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statistics informed decisions using data

Questions and Answers of Statistics Informed Decisions Using Data

InExercise5.4,133ofthe429identifyingasRepublicansand429ofthe487identifyingas Democratsstatedclimatechangeisamajorthreat.Showhowtodisplaytheresultsina
Refertothepreviousexercise.Usinguniformpriordistributions,findtheposteriorprobability that thepopulationproportionbelievinginlifeafterdeathishigherforfemalesthanformales.
Inthe2018GeneralSocialSurvey,whenaskedwhethertheybelievedinlifeafterdeath,1017 of 1178femalessaid yes, and703of945malessaid yes. Testthatthepopulationproportions are
Usingthe Anorexia data fileatthetextwebsite:(a) Test H0: μ1 = μ2 against Ha: μ1 ≠ μ2 for theweightchangeswiththefamilyandcognitive behavioraltherapies.Reportandinterpretthe
Ideally,resultsofastatisticalanalysisshouldnotdependgreatlyonasingleobservation.Ina sensitivity study, were-dotheanalysisafterdeletinganoutlierfromthedatasetorchangingits
Anexperiment40 used asampleofcollegestudentstoinvestigatewhethercellphoneuseimpairs drivers’reactiontimes.Onamachinethatsimulateddrivingsituations,atirregularperiods a
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)
FromGSSresultsat sda.berkeley.edu/archive.htm, politicalideology(POLVIEWS,with 1 = extremely liberaland7 = extremely conservative)hadmeanandstandarddeviationby
Theexamplein Section 3.1.4 describedanexperimenttoestimatethemeansaleswitha proposedmenuforanewrestaurant.Inarevisedexperimenttocomparetwomenus,onTuesday of
Usedescriptivestatisticsandsignificanceteststocomparethepopulationmeanpoliticalide-ology foreachpairofgroupsin Table5.2 using the Polid data file.Summarizeresultsusing
AstudyofsheepmentionedinExercise1.27analyzedwhetherthesheepsurvivedforayear from theoriginalobservationtime(1 = yes,0 = no) asafunctionoftheirweight(kg) atthe original
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
Youwanttoknowwhetheradultsinyourcountrythinktheidealnumberofchildrenisequal to 2,ontheaverage,orhigherorlowerthanthat.(a) Definingnotation,state H0 and Ha for investigatingthis.(b)
to determinewhichstatehasgreaterevidencesupportinga Republican victory.Explainyourreasoning.(b) ConductaBayesiananalysistoanswerthequestionin(a)byfindingineachcasethe posterior P(π < 0.50),
against Ha: π >
BeforeaPresidentialelection,pollsaretakenintwoswingstates.TheRepublicancandidate waspreferredby59ofthe100peoplesampledinstateAandby525of1000sampledinstate B.
Inthescientifictestofastrologydiscussedin Section 5.5.5, theastrologerswerecorrectwith 40 oftheir116predictions.Test H0: π = 1~3 against Ha: π > 1~3 to analyzewhethertheir predictions
Theexamplein Section 5.2.2 could notdeterminewhetheramajorityorminorityofAmericans consider climatechangetobeamajorthreat.However,thisisclearerforparticulargroups.A
Same-sexmarriagewaslegalizedacrossCanadabytheCivilMarriageActenactedin2005.Is this supportedbyamajority,oraminority,oftheCanadianpopulation?Ina2017surveyof 3402 Canadians(https://sondage.crop.ca),
against Ha: π ≠ 0.50. Interpretthe P-value.Isitappropriateto“accept H0? Whyorwhynot?
Whenagovernmentdoesnothaveenoughmoneytopayfortheservicesthatitprovides,it can raisetaxesoritcanreduceservices.WhentheFloridaPollaskedarandomsampleof1200 Floridians
Introducingnotationfor a parameter,statethefollowinghypothesesintermsoftheparameter valuesandindicatewhetheritisanullhypothesisoranalternativehypothesis.(a)
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,
Apointestimator ˆθ is locationinvariant if forallpossibledataandallconstantsc, when weadd c to eachobservation,theestimatorincreasesbyc. Itis scaleinvariant if whenwe multiplyeachobservationby c >
Explainhowusingastandardnormalpivotalquantityintheformwouldleadtoa one-sided 95% confidenceintervalfor θ.
Forthemultinomialdistribution(2.14)withcounts {yj} in c categories satisfying Σj yj = n, Bayesianmethodsoftenusethe Dirichlet distribution as apriordistributionfor (π1, ...,πc}, p(π1,
Fora continuousdistribution,explainwhythenumberofobservationsthatfallbelowthe populationmedianhasabinom(n, 0.50) distribution.Ifweordertheobservationsinmagnitude, giving the orderstatistics Y(1) ≤
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)
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
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)
Refertothedefinitionsofchi-squaredand T random variablesin Section 4.4.5.(a) Fromtherepresentation (Z2 1 +⋯+Z2d) with standardnormals,explainwhythe χ2 d distribution hasmean d.(b)
Theoryexiststhatjustifiessubstitutingtheestimatedstandarderrorforthetrueoneinforming a pivotalquantity.Hereweshowthisforaproportion.(a) The continuousmappingtheorem states
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) →
LongbeforeRonaldFisher proposedthemaximumlikelihoodmethodin1922,in1894Karl Pearsonproposedthe methodofmoments. Thisexpressesthemeanoftheprobability distribution
estimates π.(c) Usingthisapproach,200subjectsareaskedwhethertheyhaveeverknowinglycheated on theirincometax.Reporttheestimateof π if thenumberofreportedheadsequals(i)50, (ii)100. TABLE 4.5 Joint
Inestimatingtheprobability π of the yes responseonasensitivequestion,themethodof randomizedresponse can beusedtoencouragesubjectstomakeresponses.Thesubjectis
to showtheinfluenceof ρ on thelikelyvaluesfor P(D S +).
and ρ =
Inadiagnostictestforadisease,asin Section 2.1.4 let D denote theeventofhavingthe disease, andlet + denote apositivediagnosisbythetest.Letthe sensitivity π1 = P(+ S D), the false positiverate π2 =
Showthatwith y successes in n binary trials,the95%scoreconfidenceinterval(4.7)for π:(a) Hasmidpoint(4.9)thatisapproximately ˜π = (y+2)~(n+4), thatis,thesampleproportion after
Bias-variancetradeoff3: For n independentweightchangesinananorexiastudy,aresearcher believesitplausiblethat μ = 0 and toestimate μ decides tousetheestimator ˜μ = cY for a constant c with 0 ≤ c
Bias-variancetradeoff2: Somestatisticalmethodsdiscussedin Chapters 7 and 8 use a“shrinkage” or“smoothing”oftheMLestimator,suchthatastheamountofshrinkagein-creases,
Bias-variance tradeoff : Forabinomialparameter π, considertheBayesianestimator (Y +1)~(n + 2) that occurswithauniformpriordistribution.(a)
For n independentobservations {yi} from aPoissondistributionwithmean μ:(a) Findthescorestatistic U(y,μ) and theinformation I(μ).(b) Showhowtofindascoreconfidenceintervalfor μ.
Forthebootstrapmethod,explainthesimilarityanddifferencebetweenthetruesampling distribution of ˆθ and theempirically-generatedbootstrapdistributionintermsofitscenter and itsspread.
Suppose y1, ...,yn are independentobservationsfromaPoissondistributionwithmeanparam-eter μ. WithaBayesianapproach,weuseagamma(k,λ) priordistributionfor μ. Then,find the posteriordistributionfor μ
Forabinom(n, π) observation y, considertheBayesestimatorof π using abeta(α, β) prior distribution.(a) Forlarge n, showthattheposteriordistributionof π has approximatemean ˆπ = y~n. (It also
Refertothepreviousexercise.SupposethattheBayesian95%posteriorintervalis(23.5,25.0).Interpret,andcomparetotheproperinterpretationofaclassical95%confidenceinterval.
and25.0years.(d) Ifwerepeatedlysampledtheentirepopulation,then95%ofthetimethepopulationmean wouldbebetween23.5and25.0years.
Arandomsampleof50recordsyieldsa95%confidenceintervalforthemeanageatfirst marriage ofwomeninacertaincountyof23.5to25.0years.Explainwhatiswrongwitheach of thefollowinginterpretations.(a)
Let Y denote astandardCauchyrandomvariable,thatis,a t-distributed randomvariablewith df = 1.(a) Fromthedefinitionofthe t distribution in Section 4.4.5, showthat Y can beexpressed as
Inthe T pivotalquantityforcomparingtwomeans,explainwhy X2 = (n1 + n2 − 2)S2~σ2 has a chi-squareddistributionwith d = n1 + n2 − 2 degrees offreedom.
Usingthedefinitionofthechi-squareddistributionin Section 4.4.5, proveits reproductive property: If X2 1 and X2 2 are independentchi-squaredrandomvariableswith X2 1 ∼ χ2 d1 and X2 2 ∼ χ2 d2 ,
Consider n independentobservationsfromanexponential pdf f(y; λ) = λe−λy for y ≥ 0, with parameter λ > 0 for which E(Y ) = 1~λ.(a) Findthesufficient statisticforestimating λ.(b)
and8.0.(d) Ifrandomsamplesofsize1467wererepeatedlyselected,theninthelongrun95%ofthe confidence intervalsformedwouldcontainthetruevalueof μ.(e)
and8.0.(b) Wecanbe95%confidentthat μ is between6.8and8.0.(c) Inthissample,ninety-fivepercentofthevaluesof y = numberofclosefriendsarebetween
Basedonresponsesof1467subjectsinaGeneralSocialSurvey,a95%confidenceintervalforthe mean numberofclosefriendsequals(6.8,8.0).Identifywhichofthefollowinginterpretations is
Selectthecorrectresponse:Otherthingsbeingequal,quadruplingthesamplesizecausesthe width ofaconfidenceintervalforapopulationmeanorproportionto(a) double,(b) halve,(c) beonequarteraswide.
Selectthecorrectresponse:Thereasonweusea zα~2 normal quantilescoreinconstructinga confidence intervalforaproportionisthat:(a)
and draw10,000randomsamplesof size 20each.Lookatthesimulatedsamplingdistributionof ˆπ, whichishighlydiscrete.Is it bell-shapedandsymmetric?UsethistohelpexplainwhytheWaldconfidenceinterval
Usesimulationoranapp,suchasthe ExploreCoverage app at www.artofstat.com/web-apps, to exploretheperformanceofconfidenceintervalsforaproportionwhen n is notverylarge and π
Explainthereasoningbehindthefollowingstatement:Studiesaboutmorediversepopula-tions requirelargersamplesizes.Illustrateforestimatingmeanincomeforallmedicaldoctors compared
Forasimplerandomsampleof n subjects,explainwhyitisabout95%likelythatthesample proportionhaserrornomorethan 1~ºn in estimatingthepopulation proportion.(Hint: To showthis“1~ºn rule,” findtwo
Explainwhyconfidenceintervalsarenarrowerwith(a) smallerconfidencelevels,(b) larger sample sizes.
Explainwhatismeantbya pivotal quantity.
TheHardy–Weinbergformulaforthegeneticvariationofapopulationatequilibriumstatesthat with probability π of the A allele, theprobabilitiesofgenotypes(AA, Aa, aa) are (π2, 2π(1 −π), (1−π)2).
Refertotheprevioustwoexercises.Considerthesellingprices(inthousandsofdollars)inthe Houses data filementionedinExercise4.31.(a) FitthenormaldistributiontothedatabyfindingtheMLestimatesof μ and σ for
Exercise2.71mentionedthatwhen Y has positivelyskeweddistributionoverthepositivereal line, statisticalanalysesoftentreat log(Y ) as havinga N(μ, σ2) distribution. Then Y has the
Consider n independentobservationsfroma N(μ, σ2) distribution.(a) Focusingon μ, findthelikelihoodfunction.Showthatthelog-likelihoodfunctionisa concave,parabolicfunctionof μ. FindtheMLestimator
For n independentobservations {yi} from thegamma pdf (2.10), withshapeparameter k known(suchaswiththeexponentialdistributionspecialcase):(a) FindthelikelihoodfunctionandshowthattheMLestimatorof λ is
Exercise2.73introducedthe Paretodistribution, whichhas pdf f(y;α) = α~yα+1 for y ≥ 1 and a parameter α > 0. Astudyusesthisfamilytomodel n independentobservationsonincome.Find
if n = (i) 1,(ii)5,(iii) 10.Howdoestheshapeof L(λ) changeas n increases? Whataretheimplications of thenarrowinglog-likelihood?(Also,whenthelog-likelihoodisclosertoparabolic,the distribution
and λ =
Forindependentobservations y1, ...,yn from theexponential pdf f(y; λ) = λe−λy for y ≥ 0, for which μ = σ = 1~λ:(a) Findthelog-likelihoodfunction L(λ).(b) FindtheMLestimator ˆλof λ.(c)
Forindependentobservations y1, ...,yn havingthegeometricdistribution(2.1):(a) Findasufficientstatisticfor π.(b) DerivetheMLestimatorof π.
When f is acontinuous pdf with median M, forsimplerandomsamplingthesamplemedianÃM has approximatelya N(M, 1~4[f(M)]2n) sampling distribution.(a) Considersamplingfromanormalpopulation,forwhich μ =
Section 4.4.6 showedthat E(s2) = σ2. Forindependentsamplingfroma N(μ, σ2) distribution,ˆσ2 = [Σi(Yi−Y )2]~n is theMLestimatorand ˜σ2 = [Σi(Yi−Y )2]~(n+1) is theestimatorhaving minimum
Isthefollowingstatementtrueorfalse?If ˆθ is anunbiasedestimatorfor θ, thenforanyfunction g, g(ˆθ) is anunbiasedestimatorfor g(θ). Iftrue,explainwhy.Iffalse,giveacounter-example.
Usingsimulation,conductaninvestigationofwhetherthesamplemeanormedianisabetter estimator ofthecommonvalueforthemeanandmedianofauniformdistribution.
Take n = 100 observationsfromastandardnormaldistributionandfindthesamplemeanand median. Dothis100,000timesandplottheirestimatedsamplingdistributions.Estimatetheir mean
The Afterlife data fileatthebook’swebsiteshowsdatafromthe2018GeneralSocialSurvey on postlife = beliefintheafterlife(1 = yes,2 = no) andreligion(1 = Protestant,2 = Catholic, 3 =Jewish,
The Houses data fileatthebook’swebsitelists,for100homesalesinGainesville,Florida, severalvariables,includingthesellingpriceinthousandsofdollarsandwhetherthehouse is new(1 = yes,0 = no).
Whena2015PewResearchsurvey(www.pewresearch.org) asked Americans whetherthereis solid evidenceofglobalwarming,92%ofDemocratswhoidentifiedthemselvesasliberalsaid yes whereas
Astudythat Section 7.3.2 discusses aboutendometrialcanceranalyzedhowahistologygrade responsevariable(HG = 0, low;HG = 1, high)relatestothreeriskfactors,includingNV =neovasculation(1 = present,0 =
Refertothepreviousexercise.ConductBayesianmethodswithimproperpriorstoobtain a posteriorintervalforthepopulationmeanweightchangeforthefamilytherapyandthe difference
RefertoExercise4.13.ConsiderBayesianinferenceforthepopulationmeanweightchange μfor thefamilytherapy.(a) Selectdiffusepriordistributionsfor μ and σ2 and findtheposteriormeanestimateand a
Refertotheclinicaltrialexamplein Section 4.7.5. UsingtheJeffreyspriorfor π1 and for π2, simulateandplottheposteriordistributionof π1 −π2. FindtheHPDinterval.Whatdoesthat
Refertothepreviousexercise,regardingestimating π when y = 0 in n = 25 trials.(a) Withuniformpriordistribution,findthe95%highestposteriordensity(HPD)interval, and
RefertothevegetariansurveyresultinExercise4.6,with n = 25 and novegetarians.(a) FindtheBayesianestimateof π using abetapriordistributionwith α = β equal (i) 0.5,(ii) 1.0,(iii)
Youwanttoestimate the proportionofstudentsatyourschoolwhoanswer yes when asked whether governmentsshoulddomoretoaddressglobalwarming.Inarandomsampleof10 students,everystudentsays yes.
Ina2016PewResearchsurvey(www.pewresearch.org), the percentagewhofavoredallowing gaysandlesbianstomarrylegallywas84%forliberalDemocratsand24%forconservative Republicans. Suppose n = 300 for
The trimmedmean calculates thesamplemeanafterexcludingasmallpercentageofthe lowestandhighestdatapoints,tolessentheimpactofoutliers.Forexample,withthefunction mean(y, 0.05) applied toavariable y, R
Toillustratehowsamplingdistributionofthecorrelationcanbehighlyskewedwhenthe the truecorrelationisnear −1 or+1,constructandplotthebootstrapdistributionforthe correlation between GDP and CO2 for the
Withthedataonthenumberofyearssinceabookwascheckedout(variable C) inthe Library data fileatthetextwebsite,usethebootstraptofindandinterpretthe95% (a) percentile confidence intervalforthemedian, (b)
Astandardizedvariable Y dealing withafinancialoutcomeisconsideredtotakeanunusually extreme valueif y > 3. Find P(Y > 3) if (a) Y is modeledashavingastandardnormaldistri-bution Y ∼ N(0, 1), (b) Y

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