一級(jí)基礎(chǔ)-定量分析教育

,,83▁????Т????? □???????????? ??ч??TopicWeightingsinFRMPartSessionStudySessionFoundationsofRiskStudySession StudySessionStudySession2BasicHypothesisTestsandConfidenceLinearRegressionwithOneRegressor&MultipleRegressorsModelingModelingandForecastingModelingandForecastingSimulationCorrelationsand3CFA 48GTJUS5[ZIUSK 8GTJUS???)UTIKVZULProbabilityFunction&CumulativeDistributionFunctionXGTJUS\GXOGHRKY5Randomexperiment(????Anobservationormeasurementprocesswithmultiplebute(??Theresultofasingletrial.Forexample,ifwerolltwodices,an mightbe3and4;adifferent emightbe5and2.Event(ж?Theresultthatreflectsnone,one,ormore esinthesamplespace.Eventscanbesimpleorcompound.Aneventisasubsetofthesamplespace.Ifwerolltwodices,anexampleofaneventmightberolling7intotal.Mutuallyexclusiveevents(л?з?):Eventsthatcannothappenatthesametime.Exhaustiveevents(?з?):Thoseincludeall 6P(AB)P(AB);0P(AB);P(A)0PP 0≤P(E)≤IfE1,E2,……,Enismutuallyexclusiveandexhaustive,then:P(E1)+P(E2)+……+ A A 7Theprobabilitythattherandomvariables(inthiscase,bothrandomvariables)takeoncertainvaluessimultaneously,P(AB).UnconditionalProbability(????,a.k.amarginalTheexpectedvalueofthevariablewithoutanyrestrictionsConditionalProbability(????Anexpectedvalueforthevariableconditionalonpriorinformationorsomerestriction(e.g.,thevalueofacorrelatedvariable).TheconditionalexpectationofB,conditionalonA,isgivenbyP(B|A).8ProbabilityandProbability9llicafrs8utions P(A|B) P(B|A)IfAandBaremutuallyexclusiveevents,then: ProbabilitythatatleastoneoftwoeventswillP(Aor P(A)+P(B)–IfAandBaremutuallyexclusiveevents,then:P(AorB) P(A)+P(B)10ProbabilityandProbabilityTheoccurrenceofAhasnoinfluenceofontheoccurrenceof P(A)or P(A)P(Aor P(A)+P(B)–Ifexclusive,mustnotCauseexclusivemeansifAoccur,Bcannotoccur,AinfluentsB. P(A)?P(B)11ProbabilityandProbabilityIfaneventAmustresultinoneofthemutuallyexclusiveeventsA1,A2,A3,……,An,thenPP PA1PA PA2PA ...PAnPA

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PPA|PB|APPP(A| 99%10 99% 5%15FRroliylitcfafrmst8utionsDescribetheprobabilitiesofallthepossible esforarandomvariable.Discreterandomvariables:thenumberofpossible canbecounted,andforeachpossible e,thereisameasurableandpositiveprobability.Continuousvariables:thenumberofpossible esisinfinite,eveniflowerandupperboundsexist.P(x)0eventhoughxcanP(x1<X<16ProbabilityandProbabilityProbabilityfunction:p(x)=P(X=Fordiscreterandom0≤p(x)≤ Probabilitydensityfunction(p.d.f): P(X P(X≤x x17ProbabilityandProbabilityProbabilityDistributionofaDiscreteRandomVariableProbabilityMassFunction(PMF)orProbabilityFunction(PF) xi xi), Propertiesofthe1.1. xi 0, 2. f(xi

Forn=3p=

Binomial:n=3,p=3.f3.f(xi x

X18 x2ProbabilityandProbabilityProbabilitydensityfunction(PDF)between60and680XAPDFhasthefollowing HeightinThetotalareaunderthecurvef(x)P(x1<X<x2)istheareaunderthecurvebetweenx1andP(x1Xx2)P(x1Xx2)P(x1Xx2)P(x1X19ProbabilityandProbabilityP(a≤X≤b)=F(b)–0abxP(a≤X≤b)=Areaunderf(x)betweenaandb=F(b)–0abx2021ProbabilityandProbabilityPropertiesofF(- 0and F(X)isanon-decreasingfunctionsuchthatifx2>x1thenF(x2)P(X≥ 1–P(x1≤X≤ F(x2)–F(X) FRroliycfafrmst8utionsMultivariateprobabilitydensity 1212We 1212Definition:f(X,Y)=P(X=xandPropertiesofthebivariateorjointprobabilitymassfunctionf(X,Y)≥0forallpairsofXandY.Thisisbecauseallprobabilities∑∑f(X,Y)22ProbabilityandProbabilityMarginalprobabilitydistributionofXandValueofValueof1122f(X) ff(X) f(X,Y)forally f(X,Y)forallx23ProbabilityandProbability XY123123124ExampleTheThejointprobabilitydistributionofrandomvariablesXandYisbyf(x,y)=kxyforx=1,2,3,y=1,2,3andkisapositiveconstant,istheprobabilitythatX+YwillexceedCannotbeCorrectAnswer?B25ExampleHalfofthemortgagesHalfofthemortgagesinaportfolioareconsideredsubprime.Theprincipalbalanceofhalfofthesubprimemortgagesandone-quarterofthenon-subprimemortgagesexceedsthevalueofthepropertyusedascollateral.Ifyourandomlyselectamortgagefromtheportfolioforreviewanditsprinciplebalanceexceedsthevalueofthecollateral,whatistheprobabilitythatitisasubprimemortgage?CorrectAnswer?D26Basic27CFA MomentandCentralExpectedSampleMeanandCoskewnessand28CentralThek-thmomentofXisdefinedas: EIf 1,then E[X],itistheCentralKThek-thcentralmomentofXisdefinedas:KEIf 1,thefirstcentralmomentisequaltoIf 2,thesecondcentralmomentistheIf 3,thenthethirdcentralmomentdividedbythecubethestandarddeviationistheIfk 4,thenthefourthcentralmomentdividedbythesquareofthevarianceisthekurtosis.29ExpectedExpectedAmeasureofcentraltendency–thefirst P(xi)xiP(x1)x1P(x2)x2...P(xn PropertiesofExpectedIfbisaconstant,E(b)=Ifaisaconstant,E(aX)=Ifaandbareconstants,thenE(aX+b)=aE(X)+E(b)=aE(X)+E(X2)≠E(X+Y)=E(X)+Ingeneral,E(XY)≠E(X)E(Y);IfXandYareindependentrandomvariables,thenE(XY)=E(X)E(Y).302E2E2EE2EμXnii1Ameasureofdispersion–thesecondAboveformulaisthedefinitionofvariance.Tocomputethevariance,weusethefollowingformula:xThepositivesquarerootofσ2,σx,isknownasthex31PropertiesofIfcisconstant,then: Ifaisconstant,then: Ifbisaconstant,then:σ2(X+ Ifaandbareconstant,then:σ2(aX+ IfXandYareindependentrandomvariablesandaandbareconstants,thenσ2(aX+bY) a2σ2(X)+b2σ2(Y). E(X2)–32SampleThesamplemeanofarandomvariable,X,isdefinedThesamplemeanisknownasanestimatorofE(X),whichwenowcallthepopulationAnestimateofthepopulationissimplythenumericalvaluetakenbyanestimator.33CFAFRa leiac籍，添 Thesamplevariance,denotedbyS2whichisanestimatorofσ2 whichwecannowcallthepopulationvariance.TheSn XS xi1nTheexpression(n–1)isknownasthedegreesofIfthesamplesizeisreasonablylarge,wecandividebyninsteadof(n–1).XS(thepositivesquarerootofS2),iscalledthesampleXx34SampleMeanandμN(yùn)Xi NXnXi nNi XNμ2nii nX21σs35Chebyshev’sForanysetofobservations(samplesorpopulation),theproportionofthevaluesthatliewithinkstandarddeviationsofthemeanisatleast1–1/k2,k>1.Thisrelationshipappliesregardlessoftheshapeof23423411134411189911PPX 1,1

36CovCovX, XE YE E EXECovariancemeasureshowonerandomvariablemoveswithanotherrandomvariable.CovariancerangesfromnegativeinfinitytopositivePropertiesofIfXandYareindependentrandomvariables,theircovarianceisCovX, EXE XE 2 bX, bcCov(X,Therelationshipbetweencovariancevariance:σ2(XrY)=σ2(X)+σ2(Y)r37CorrelationCorrelationhasnounits,rangesfrom-1toCorrelationmeasuresthelinearrelationshipbetweentworandomvariables.Iftwovariablesareindependent,theircovarianceiszero,therefore,thecorrelationcoefficientwillbezero.Theconverse,however,isnottrue.Forexample,YX2.σ2(XrY)σ2(X)+σ2(Y)r38Correlationr=perfectpositive0<r<positivelinearr=nolinear-1<r<r=-perfectnegativeperfect perfect perfectcorrelationr=+1correlationr=0.8correlationr=

perfect perfectcorrelationr=0.7correlationr=ρCovρCovσx39Mst NN wiwjCov(Ri,Rji1j40AmeasureofasymmetryofaPDF–thethirdSymmetricalandnonsymmetricalPositivelyskewed(rightskewed)andnegativelyskewed(leftNegative-MeanMedian

Mean=Median=

Positive-ModeMedianE(RE(RPwiE(Ri inRNwimarketvalueofinvestmentinassetmarketvalueoftheSEX3x3thirdmomentaboutmeancubeofstandardKEμxfourthEX22 squareofsecondxPositiveskewed:Mode<median<mean,havingarightfatNegativeskewed:Mode>media>mean,havingaleftfat41AmeasureoftallnessorflatnessofaPDF–thefourthForanormaldistribution,theKvalueis kurtosis–>=<Excess>=<(assumingsamefatthin42

FatAAleptokurticdistributionhasmorefrequentextremelylargefromthemeanthananormal43CoskewnessandThethirdcrosscentralmomentisreferredtoasThefourthcrosscentralmomentisreferredtoasRiskmodelswithtime-varyingvolatilityortime-varyingcandisplayawiderangeofbehaviorswithveryfewfreeCopulasc sobeusedtodescribecomplexinteractionsbetweenvariablesthatgobeyondcovariances,and epopularinriskmanagementinrecent44CoskewnessandABCD1234567A+ABCD1234567A+C+1234567Thetwoportfolios(A+BandC+D)havethesamemeanandstandarddeviation,buttheskewsoftheportfoliosaredifferent.45EXμx2Yμσ2XEXμx3YμXFonScatterplotsshowthedifferencebetweenBversusAandDversusAandB:theirbestpositivereturnsoccurduringthesametimeperiod,buttheirworstnegativereturnsoccurindifferentperiods.Thiscausesthedistributionofpointstobeskewedtowardthetop-rightofthechart.CandD:theirworstnegativereturnsoccurinthesameperiod,buttheirbestpositivereturnsoccurindifferentperiods.Inthesecondchart,thepointsareskewedtowardthebottom-leftofthechart.46CoskewnessandAandCandAandCand--Thenontrivialcoskewnessoftwovariables:SXXYandSXYYForexampleThenontrivialcokurtosisoftwovariables:KXXXY澝KXXYYandFor47BestLinearUnbiasedEstimatorAnotherpropertyofapointestimateislinearity.Apointestimateshouldbealinearestimator(i.e.,itcanbeusedasalinearfunctionofthesampledata).Iftheestimatoristhebestavailable(i.e.,hastheminimumvariance),exhibitslinearity,andisunbiased,itissaidtobethebestlinearunbiasedestimatorAlloftheestimatorsthatweproducedinthischapterforthemean,variance,covariance,skewness,andkurtosisareeitherBLUEortheratioofBLUEestimators.48ExampleSupposeSupposethatAandBarerandomvariables,eachfollowsanormaldistribution,andthecovariancebetweenAandBis0.35.isthevarianceof(3A+Correct49ExampleGivenGiventhatxandyarerandomvariables,anda,b,canddarewhichoneofthefollowingdefinitionsisE(ax+by+c)=aE(x)+bE(y)+c,ifxandyareσ2(ax+by+c)=σ2(ax+by)+c,ifxandyareCov(ax+by,cx+dy)=acσ2(x)+bdσ2(y)+(ad+bc)Cov(x,y),ifxandyarecorrelated.σ2(x–y)=σ2(x+y)=σ2(x)+σ2(y),ifxandyareCorrectAnswer?B50ExampleWhichoneofthefollowingstatementsWhichoneofthefollowingstatementsaboutthecorrelationcoefficientisfalse?Italwaysrangesfrom-1toAcorrelationcoefficientofzeromeansthattworandomvariablesareindependent.ItisameasureoflinearrelationshipbetweentworandomItcanbecalculatedbyscalingthecovariancebetweentworandomvariables.CorrectAnswer?B51p P Cxpx1pp P Cxpx1pnn px1pnxx! x52DiscreteProbabilityBernoulliBinomialPoissonContinuousProbabilityContinuousUniformNormalLognormalOtherCommonlyusedProbabilityChiSquaretFParametricandNonparametricMixture53Binomial 1–TheprobabilityofxsuccessesinnBernoullirandompp(1–Binomialrandomnp(1–54SomeImportantProbabilityTheBinomialDistribution–p= p= p=n=n=

p P λkek! npBinomialp P λkek! np=55PoissonPoissonWhentherearealargenumberoftrialsbutasmallprobabilityofsuccess,Binomialcalculations eimpractical.Ifwesubstituteλ/nforp,andletnverylarge,theBinomial esthePoissonDistribution.Xreferstothenumberofsuccessperλindicatestherateofoccurrenceoftherandomevents;i.e.,itlsushowmanyeventsoccuro ageperunitoftime.Thenumberoffishcaughtinaday;thenumberofpotholeson1kmstretchofroad;thenumberof sappearedina mall;thenumberofphonecallsinaday.56Poisson ThesumofindependentPoissonvariablesisafurtherPoissonvariablewithmeanequaltothesumoftheindividualmeans.ThePoissonDistributionisthelimitingcaseoftheBinomialDistributionasngoestoinfinityandpgoestozero,whilenpλremainsfixed.Inaddition,whenλislargethePoissonDistributioniswellapproximatedbytheNormalDistributionwithmeanandvarianceofλ,throughthecentrallimittheorem.57CFAFRo Ibiitfari968butionsAcompanyreceivesthreecomplaintsperdayoage.Whatistheprobabilityofreceivingmorethanonecomplaintonaparticularday?λ 2or3or4orP(‘morethan P(2)+P(3)+P(4)+P(‘morethanone’) 1–{P(0)+P(1)} e3?30/0! e3?31/1! P(0)+P(1) 1–{P(0)+ 1– 58ContinuousUniform 59ContinuousUniform (a+ (b–Foralla≤x1<x2≤b,weTherandomvariableXwithdensityfunction≤8,and0otherwise.Calculateits

k/3for2≤1 0a 0xabaforxaforaxbforxP fx x1 b60NormalAsnincreases,thebinomialdistributionapproachesNormalf -1x-uf e22ThenormalcurveisxtendsThecurveextendstoxtends

ThecueThecurveThecue61NormalX~N(μ,σ2),fullydescribedbyitstwoparametersμand 0; Alinearcombination(function)oftwo(ormore)normallydistributionrandomvariablesisitselfnormallydistributed.Thetailsgetthinandgotozerobutextend 62NormalApproximay68%ofallobservationsfallintheintervalμrσApproximay90%ofallobservationsfallintheintervalApproximay95%ofallobservationsfallintheintervalApproximay99%ofallobservationsfallintheinterval-

-

- +1.65+1.9663

+2.58 Xμ XμN(yùn)Standardization:ifX~N(μ,σ2),σZ-HowweusethestandardnormaldistributiontoExample:X~N(70,9),computetheprobabilityofX≤ZX 64.1270- P(Z≤- Question1:computetheprobabilityofX≥Question2:computetheprobabilityof64.12≤X≤64TheStandardNormal65ExampleLetZbeastandardnormalrandomvariable,andeventXisdefinedtohappenifeitherZtakesavaluebetween-0.5and+0.5orZtakesanyLetZbeastandardnormalrandomvariable,andeventXisdefinedtohappenifeitherZtakesavaluebetween-0.5and+0.5orZtakesanyvaluegreaterthen1.5.WhatistheprobabilityofeventXhappeningifN(0.5)=0.6915andN(-1.5)=0.0668,whereN(.)isthecumulativedistributionfunctionofastandardnormalvariable?Correctanswer?C66ExampleWhichofthefollowingstatementWhichofthefollowingstatementaboutthenormaldistributionisnotSkewnessequalsTheentiredistributioncanbecharacterizedbytwomoments,meanandvariance.Thenormaldensityfunctionhasthefollowingf(x) exp2212x2Correct67ExampleWhichWhichtypeofdistributionproducesthelowestprobabilityforavariabletoexceedaspecialextremevaluewhichisgreaterthanthemean,assumingthedistributionallhavethesamemeanandvariance?AleptokurticdistributionwithakurtosisofAleptokurticdistributionwithakurtosisofAnormalAplatykurticCorrectAnswer?D68ExampleA$50millionprudentfund(PF)ismergedA$50millionprudentfund(PF)ismergedwitha$200millionaggressivefund(AF).ThereturnofPF~N(0.03,0.072)andthereturnofAF~N(0.07,0.152).Seniormanageraskedyoutoestimatethelikelihoodthatthereturnsofthecombinedportfoliowillexceed26%.Assumingthereturnsareindependent,whatistheprobabilitythatthereturnwillexceed26%?CorrectAnswer?C69f(x) exp 1lnxx222,xlnX~NEf(x) exp 1lnxx222,xlnX~NE 1σ2 2D exp2μ expn(n20i Z (kLognormalTheBlack-ScholesModelassumesthatthepriceoftheunderlyingIflnXisnormal,thenXislognormal;ifavariableislognormal,itsnaturallogisnormal.ItisusefulformodelingassetpriceswhichnevertakenegativeRightBoundedfrombelowby70Chi-SquareChi-Square(2)Chi-Squareteststatistic,2,withn–1degreesoffreedom,iscomputedas:df=71Chi-SquareTheChi-Squaredistributiontakeonlypositivevalueandrangesfrom0toinfinity(afterall,itisthedistributionofasquaredTheChi-Squaredistributionisapositiveskeweddistribution,thedegreeoftheskewnessdependingonthed.f.Forcomparativelyfewd.f.thedistributionishighlyskewedtotheright,butasthed.f.increase,thedistribution increasinglysymmetricalandapproachesthenormaldistribution.E(X)=k,D(X)=2k,wherekistheIfZ1andZ2aretwoindependentChi-Squarevariableswithk1andk2d.f.,thentheirsum(Z1+Z2)isalsoaChi-Squarevariablewithd.f.=(k1+k2).72tRecallthat,ZXμX~N0,1,bothμXandσ2arenσ nXxSupposeweonlyknow andestimateσ2byitsx X ,weobtainanewtXμtXμX~Sx/nExplainthed.f.(degreesof Beforewecomputethe (andhence ),wemust ScomputeX.ButsinceweusethesamesampletocomputeX,wehave(n-1),notn,independentobservationstocompute2Sxsotospeak,welose73tThemeanoftdistributioniszero,anditsvariancen/(n–Thevarianceoftdistributionislargerthanthevarianceofthestandardnormaldistribution,sotdistributionisflatterthanthenormaldistribution,butasnincreases,thevarianceoftdistributionapproachesthevarianceofthestandardnormaldistribution,namely1.74T-distribution75F-ii F-IfU1andU2aretwoindependentChi-Squareddistributionswithk1andk2degreesoffreedom,respectively,thenX: U1k1~FkU2 1followsanF-distributionwithparametersk1andAsd.f.increase,theF-DistributionapproachesNormalX2~X2~F1,76F-F-

ProbabtyProbabtyDensTheFdistributionforvariousSkewedtotherightandalsorangesbetween0andApproachestheNormalDistributionask1andk2,thed.f.elarge.77ExampleTheannualmarginalprobabilityofdefaultofabondis15%inyearTheannualmarginalprobabilityofdefaultofabondis15%inyear1and20%inyear2.Whatistheprobabilityofthebondsurviving(i.e.noCorrectAnswer?Probability(nodefault)=(1–15%)?(1–20%)=78ExampleOnOnamultiplechoiceexamwithfourchoicesforeachofsixwhatistheprobabilitythatastudentgetslessthantwocorrectsimplybyCorrectAnswer?Dp(X=0)=(3/4)6=Theprobabilityofgettinglessthantwoquestionscorrectis+p(X=1)=79Example

AcallcenterreceivesageoftwoAcallcenterreceivesprobabilitythattheywillreceive20callsinan8-hourdayisclosestCorrectAnswer?Tosolvethisquestion,wefirstneedtorealizethattheexpectednumberofphonecallsinan8-hourdayis16.UsingthePoissondistribution,wesolvefortheprobabilitythatXwillbe20.

80ExampleIfIfwesaythatcommoditypricefollowalognormaldistribution,meanthatoverThenaturallogarithmofthepriceisnormallyThechangeinthepriceisnormallyThechangeinthenaturallogarithmofthepriceisnormallydistributedovertime.ThereciprocalofthepriceisnormallyCorrectAnswer?Arandomvariablehasalognormaldistributionifitslogarithmisitselfnormallydistributed.81Mitii Thedistributionthatresultsfromaweightedaveragedistributionofdensityfunctionsisknownasamixturedistribution.Moregenerally,wecancreateadistribution:??σ??????σ???? ??wherethevariousfi(x)'sareknownasthecomponentorweights.82Mixture83HypothesisTestsandConfidence84SamplingandPointEstimation澝ConfidenceIntervalTheCentralLimitPropertiesofpointHypothesisThebasisofTheapplicationof85SampleandDescriptivestatistics:Summarizetheimportantcharacteristicsoflargedatasets.Inferentialstatistics:Makeforecasts,estimates,orjudgmentsaboutalargesetofdataonthebasisofthestatisticalcharacteristicsofasmallerset(asample). 86StatisticalInference:EstimationandHypothesisStratifiedrandomsampling:toseparatethepopulationintosmallergroupsbasedononeormoredistinguishingcharacteristics.StratumandcellsM?N.samplingerrorofthemeansamplemean–populationThesamplestatisticitselfisarandomvariableandhasaprobabilitydistribution.87CtiithcIfX1,X2, arandomsamplefromanypopulation(i.e.,probabilitydistribution)withmeanμandσ2,thesamplemeanXtendstobenormallydistributedwithXmean

andσ2/nxasthesamplesizeincreases y)Ofcourse,iftheXihappentobefromthenormalpopulation,thesamplemeanfollowsthenormaldistributionregardlessofthesamplesize. StandardError(SE)ofmeanX:SE However,thepopulation’sstandarddeviationisalmostneverknown.Instead,weusethestandarddeviationofthesample88PropertiesofpointThemeanoftheestimatorscoincideswiththetrueparametere.g.E(X)Anunbiasedestimatorisalsoefficientifthevarianceofitssamplingdistributionissmallerthltheotherunbiasedestimatorsoftheparameteryouaretryingtoestimate.

X~N(X,2Theaccuracyoftheparameterestimateincreasesasthesamplesizeincreases(seethestandarderror).

n

Xin89PointEstimationandConfidenceIntervalUsingasinglenumericalvaluetoestimatetheparameterof

Degreeofconfidence(1–

Thetdistributionfor27ConfidenceIntervalstandarderror]

[PointEstimate+/-(reliabilityfactor)XXS2X90ConfidenceConfidenceIntervalThepopulationhasanormaldistributionwithaknownXX α Point StandardThepopulationhasanormaldistributionwithaunknownConfidenceWhensamplingformsmallsample(n<largersample(n≥Normaldistribution ownzzNormaldistributionwithunknownttStatisticorzNonnormaldistributionwitownnotzNonnormaldistributionwithnottStatisticorz91StatisticalInference:EstimationandHypothesisEstimationandHypothesisTesting:TwinBranchesOfexchange(NYSE)TCMean=23.25Variance=90.13Standarddeviation=92StatisticalInference:EstimationandHypothesis X x~t(n1) P(-2.052 2.052)

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criticaltObtainaRandom

Thetdistributionfor27PX2.052 X2.052SX

Xα/Xα/snnn Importantpoint:oneshouldsaythattheprobabilityis0.95thatrandomintervalcontainsthetrueμX.93eicfmpothesisUsingsampleestimator,obtainingaconfidenceinterval.ReturningtoourP/Eexample,wehaven 28,X23.25,and 9.49.Weobtain19.57≤μX≤26.93asthe95%confidenceintervalforμX.X2.052SXnXX2.052SXn94HypothesisStep Step StepStatenullStatenullalternativeIdentifythetestSelectalevelFormulateTakeasample,arriveFormulateTakeasample,arriveatdecisionDonot95HypothesisStatisticalassessmentofastatementoridearegardingapopulationThenullhypothesis(H0)andalternativehypothesisOne-tailedtestvs.Two-tailedH0:μH0:μ≥ Ha:μ<H0:μ≤ Ha:μ>Two-tailedHH0:μ= Ha:μ≠CriticalThedistributionofteststatistic(z,t,2,SignificancelevelOne-tailedortwo-tailed96StatisticalInference:EstimationandHypothesis H0:μX H1:μX≠18.5 (samplestatistic–hypothesizederrorofthesample X)~

Obtaincriticaltvaluet(n 297StatisticalInference:EstimationandHypothesis Theregionof

Reject

FailtoReject

Reject

FailtoReject

RejectFailtorejectH0if|teststatistic|<criticalvalue.Wecanneversay“accept”H0.98StatisticalInference:EstimationandHypothesisTheRuleofPPvalueisthesmallestlevelofsignificanceforwhichthenullcanbeReturntoourP/E X (23.25 P

RejectH0ifthep-valueislessthanthesignificancelevelofthehypothesistest.DonotrejectH0ifthep-valueisgreaterthanthesignificance99F-tn NegativeofCriticalvalueforPositiveoftestsignificancetestIfP-value<alpha,werejectnull100TestofSinglePopulationH0:μ=z-testvs.t-Normalpopulation,n<n≥Knownpopulationvariancez-z-Unknownpopulationt-t-testorz-z-101TestofSinglePopulationH0:σ2=TheChi-Square

(α=0.05,df=

0 RejectH0FailtoRejectH0Reject0σ 0zxzxσ/txμs/(n20 n

F1F11 112 22103TestofVariancesH0:σ12=TheF- Alwaysputthelargervarianceinthenumerator( s2 Therejectionregionisalwaystheright-sidetail,nomattertheone-tailedortwo-tailed.104F-Distribution105CFAFRu rpefaf9688population,knownpopulationvariance0zxσ/Normallydistributedpopulationvariance0txs/t(n-Normally2202n1202nTwoindependent2122Fs2 Fn11,n2106TypeIandTypeIITypeIRejectthenullhypothesiswhenit’sactuallyTypeIITheprobabilityofmakingaTypeIerror:Significancelevel P(TypeIerror)PowerofaTheprobabilityofcorrectlyrejectingthenullhypothesiswhenisPowerofa 1–P(TypeII107TypeIandTypeIITrueH0isH0isDonotrejectSignificancelevel,=1–P(Type=P(TypeI108ExampleConsiderConsiderastockwithaninitialpriceof$100.ItspriceoneyearnowisgivenbyS=100?er,wheretherateofreturnrisnormallydistributedwithameanof0.1andastandarddeviationof0.2.With95%confidence,afterrounding,Swillbebetween:$67.57and$70.80and$74.68and$102.18andCorrectAnswer?The95%confidenceintervalforris-0.292tor=0.1–0.2?1.96=-0.2920orr=0.1+(0.2?1.96)=The95%confidenceintervalforSis$74.68toS=100?e-0.292=74.68orS=100?e0.492=109ExampleAccordingtotheBaselback-testingframeworkguidelines,penaltiesstarttoapplyistherearefiveormoreexceptionsduringtheprevious year.TheTypeIerrorrateofthis 11percent.AccordingtotheBaselback-testingframeworkguidelines,penaltiesstarttoapplyistherearefiveormoreexceptionsduringtheprevious is97percentofexceptionsinsteadoftherequired99percent, powerofthe 87percent.Thisimpliesthatthereisa(an): 89%89%probabilityregulatorswillrejectthecorrect87%probabilityregulatorswillnotrejectthecorrect13%probabilityregulatorswillnotrejecttheincorrectCorrectAnswer?D110ExampleWhichofthefollowingstatementsWhichofthefollowingstatementsregardinghypothesistestingisTypeIIerrorreferstothefailuretorejectthenullhypothesisitisactuallyHypothesistestingisusedtomakeinferencesabouttheparametersofagivenpopulationonthebasisofstatisticscomputedforasamplethatisdrawnfromthatpopulation.Allelsebeingequal,thedecreaseinthechanceofmakingaTypeIerrorcomesatthecostofincreasingtheprobabilityofmakingaTypeIIerror.Thep-valuedecisionruleistorejectthenullhypothesisifthevalueisgreaterthanthesignificanceCorrectAnswer?D111xl4 WhatWhatdoesahypothesistestatthe5%significancelevelP(notrejectH0|H0istrue)=P(notrejectH0|H0isfalse)=P(rejectH0|H0istrue)=P(rejectH0|H0isfalse)=CorrectAnswer?C112ExampleSunstarisamutualfundwithastatedobjectiveofcontrollingvolatility,asmeasuredbythestandarddeviationofmonthlyreturns.Giventheinformationbelow,Sunstarisamutualfundwithastatedobjectiveofcontrollingvolatility,asmeasuredbythestandarddeviationofmonthlyreturns.Giventheinformationbelow,