100s)13505.503.324607.503.333508.003.044308.004.553506.803.063807.504.074304.503.084706.403.794507.003.5104905.004.0113407.203.5123007.903.2134405.904.0144505.003.5153007.002.7PieSalesPriceAdvertisingPieSales1Price-0.443271Advertising0.556320.030441Correlationmatrix:Multipleregressionmodel:63InterpretationofEstimatedCoefficientsSlope(bi)Estimatesthattheaveragevalueofychangesbybiunitsforeach1unitincreaseinXiholdingallothervariablesconstantExample:ifb1=-20,thensales(y)isexpectedtodecreasebyanestimated20piesperweekforeach$1increaseinsellingprice(x1),netoftheeffectsofchangesduetoadvertising(x2)y-intercept(b0)Theestimatedaveragevalueofywhenallxi=0(assumingallxi=0iswithintherangeofobservedvalues)64BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-33PieSalesCorrelationMatrixPricevs.Sales:r=-0.44327ThereisanegativeassociationbetweenpriceandsalesAdvertisingvs.Sales:r=0.55632ThereisapositiveassociationbetweenadvertisingandsalesPieSalesPriceAdvertisingPieSales1Price-0.443271Advertising0.556320.03044165ScatterDiagramsSalesvs.Price01002003004005006000246810Salesvs.Advertising0100200300400500600012345SalesSalesPriceAdvertising66BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-34EstimatingaMultipleLinearRegressionEquationComputersoftwareisgenerallyusedtogeneratethecoefficientsandmeasuresofgoodnessoffitformultipleregressionExcel:Data/DataAnalysis/RegressionMinitab:Stat/Regression/Regression…67EstimatingaMultipleLinearRegressionEquationExcel:68BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-35EstimatingaMultipleLinearRegressionEquationMinitab:69MultipleRegressionOutputRegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVAdfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888ertising)74.131(Advce)24.975(Pri-306.526Sales70BusinessStatistics:ADecision-MakingApproach,7e©2008Prentice-Hall,Inc.Chapter9StudentLectureNotes9-36TheMultipleRegressionEquationertising)74.131(Advce)24.975(Pri-306.526Salesb1=-24.975:saleswilldecrease,onaverage,by24.975piesperweekforeach$1increaseinsellingprice,netoftheeffe