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statistics informed decisions using data
Foundations Of Statistics For Data Scientists With R And Python 1st Edition Alan Agresti - Solutions
Explainthelogicunderlyinginvertingpermutationteststoobtainaconfidenceintervalforthe difference betweentwopopulationmeans.Illustratethemethodbyfindingthe95%confidence intervalforthedifferencebetweenmeaninteractiontimesforpettingandpraiseofdogsusing the datain Section 5.8.2. Comparewithresultsfroma t
RefertoExercise5.44anditsscenarioof n1 = n2 = 4 with y1 = 5 and y2 = 10.(a) Constructtwo scenarios suchthatthetwo-sidedpermutationtestcomparingmeanswould have(i) P-value < 0.05, (ii) P-value > 0.05.(b) Createascenariosuchthatthe P-valueforthepermutationtestwouldbetheminimum possiblebutthe
Explainthelogicunderlyingthepermutationtesttocomparetwodistributions.Compareits assumptions withthoseofthetwo-sample t test.
ExplainwhytheconfidenceintervalbasedontheWaldtestof H0: θ = θ0 is symmetricaroundˆθ (i.e., havingcenterexactlyequalto ˆθ. Thisisnottruefortheconfidenceintervalsbasedon the likelihood-ratioandscoretests.)Explainwhysuchsymmetrycanbeproblematicwhen θand ˆθ are
Foralargenumber n of independentPoissonrandomvariables {Yi}, with μ = E(Yi), consider testing H0: μ = μ0.(a) Showthatthescoreteststatisticis Z =ºn( ¯ Y − μ0)~ºμ0.(b) Showthatthe Waldteststatisticis Z =ºn( ¯ Y − μ0)~º¯ Y . Under H0, why wouldyou
Fortwocategoricalvariables X and Y , let πij = P(X = i, Y = j), i = 1, ...,r, j = 1, ...,c.Consider H0: πij = P(X = i)P(Y = j) for all i and j with n observationshavingcellcounts{yij}. Usingthemultinomialdistributionfor {yij} and theMLestimatefor πij of ˆπij,0 =(yi+~n)(y+j~n) under H0 and
Fora c-category variable,considertesting H0: π1 = π10, ...,πc = πc0 when counts (y1, ...,yc)haveamultinomialdistribution(2.14)with n = Σj yj .(a) UsingtheresultthattheMLestimateof πj is the jth sampleproportion yj~n, showthat the likelihood-ratiostatisticfortesting H0 iswith df = c−1 for
Derivethelikelihood-ratiotestof H0: π1 = π2 for independent Y1 ∼ binom(n1, π1) and Y2 ∼binom(n2, π2).
Adataanalystassumesthatthe n independentobservationsinadatafilecomefromaPoisson distribution.(a) Derivethelikelihood-ratiostatisticfortesting H0: μ = μ0 against Ha: μ ≠ μ0.(b) Simulatetheexactdistributionoftheteststatisticin(a)for n = 25, when μ0 = 3.0.(Section A.5.2 of the R
Whenarticlesinthemassmediaaboutmedicalstudiesreportlargedangersofcertainagents(e.g., coffeedrinking),laterresearchoftensuggeststhattheeffectsaresmallerthanfirstbe-lieved,ormaynotevenexist.Explainwhy.
Somejournalspublishresearchresultsonlyiftheyachievestatisticalsignificanceatthe0.05α-level.Explain publicationbias and itsdangers.
Inanevaluationof32schoolsinacountyoverthepastfiveyearsaccordingtothemean score ofseniorstudentsonastandardizedachievementtest,onlyoneschoolperformedabove the medianinallfiveyears.Explainwhatismisleadingaboutaconclusionthatthatschool
Aresearchstudyconducts40significancetests.Ofthese,onlytwoaresignificantatthe0.05 level.Theauthorswriteareportaboutthosetworesults,notmentioningtheother38tests.Explain whatismisleadingabouttheirreport.
Aresearcherconductsasignificancetesteverytimesheanalyzesanewdataset.Overtime, she conducts100significancetests,eachatthe0.05level.If H0 is trueineverycase,whatis the probabilitydistributionofthenumberoftimessherejects H0, andhowmanytimeswould weexpect H0 to berejected?
Youplantotest H0: μ1 = μ2. Whenyourresearchhypothesisisthat μ1 > μ2, ifyouarecorrect, explain whyyouwillhavegreaterpowerifyouuse Ha: μ1 > μ2 instead of Ha: μ1 ≠ μ2.
to explainwhy P(TypeII error) increasestoward0.95as μ decreases toward0.
Fortesting H0: μ = 0 against Ha: μ > 0 with α = 0.05, use Figure
as π gets closertothe H0 valueof0.50.(c) Set π = 0.60. Report P(TypeIIerror)for n equal to(i)50,(ii)100,(iii)200,and summarize theimpactof n on P(TypeIIerror).
with samplesize100, and set P(TypeIerror) = α = 0.05. Theappshowsthenullsamplingdistributionof ˆπ and the actual samplingdistributionof ˆπ for varioustruevaluesof π. Clickon Show TypeIIerror, and it alsodisplays P(TypeIIerror).(a)
and Ha: π >
Usethe ErrorsandPower app at www.artofstat.com/web-apps to investigatetheperformance of significancetests.Setthehypothesesas H0: π =
Medicaltestsfordiagnosingconditionssuchasbreastcancerarefallible,justlikedecisions in significancetests.Identify(H0 true, H0 false) withdisease(absent,present),and(Reject H0, Donotreject H0) withdiagnostictest(positive,negative).Inthiscontext,explainthe difference
Criminaldefendantsareconvictedifthejuryfindsthemtobeguilty“beyondareasonable doubt.”Ajuryinterpretsthistomeanthatifthedefendantisinnocent,theprobabilityof beingfoundguiltyshouldbeonly1inabillion.Describeanydisadvantagethisstrategyhas.
Foramatched-pairs t test (Exercise5.16),let σ2 = var(Yi1) = var(Yi2) and ρ = corr(Yi1, Yi2).Using theresultfromExercise2.63thatfortworandomvariables Y1 and Y2, var(Y1 − Y2) =var(Y1) + var(Y2) − 2cov(Y1, Y2), showthatvar(Y 1 − Y 2) = var(d) = 2σ2(1 − ρ)~n. Explainhow this indicates that
Refertothepreviousexerciseandthe P-valueof0.057.(a) Explainwhythe P-valueisthesmallest α-levelatwhich H0 can berejected.(b) Explainwhythe94.3%confidenceintervalisthenarrowestconfidenceintervalfor μ that contains μ0 = 100.
level.
Arandomsampleofsize40has y = 120. The P-valuefortesting H0: μ = 100 against Ha:μ ≠ 100 is 0.057. Explainwhatisincorrectabouteachofthefollowinginterpretationsofthis P-value,andprovideaproperinterpretation.(a) Theprobabilitythat H0 is correctequals0.057.(b) Theprobabilitythat y = 120 if H0 is
at μ = 4. Then:(a) At μ = 5, β > 0.36.(b) If α = 0.01, thenat μ = 4, β > 0.36.(c) If n = 50, thenat μ = 4, β > 0.36.(d) Thepowerofthetestis0.64at μ = 4.(e) Thismustbefalse,becausenecessarily α + β = 1.
Let β denote P(TypeIIerror).Foran α = 0.05-leveltestof H0: μ = 0 against Ha: μ > 0 with n = 30 observations, β =
Weanalyzewhetherthedischargeofarsenicintheliquideffuentfromanindustrialplant exceeds thecompanyclaimofameanof10 mg perliter.Forthedecisionintheone-sidedtest using α = 0.05:(a) Iftruly μ = 10, withprobability0.05wewillconcludethattheyareexceedingthelimit.(b) Iftruly μ = 10,
Resultsof99%confidenceintervalsformeansareconsistentwithresultsoftwo-sidedtestswith which α-level?Explaintheconnection.Selectthecorrectresponse(s)inthenexttwoexercises.(Morethanonemaybecorrect.)
Fortesting H0: P(Y = j S X = i) = P(Y = j) for all i and j, thatis, homogeneity of the conditional distributionsofa c-category responsevariableatthe r categories ofanexplanatory variable,comparethenumberofparametersunder Ha and under H0 to find df for thechi-squared test.
FortheBayesianmodelforcomparingmeansin Section 5.3.4, explainwhythepriorand posterior P(μ1 = μ2) = 0.
Constructtwoscenariosofindependentsamplesoffourmenandfourwomenwith y = number of hoursspentonInternetinpastweekhaving y1 = 5 and y2 = 10, suchthatfor testing H0 ∶μ1 = μ2 against Ha ∶ μ1 ≠ μ2, (a) P-value < 0.05, (b) P-value > 0.05. Howdothewithin-groups variabilitydifferinthetwocases?
Explainwhy the terminology“donotreject H0” ispreferableto“accept H0.”
Small P-valuesindicatestrongevidenceagainst H0, becausethedatawouldthenbeunusual if H0 weretrue.Whydoesitnotmakesensetodefinea P-valueastheprobabilitythatthe test statisticequalsthe observedresult (when H0 is true)ratherthanasthetailprobability that theteststatisticequalsthe
Abook44 on methodsformodelingsurvivaltimesdiscussedanexamplecomparingtimesof remission (inweeks)ofleukemiapatientstakingadrugorcontrol.Thedata,withcensored observationsindicatedbythe“+”sign,are:Treatment: 6+, 6, 6, 6, 7, 9+, 10+, 10, 11+, 13, 16, 17+, 19+, 20+, 22, 23, 25+, 32+, 32+, 34+,
Fortheexamplein Section 5.8.4, thesubjecttakingthedrugwhowascensoredafter4months is nowfoundtohavehadasurvivaltimeof11months.(a) Conductasignificancetestofidenticalsurvivaldistributions.Interpretthe P-value.(b) Giveanexampleofanadditionalchangeinthedatathatwouldhavenofurthereffecton the testresult.
and showtherelationtothe95%confidence intervalforthecomparison.(b) Sincethedistributionofsellingpriceseemsskewedright,conductapermutationtestto compare thepopulationmedians.Interpret.
The Houses data fileatthebook’swebsitelists,for100homesalesinGainesville,Florida, severalvariables,includingthesellingprice(inthousandsofdollars)andwhetherthehouseis new (1 = yes,0 = no).(a) Basedongraphicalornumericaldescriptivestatistics,useanappropriatemethodwith the t distribution tofindthe
Refertothepettingversuspraiseofdogsexamplein Section 5.8.2.(a) Forthe14timesobserved,showthepartitioningofthevaluestothetwogroupsforwhich the P-valuewouldbesmallest.Whatisthat P-value?(b) Forthispartitioning,usethebootstraptoconstructa95%confidenceintervalforthe difference
Section 5.3.2 used a t test tocomparecognitivebehavioralandcontrolgroupsforanorexia patients.Usingsimulationwithsoftwareorwiththe Permutation Test app at www.artofstat.com/web-apps, conducttheanalogouspermutationtestcomparing(a) means,(b) meanranks(Wilcoxontest).Statehypotheses,interpret
Foran α = 0.05-levellikelihood-ratiotestof H0: θ = θ0 using thelikelihoodfunctionvaluesℓ(ˆθ) and ℓ0 = ℓ(θ0), explainwhythecorresponding95%confidenceintervalfor θ is thesetofθ0 for which ℓ(ˆθ)~ℓ(θ0) ≤ exp(3.84~2) = 6.8, where3.84isthe0.95quantileforachi-squared distribution
Aremanymedical“discoveries”actuallyTypeIerrors?Inmedicalresearch,suppose43 that an actualpopulationeffectexistsonly10%ofthetimeandthatwhenaneffecttrulyexists, the chanceofmakingaTypeIIerrorandfailingtodetectit(perhapsbecauseofinsufficient sample
JonesandSmithseparatelyconductstudiestotest H0: μ = 500 against Ha: μ ≠ 500, eachwith n = 1000. Jonesgets y = 519.5, with se = 10.0. Smithgets y = 519.7, with se = 10.0.(a) Showthat the P-valueis0.051forJonesand0.049forSmith.Using α = 0.050, foreach study
against Ha: π ≠ 0.50, each with n = 400. Jonesgets ˆπ = 220~400 = 0.550. Smithgets ˆπ = 219~400 = 0.5475.(a) Showthatthe P-valueis0.046forJonesand0.057forSmith.Using α = 0.05, indicatein eachcasewhethertheresultis“statisticallysignificant.”Usingthis,explainthemisleading
JonesandSmithseparatelyconductstudiestotest H0: π =
Section 5.5.5 mentionedastudyaboutwhetherastrologerscanpredictthecorrectpersonality chartforagivenhoroscopebetterthanbyrandomguessing.(a) Inthewordsofthatstudy,whatwouldbea(i)TypeIerror?(ii)TypeIIerror?(b) Ifwedecrease α from 0.05to0.01,towhatvaluedoes P(TypeIIerror)increase?
Astudy42 compared populationdynamicsofthethreatenedspeciesSootyFalcononFahal Island andtheDaymaniyatislandsintheSeaofOmanduring2007-2014.Theclutchsizeshad mean andstandarddeviation2.660and0.618onFahalIsland(n = 100) and2.920and0.787 on theDaymaniyatislands(n = 53).
The2018General SocialSurveyasked1136subjectswhethertheybelieveinheavenandwhether they believeinhell.Ofthem,804said yes to both,209said no to both,113said yes to heaven and no to hell,and10said no to heavenand yes to hell.(a) Showthedataina2×2 contingencytable.Denotethepopulationproportionssaying
Astudyof100womensufferingfromexcessivemenstrualbleedingconsiderswhetheranew analgesic providesgreaterreliefthanthestandardanalgesic.Ofthewomen,40reportedgreater relief withthestandardanalgesicand60reportedgreaterreliefwiththenewone.Testthe
The Afterlife data fileatthebook’swebsiteshowsdatafromthe2018GeneralSocialSurvey on postlife = beliefintheafterlife(1 = yes,2 = no), religion(1 = Protestant,2 = Catholic, 3 =Jewish, othercategoriesexcluded),andgender(1 = male, 2 = female). Analyzethesedatawith
Forthedatainthe PartyID data fileatthebook’swebsite,usesignificancetestingandesti-mation methodstoanalyzetherelationshipbetweenpoliticalpartyaffiliationandrace.
Usingthe GSS2018 data file,crossclassifythe2016voteforPresident(PRES16, with1 = Clinton, 2 = Trump,3 = Other, 4 = Never)bysex(1 = male, 2 = female).(a) Formthecontingencytableandreporttheconditionaldistributionsonthevote.(b)
Withthedataintheexamplein Section 5.4.5, conductandinterpretthePearsonchi-squared test (a) comparingdivorced/separatedwithnevermarriedonhappiness;(b) comparingmarried with divorced/separatedandnevermarriedcombined.(Thesumofthetwo X2 statistics, with df = 2 +2 = 4, approximatelyequals X2 for
Fortheexamplein Section 5.4.5, interpretthestandardizedresidualsforthe not toohappy category.
Usethe Happy data filefromthe2018GeneralSocialSurveyatthetextwebsitetoforma contingencytablethatcrossclassifieshappinesswithgender.For H0: independencebetween happiness andgender:(a) Conductandinterpretthechi-squaredtest.(b) Showtheestimatedexpectedfrequenciesandstandardizedresiduals,andformamosaic
The Substance data fileatthebook’swebsiteshowsacontingencytableformedfromasurvey that askedasampleofhighschoolstudentswhethertheyhaveeverusedalcohol,cigarettes, and marijuana.Findthe P-valuefortestingwhetheradifferenceexistsbetweenthosewho
InExercise5.4,133ofthe429identifyingasRepublicansand429ofthe487identifyingas Democratsstatedclimatechangeisamajorthreat.Showhowtodisplaytheresultsina contingencytable,andusechi-squaredtotestwhetheropinionisindependentofpolitical party.Interpret.
Refertothepreviousexercise.Usinguniformpriordistributions,findtheposteriorprobability that thepopulationproportionbelievinginlifeafterdeathishigherforfemalesthanformales.
Inthe2018GeneralSocialSurvey,whenaskedwhethertheybelievedinlifeafterdeath,1017 of 1178femalessaid yes, and703of945malessaid yes. Testthatthepopulationproportions are equalforfemalesandmales.Reportandinterpretthe P-value.
Usingthe Anorexia data fileatthetextwebsite:(a) Test H0: μ1 = μ2 against Ha: μ1 ≠ μ2 for theweightchangeswiththefamilyandcognitive behavioraltherapies.Reportandinterpretthe P-value,andgivethedecisionfor α = 0.05.If thedecisionisinerror,whattypeoferrorisit?(b)
Ideally,resultsofastatisticalanalysisshouldnotdependgreatlyonasingleobservation.Ina sensitivity study, were-dotheanalysisafterdeletinganoutlierfromthedatasetorchangingits valuetoamoretypicalvalueandcheckingwhetherresultschangemuch.Fortheanorexiadata analysis in Section 5.3.2,
Anexperiment40 used asampleofcollegestudentstoinvestigatewhethercellphoneuseimpairs drivers’reactiontimes.Onamachinethatsimulateddrivingsituations,atirregularperiods a targetflashedredorgreen.Participantswereinstructedtopressabrakebuttonassoonas
cm (63.7 inches)forwomen,withstandarddeviationabout7 cm for eachgroup.For all finishersintheBostonMarathonsince1972,thetimetofinishhasameanof221minutes for
Arecentreport39 estimated meanadultheightsintheU.S.of175.4 cm (69.1 inches)formen and
The Income data fileatthebook’swebsiteshowsannualincomesinthousandsofdollarsfor subjectsinthreeracial-ethnicgroupsintheU.S.(a) Statingallassumptionsincludingtherelativeimportanceofeach,showallstepsofasig-nificance testforcomparingpopulationmeanincomesofBlacksandHispanics.Interpret.(b)
FromGSSresultsat sda.berkeley.edu/archive.htm, politicalideology(POLVIEWS,with 1 = extremely liberaland7 = extremely conservative)hadmeanandstandarddeviationby politicalpartyidentification(PARTYID)(3.74,1.39)forthe229strongDemocratsin1974,(4.76,
Theexamplein Section 3.1.4 describedanexperimenttoestimatethemeansaleswitha proposedmenuforanewrestaurant.Inarevisedexperimenttocomparetwomenus,onTuesday of theopeningweektheownergivescustomersmenuAandonWednesdayshegivesthemmenu B. Thebillsaverage$22.30forthe43customersonTuesday(s = 6.88)
Usedescriptivestatisticsandsignificanceteststocomparethepopulationmeanpoliticalide-ology foreachpairofgroupsin Table5.2 using the Polid data file.Summarizeresultsusing P-valuesandusinganon-technicalexplanation.
AstudyofsheepmentionedinExercise1.27analyzedwhetherthesheepsurvivedforayear from theoriginalobservationtime(1 = yes,0 = no) asafunctionoftheirweight(kg) atthe original observation.Statinganyassumptionsincludingtheconceptualpopulationofinterest, use a t test withthedatainthe Sheep data
than P-value = 0.4173545.
Theoutputofa t significance testreportsa P-valueof0.4173545.Insummarizingthetest, explain whyitismoresensibletoreport P-value =
Forthe Students data fileatthetextwebsite,analyzepoliticalideology.(a) Testwhetherthepopulationmean μ differs from4.0,themoderateresponse.Reportthe P-value,andinterpret.Makeaconclusionusing α-level = 0.05.(b) Constructthe95%confidenceintervalfor μ. Explainhowresultsrelatetothoseofthe test in(a).
Youwanttoknowwhetheradultsinyourcountrythinktheidealnumberofchildrenisequal to 2,ontheaverage,orhigherorlowerthanthat.(a) Definingnotation,state H0 and Ha for investigatingthis.(b) SoftwareshowstheseresultsforresponsesinarecentGSStothequestion,“Whatdoyou think
to determinewhichstatehasgreaterevidencesupportinga Republican victory.Explainyourreasoning.(b) ConductaBayesiananalysistoanswerthequestionin(a)byfindingineachcasethe posterior P(π < 0.50), correspondingtothe P-valuein(a).Usebeta(50,50)priors,which
against Ha: π >
BeforeaPresidentialelection,pollsaretakenintwoswingstates.TheRepublicancandidate waspreferredby59ofthe100peoplesampledinstateAandby525of1000sampledinstate B. Treattheseasindependentbinomialsamples,wheretheparameter π is thepopulation proportionvotingRepublicaninthestate.(a)
Inthescientifictestofastrologydiscussedin Section 5.5.5, theastrologerswerecorrectwith 40 oftheir116predictions.Test H0: π = 1~3 against Ha: π > 1~3 to analyzewhethertheir predictions werebetterthanexpectedmerelybychance.Findthe P-value,makeadecision using α = 0.05, andinterpret.
Theexamplein Section 5.2.2 could notdeterminewhetheramajorityorminorityofAmericans consider climatechangetobeamajorthreat.However,thisisclearerforparticulargroups.A surveybyPewResearchCenterinMarch2020reportedthat88%ofthe487Democrats and
Same-sexmarriagewaslegalizedacrossCanadabytheCivilMarriageActenactedin2005.Is this supportedbyamajority,oraminority,oftheCanadianpopulation?Ina2017surveyof 3402 Canadians(https://sondage.crop.ca), 73%supported the act.Analyzetheresultswith a significancetest,statinganyassumptions,andinterpretthe
against Ha: π ≠ 0.50. Interpretthe P-value.Isitappropriateto“accept H0? Whyorwhynot?
Whenagovernmentdoesnothaveenoughmoneytopayfortheservicesthatitprovides,it can raisetaxesoritcanreduceservices.WhentheFloridaPollaskedarandomsampleof1200 Floridians whichtheypreferred,52%(624ofthe1200)chose raisetaxes and 48%chose reduce services. Let π denote
Introducingnotationfor a parameter,statethefollowinghypothesesintermsoftheparameter valuesandindicatewhetheritisanullhypothesisoranalternativehypothesis.(a) TheproportionofalladultsintheUKwhofavorlegalizedgamblingequals0.50.(b)
Refertothepreviousexercise.Fortheabsolute-errorlossfunction L(θ, ˆθ) = Sθ − ˆθS, showthattheBayesestimatorof θ is themedianoftheposteriordistribution.
Abasicelementin statisticaldecisiontheory is the loss function for astatisticaldecision.In thecontextofestimation,acommonlossfunctionforanestimator ˆθ of aparameter θ is the squared-error loss, L(θ, ˆθ) = (ˆθ − θ)2.The lossfunctionreferstoasinglesample,andtoevaluate ˆθ,
Apointestimator ˆθ is locationinvariant if forallpossibledataandallconstantsc, when weadd c to eachobservation,theestimatorincreasesbyc. Itis scaleinvariant if whenwe multiplyeachobservationby c > 0, theestimatormultipliesby c.(a) Showthat Y is locationinvariantand S is scaleinvariant.(b)
Explainhowusingastandardnormalpivotalquantityintheformwouldleadtoa one-sided 95% confidenceintervalfor θ. UsethisideatogetaWald95%lowerconfidenceboundforthepopulationproportionsupportinglegalizationofmarijuanain Exercise 4.5. T(Y)-8 T(Y)-0 P
Forthemultinomialdistribution(2.14)withcounts {yj} in c categories satisfying Σj yj = n, Bayesianmethodsoftenusethe Dirichlet distribution as apriordistributionfor (π1, ...,πc}, p(π1, ...,πc;α1, ...,αc) ∝ πα1−1 1 πα2−1 2 ⋯παc−1 c , 0 ≤ πj ≤ 1, Σjπj = 1, for
Fora continuousdistribution,explainwhythenumberofobservationsthatfallbelowthe populationmedianhasabinom(n, 0.50) distribution.Ifweordertheobservationsinmagnitude, giving the orderstatistics Y(1) ≤ Y(2) ≤ ⋯ ≤ Y(n), explainwhytheprobabilityisabout0.95that the interval (Y(a), Y(b))
Themomentgeneratingfunctionofa χ2 d random variableis md(t) = (1 − 2t)−d~2 for t < 1~2.Supposethat U and V are independent,with U ∼ χ2 d1 and U + V ∼ χ2 d1+d2 . FromExercise3.45, mU+V (t) = mU(t)mV (t). Showthat V ∼ χ2 d2 .
Thefamilyofprobabilitydistributionshaving pdf of form f(y; θ) = B(θ)h(y) exp[Q(θ)R(y)]is calledthe exponentialfamily. The naturalexponentialfamily is thespecialcase Q(θ) = θand R(y) = y, with θ called the naturalparameter.(a)
Let Z denote astandardnormalrandomvariable,whichhas pdf ϕ(z) = (1~º2π) exp(−z2~2)and cdf Φ. Recallthat Y = Z2 has a χ21 distribution.(a) Explainwhythe cdf of Y for y ≥ 0 is F(y) = Φ(ºy) − Φ(−ºy).(b) Takingthederivative, showthatthe pdf of Y is24 f(y) = (1~ºy)[ϕ(ºy) + ϕ(−ºy)]
Refertothedefinitionsofchi-squaredand T random variablesin Section 4.4.5.(a) Fromtherepresentation (Z2 1 +⋯+Z2d) with standardnormals,explainwhythe χ2 d distribution hasmean d.(b) Fromtherepresentation T = Z~»X2~d, explainwhythe t distribution convergestothe standard normalas d → ∞. (Hint:
Theoryexiststhatjustifiessubstitutingtheestimatedstandarderrorforthetrueoneinforming a pivotalquantity.Hereweshowthisforaproportion.(a) The continuousmappingtheorem states thatcontinuousfunctionspreservelimits eveniftheirargumentsaresequencesofrandomvariables.Inparticular,if Xn p→c and if g() is
UsetheMarkovinequality(Exercise2.50)toshowthatas n → ∞:(a) If ˆθ is mean-squared-errorconsistent in thesensethat E(ˆθ − θ)2 → 0, then ˆθp→θ.(b) If E(ˆθ) → θ and var(ˆθ) → 0, then ˆθp→θ. (Here,useMSE(ˆθ) = var(ˆθ) + (bias)2.)
LongbeforeRonaldFisher proposedthemaximumlikelihoodmethodin1922,in1894Karl Pearsonproposedthe methodofmoments. Thisexpressesthemeanoftheprobability distribution intermsoftheparameterandequatesittothesamplemean.Fortwoparameters, youequatethefirsttwomoments.Let Y1, ...,Yn be n
estimates π.(c) Usingthisapproach,200subjectsareaskedwhethertheyhaveeverknowinglycheated on theirincometax.Reporttheestimateof π if thenumberofreportedheadsequals(i)50, (ii)100. TABLE 4.5 Joint probabilities for randomized response outcomes. Second Response First Coin Head Tail Head 0.25 0.25
Inestimatingtheprobability π of the yes responseonasensitivequestion,themethodof randomizedresponse can beusedtoencouragesubjectstomakeresponses.Thesubjectis askedtoflipacoin,insecret.Ifitisahead,thesubjecttossesthecoinoncemoreandreports the
to showtheinfluenceof ρ on thelikelyvaluesfor P(D S +).
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