Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann?2020CengageLearning?2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.1Chapter19-DecisionAnalysis19.1-ProblemFormulation19.2-DecisionMakingwithProbabilities19.3-DecisionAnalysiswithSampleInformation19.4-ComputingBranchProbabilitiesusingBayes’Theorem2ProblemFormulation(1of2)Thefirststepinthedecisionanalysisprocessisproblemformulation.Webeginwithaverbalstatementoftheproblem.Thenweidentify:thedecisionalternativesthestatesofnature(uncertainfutureevents)thepayoffs(consequences)associatedwitheachspecificcombinationof:decisionalternativestateofnature3ProblemFormulation(2of2)Adecisionproblemischaracterizedbydecisionalternatives,statesofnature,andresultingpayoffs.Thedecisionalternativesarethedifferentpossiblestrategiesthedecisionmakercanemploy.Thestatesofnaturerefertofutureevents,notunderthecontrolofthedecisionmaker,whichmayoccur.Statesofnatureshouldbedefinedsothattheyaremutuallyexclusiveandcollectivelyexhaustive.4PayoffTablesTheconsequenceresultingfromaspecificcombinationofadecisionalternativeandastateofnatureisapayoff.Atableshowingpayoffsforallcombinationsofdecisionalternativesandstatesofnatureisapayofftable.Payoffscanbeexpressedintermsofprofit,cost,time,distanceoranyotherappropriatemeasure.5DecisionTrees(1of2)Adecisiontreeprovidesagraphicalrepresentationshowingthesequentialnatureofthedecision-makingprocess.Eachdecisiontreehastwotypesofnodes:roundnodescorrespondtothestatesofnaturesquarenodescorrespondtothedecisionalternatives6DecisionTrees(2of2)Thebranchesleavingeachroundnoderepresentthedifferentstatesofnaturewhilethebranchesleavingeachsquarenoderepresentthedifferentdecisionalternatives.Attheendofeachlimbofatreearethepayoffsattainedfromtheseriesofbranchesmakingupthatlimb.7DecisionMakingwithProbabilities(1of2)Oncewehavedefinedthedecisionalternativesandstatesofnatureforthechanceevents,wefocusondeterminingprobabilitiesforthestatesofnature.Theclassicalmethod,relativefrequencymethod,orsubjectivemethodofassigningprobabilitiesmaybeused.BecauseoneandonlyoneoftheNstatesofnaturecanoccur,theprobabilitiesmustsatisfytwoconditions:8DecisionMakingwithProbabilities(2of2)Thenweusetheexpectedvalueapproachtoidentifythebestorrecommendeddecisionalternative.Theexpectedvalueofeachdecisionalternativeiscalculated(explainedonthenextslide).Thedecisionalternativeyieldingthebestexpectedvalueischosen.9ExpectedValueApproach(1of6)

10ExpectedValueApproach(2of6)Example:BurgerPrinceBurgerPrinceRestaurantisconsideringopeninganewrestaurantonMainStreet.Ithasthreedifferentrestaurantlayoutmodels(A,B,andC),eachwithadifferentseatingcapacity.BurgerPrinceestimatesthattheaveragenumberofcustomersservedperhourwillbe80,100,or120.Thepayofftableforthethreemodelsisonthenextslide.11ExpectedValueApproach(3of6)PayoffTableAverageNumberofCustomersPerHourssubscript1baselineequals80ssubscript2baselineequals100ssubscript3baselineequals120ModelA$10,000$15,000$14,000ModelB$8,000$18,000$12,000ModelC$6,000$16,000$21,00012ExpectedValueApproach(4of6)Calculatetheexpectedvalueforeachdecision.Thedecisiontreeonthenextslidecanassistinthiscalculation.Hered1,d2,d3representthedecisionalternativesofmodelsA,B,andC.Ands1,s2,s3representthestatesofnatureof80,100,and120customersperhour.13ExpectedValueApproach(5of6)DecisionTree14ExpectedValueApproach(6of6)DecisionTree15ExpectedValueofPerfectInformation(1of4)Frequently,informationisavailablethatcanimprovetheprobabilityestimatesforthestatesofnature.Theexpectedvalueofperfectinformation(EVPI)istheincreaseintheexpectedprofitthatwouldresultifoneknewwithcertaintywhichstateofnaturewouldoccur.TheEVPIprovidesanupperboundontheexpectedvalueofanysampleorsurveyinformation.16ExpectedValueofPerfectInformation(2of4)TheexpectedvalueofperfectinformationisdefinedasEVPI=|EVwPI–EVwoPI|where:EVPI=expectedvalueofperfectinformationEVwPI=expectedvaluewithperfectinformationaboutthestatesofnatureEVwoPI=expectedvaluewithoutperfectinformationaboutthestatesofnature17ExpectedValueofPerfectInformation(3of4)EVPICalculationStep1:Determinetheoptimalreturncorrespondingtoeachstateofnature.Step2:Computetheexpectedvalueoftheseoptimalreturns.Step3:SubtracttheEVoftheoptimaldecisionfromtheamountdeterminedinstep(2).18ExpectedValueofPerfectInformation(4of4)CalculatetheexpectedvaluefortheoptimumpayoffforeachstateofnatureandsubtracttheEVoftheoptimaldecision.EVPI=0.4(10,000)+0.2(18,000)+0.4(21,000)–14,000=$2,00019DecisionAnalysisWithSampleInformation(1of7)Knowledgeofsample(survey)informationcanbeusedtorevisetheprobabilityestimatesforthestatesofnature.Priortoobtainingthisinformation,theprobabilityestimatesforthestatesofnaturearecalledpriorprobabilities.Withknowledgeofconditionalprobabilitiesfortheoutcomesorindicatorsofthesampleorsurveyinformation,thesepriorprobabilitiescanberevisedbyemployingBayes'Theorem.Theoutcomesofthisanalysisarecalledposteriorprobabilitiesorbranchprobabilitiesfordecisiontrees.20DecisionAnalysisWithSampleInformation(2of7)DecisionStrategyAdecisionstrategyisasequenceofdecisionsandchanceoutcomes.Thedecisionschosendependontheyettobedeterminedoutcomesofchanceevents.Theapproachusedtodeterminetheoptimaldecisionstrategyisbasedonabackwardpassthroughthedecisiontree.21DecisionAnalysisWithSampleInformation(3of7)BackwardPassThroughtheDecisionTreeAtChanceNodes:Computetheexpectedvaluebymultiplyingthepayoffattheendofeachbranchbythecorrespondingbranchprobability.AtDecisionNodes:Selectthedecisionbranchthatleadstothebestexpectedvalue.Thisexpectedvaluebecomestheexpectedvalueatthedecisionnode.22DecisionAnalysisWithSampleInformation(4of7)

23DecisionAnalysisWithSampleInformation(5of7)DecisionTree(tophalf)24DecisionAnalysisWithSampleInformation(6of7)DecisionTree(bottomhalf)25DecisionAnalysisWithSampleInformation(7of7)26ExpectedValueofSampleInformation

(1of4)Theexpectedvalueofsampleinformation(EVSI)istheadditionalexpectedprofitpossiblethroughknowledgeofthesampleorsurveyinformation.EVSI=|EVwSI–EVwoSI|where:EVSI=expectedvalueofsampleinformationEVwSI=expectedvaluewithsampleinformationaboutthestatesofnatureEVwoSI=expectedvaluewithoutsampleinformationaboutthestatesofnature27ExpectedValueofSampleInformation(2of4)EVwSICalculationStep1:Determinetheoptimaldecisionanditsexpectedreturnforthepossibleoutcomesofthesampleusingtheposteriorprobabilitiesforthestatesofnature.Step2:Computetheexpectedvalueoftheseoptimalreturns.28DecisionAnalysisWithSampleInformation(1of2)29ExpectedValueofSampleInformation(3of4)Iftheoutcomeofthesurveyis"favorable,”chooseModelC.Iftheoutcomeofthesurveyis“unfavorable,”chooseModelA.EVwSI=0.54($17,855)+0.46($11,433)=$14,900.8830ExpectedValueofSampleInformation(4of4)EVSICalculationSubtracttheEVwoSI(thevalueoftheoptimaldecisionobtainedwithoutusingthesampleinformation)fromtheEVwSI.EVSI=0.54($17,855)+0.46($11,433)–$14,000=$900.88ConclusionBecausetheEVSIislessthanthecostofthesurvey,thesurveyshouldnotbepurchased.31ComputingBranchProbabilitiesUsingBayes’Theorem(1of2)Bayes’Theoremcanbeusedtocomputebranchprobabilitiesfordecisiontrees.Forthecomputationsweneedtoknow:theinitial(prior)probabilitiesforthestatesofnature,theconditionalprobabilitiesfortheoutcomesorindicatorsofthesampleinformationgiveneachstateofnature.Atabularapproachisaconvenientmethodforcarryingoutthecomputations.32ComputingBranchProbabilitiesUsingBayes’Theorem(2of2)Step1Foreachstateofnature,multiplythepriorprobabilitybyitsconditionalprobabilityfortheindicator.Thisgivesthejointprobabilitiesforthestatesandindicator.Step2Sumthesejointprobabilitiesoverallstates.Thisgivesthemarginalprobabilityfortheindicator.Step3Foreachstate,divideitsjointprobabilitybythemarginalprobabilityfortheindicator.Thisgivestheposteriorprobabilitydistribution.33DecisionAnalysisWithSampleInformation(2of2)

34PosteriorProbabilities(1of2)StatePriorConditionalJointPosterior80.4.2.08.148100.2.5.10.184120.4.9.36.66735PosteriorProbabilities(2of2)StatePriorConditionalJointPosterior80.4.8.32.696100.2.5.10.217120.4.1.04.08736Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann?2020CengageLearning?2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.37Chapter20-IndexNumbers20.1-PriceRelatives20.2-AggregatePriceIndexes20.3-ComputinganAggregatePriceIndexfromPriceRelatives20.4-SomeImportantPriceIndexes20.5-DeflatingaSeriesbyPriceIndexes20.6-PriceIndexes:OtherConsiderations20.7-QuantityIndexes38PriceRelatives(1of3)

39PriceRelatives(2of3)Example:BescoProductsThepricesBescopaidfornewspaperandtelevisionadsin2004and2014areshownbelow.Using2004asthebaseyear,computea2014priceindexfornewspaperandtelevisionadprices.

2004

2014Newspaper$14,794$29,412Television11,46923,90440PriceRelatives(3of3)Newspaper: Television: Televisionadvertisingcostincreasedatagreaterrate.41AggregatePriceIndexes(1of13)

42AggregatePriceIndexes(2of13)

wherethesumsareoverallitemsinthegroup.43AggregatePriceIndexes(3of13)Whenthefixedquantityweightsaredeterminedfromthebase-yearusage,theindexiscalledaLaspeyresindex.Whentheweightsarebasedonperiodtusage,theindexisaPaascheindex.Example:CityofRockdaleDataonenergyconsumptionandexpendituresbysectorforthecityofRockdalearegivenonthenextslide.Constructanaggregatepriceindexforenergyexpendituresin2014using1993asthebaseyear.44AggregatePriceIndexes(4of13)Example:CityofRockdaleSectorQuantity(BTU)1993Quantity(BTU)2014UnitPrice($/BTU)1993UnitPrice($/BTU)2014Residential9,4738,804$2.12$10.92Commercial5,4166,0151.9711.32Industrial21,28717,832.795.13Transport15,29320,2622.326.1645AggregatePriceIndexes(5of13)Example:CityofRockdaleUnweightedAggregatePriceIndex46AggregatePriceIndexes(6of13)Example:CityofRockdaleWeightedAggregateIndex(LaspeyresMethod)WeightedAggregateIndex(PaascheMethod)ThePaaschevaluebeinglessthantheLaspeyresindicatesusagehasincreasedfasterinthelower-pricedsectors.47AggregatePriceIndexes(7of13)Example:AnnualCostofLawnCareDinaEversispleasedwithherlovelylawn,butsheisconcernabouttheincreasingcostofmaintainingit.Thecostincludesmowing,fertilizing,watering,andmore.Dinawantsanindexthatmeasuresthechangeintheoverallcostofherlawncare.Priceandquantitydataforherannuallawnexpensesarelistedonthenextslide.48AggregatePriceIndexes(8of13)Example:AnnualCostofLawnCareItemQuantity(Units)UnitPrice($)2010UnitPrice($)2014Mowing3257.0079.00LeafRemoval356.0071.00Watering(1000sgal.)401.832.78Fertilizing256.0067.00SprinklerRepair1109.00128.0049AggregatePriceIndexes(9of13)UnweightedAnnuallawncareexpensesincreased24%from2010to2014.50AggregatePriceIndexes(10of13)Weighted(FixedQuantity)

Annuallawncareexpensesincreased36%from2010to2014.51AggregatePriceIndexes(11of13)Weighted(Base-PeriodQuantity)LaspeyresIndexSpecialcaseofthefixedquantityindex:

MorewidelyusedthanthePaascheindex.Weighted(PeriodtQuantity)PaascheIndex Avariable-quantityindex: Pro:Reflectscurrentusage.Con:Weightsrequirecontinualupdating.52AggregatePriceIndexes(12of13)ItemiUnitPrice($)2010Psubscripti0baselineUnitPrice($)2014PsubscriptitbaselinePriceRelativeleftparenthesisstartfractionPsubscriptitbaselineoverPsubscripti0endfractionrightparenthesis100Mowing57.0079.00138.6LeafRemoval56.0071.00126.8Water(1000sgal)1.832.78151.9Fertilizing56.0067.00119.6SprinklerRepair109.00128.00117.4The5-yearincreasesinunitpricerangedfromalowof17.4%forsprinklerrepairtoahighof51.9%forwater.53AggregatePriceIndexes(13of13)Item252PriceRelativeleftparenthesisstartfractionPsubscriptitbaselineoverPsubscripti0endfractionrightparenthesis100BasePriceleftparenthesis$rightparenthesisPsubscripti0baselineQuantityQsubscriptibaselineWeightwsubscriptiequalsPsubscripti0baselineQsubscriptiWeightedPriceRelativeleftparenthesisstartfractionPsubscriptitbaselineoverPsubscripti0endfractionrightparenthesisleftparenthesis100rightparenthesiswsubscriptiMowing138.657.00321824.0252,806.40Leaves126.856.003168.021,302.40Water151.91.834073.211,119.08Fertilize119.656.002112.013,395.20Sprinkler117.4109.001109.012,796.60Thisvalueisthesameastheoneidentifiedbytheweightedaggregateindexcomputation.54SomeImportantPriceIndexes(1of5)ConsumerPriceIndex(CPI)PrimarymeasureofthecostoflivinginU.S.Basedon400itemsincludingfood,housing,clothing,transportation,andmedicalitems.Weightedaggregatepriceindexwithfixedweightsderivedfromausagesurvey.PublishedmonthlybytheU.S.BureauofLaborStatistics.Itsbaseperiodis1982-1984withanindexof100.55SomeImportantPriceIndexes(2of5)ConsumerPriceIndex(CPI)Base1982-1984=100.0YearCPIYearCPIYearCPIYearCPI198082.41989124.01998163.02007207.3198190.91990130.71999166.62008215.3198296.51991136.22000172.22009214.5198399.61992140.32001177.12010218.11984103.91993144.52002179.92011224.91985107.61994148.22003184.02012229.61986109.61995152.42004188.92013233.01987113.61996156.92005195.32014236.71988118.31997160.52006201.62015???Note:For1982–1984,(96.5+99.6+103.9)/3=100.0Alsonote:CPIfor2009waslowerthanCPIfor2008.56SomeImportantPriceIndexes(3of5)ProducerPriceIndex(PPI)MeasuresthemonthlychangesinpricesinprimarymarketsintheU.S.Usedasaleadingindicator

ofthefuturetrendofconsumerpricesandthecostofliving.Coversraw,manufactured,andprocessedgoodsateachlevelofprocessing.Includestheoutputofmanufacturing,agriculture,forestry,fishing,mining,gasandelectricity,andpublicutilities.IsaweightedaverageofpricerelativesusingtheLaspeyresmethod.57SomeImportantPriceIndexes(4of5)DowJonesAveragesIndexesdesignedtoshowpricetrendsandmovementsontheNewYorkStockExchange.TheDowJonesIndustrialAverage

(DJIA)

isbasedoncommonstockpricesof30industrialfirms.TheDJIAisnotexpressedasapercentageofbase-yearprices.Anotheraverageiscomputedfor20transportationstocks,andanotherfor15utilitystocks.58SomeImportantPriceIndexes(5of5)DowJonesIndustrialAverage(DJIA)30Companies(asof03/2015)3MGeneralElectricNikeAmericanExpressGoldmanSachsPfizerAppleTheHomeDepotProcter&GambleBoeingIBMTravelsCaterpillarIntelUnitedHealthGroupChevronCorp.Johnson&JohnsonUnitedTechnologiesCiscoSystemsJPMorganChaseVerizonCoca-ColaMcDonald’sVisaDuPontMerckWal-MartExxonMobilMicrosoftWaltDisney59DeflatingaSeriesbyPriceIndexes(1of5)Inordertocorrectlyinterpretbusinessactivityovertimewhenitisexpressedindollaramounts,weshouldadjustthedatafortheprice-increaseeffect.Removingtheprice-increaseeffectfromatimeseriesiscalleddeflatingtheseries.Deflatingactualhourlywagesresultsinrealwagesorthepurchasingpower

ofwages.60DeflatingaSeriesbyPriceIndexes(2of5)Example:McNeerCleanersMcNeerCleaners,with46branchlocations,hashadthetotalsalesrevenuesshownonthenextslideforthelastfiveyears.Deflatethesalesrevenuefiguresonthebasisof1982-1984constantdollars.Istheincreaseinsalesdueentirelytotheprice-increaseeffect?61DeflatingaSeriesbyPriceIndexes(3of5)Example:McNeerCleanersYearTotalSales($1000)CPI20108,446218.120119,062224.920129,830229.6201310,724233.0201411,690236.762DeflatingaSeriesbyPriceIndexes(4of5)AdjustingRevenueForthePrice-IncreaseEffectYearDeflatedSales($1000)AnnualChange(%)2010Leftparenthesis8446dividedby218.1rightparenthesisleftparenthesis100rightparenthesisequals3873EMPTYCELL2011Leftparenthesis9062dividedby224.9rightparenthesisleftparenthesis100rightparenthesisequals4029positive4.02012Leftparenthesis9830dividedby229.6rightparenthesisleftparenthesis100rightparenthesisequals4281positive6.32013Leftparenthesis10,724dividedby233.0rightparenthesisleftparenthesis100rightparenthesisequals4603positive7.52014Leftparenthesis11,690dividedby236.7rightparenthesisleftparenthesis100rightparenthesisequals4939positive7.3Afteradjusting,revenueisstillincreasingatanaveragerateof6.3%peryear.63DeflatingaSeriesbyPriceIndexes(5of5)Arealsalesincreaseof27.5%from2010to201464PriceIndexes:OtherConsiderations(1of3)SelectionofItemsWhentheclassofitemsisverylarge,arepresentativegroup(usuallynotarandomsample)mustbeused.Thegroupofitemsintheaggregateindexmustbeperiodicallyreviewedandrevisedifitisnotrepres