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statistics informed decisions using data
Foundations Of Statistics For Data Scientists With R And Python 1st Edition Alan Agresti - Solutions
Refertotheanalysesoftheendometrialcancerdatain Section 7.3.2.(a) Explainthedifferencebetweentheinterpretationsoftheclassicalone-sided P-valueof 0.0011 andtheBayesianposterior P(β1 ≤ 0) = 0.0002 that weobtainusing σ =
Suppose y = 0 at x = 10, 20, 30, 40 and y = 1 at x = 60, 70, 80, 90.(a) WhatisthetrueMLestimate ˆβ for logisticregression?Reporttheestimateandstandard error providedbyyoursoftware.Whyisthereportedstandarderrorsolarge?(b) UseaBayesianmethodortheFirthpenalizedlikelihoodmethodtofitthemodel.Compare
Fortheclinicaltrialsdatamentionedin Section 7.2.6 of 10successesin10trialsforthenew drug and5successesin10trialsforplacebo,fitthelogisticregressionmodel.(a) Whatdoessoftwarereportfor ˆβ1? Whatistheactualvalue?(b) GivetheresultoftheWaldtestof H0: β1 = 0? Comparewiththeresultofthelikelihood-ratio
Comparisonof oddsratio with ratioofproportions:(a) Accordingto ourworldindata.org, in2017theproportionofthepopulationthatdied due tosuicidesdonebyfirearmswas0.00006291(i.e.,about6.29per100,000)inthe United Statesand0.00000168(i.e.,about0.17per100,000)intheUnitedKingdom.Use these tofindan oddsratio
“Standyourground”(SYG)lawsempowerindividualstouseanyforcenecessarytodefend themselvesagainstanyonetheybelievetobeanimminentthreat.Forcourttrialsinwhichthe defendantclaimedSYG,logisticregressionwasusedtoanalyzetheprobabilityofconviction.26 The
FortheItaliansurveyintroducedin Section 7.2.5, resultsforthe27,775subjectshavingage over65areinthe Employment2 data fileatthebook’swebsite.Sampleproportionsemployed at thevariouscombinationsofvaluesofexplanatoryvariablessuggestthatthegendereffect
Fittheinteractionmodelreferredtoin Section 7.2.5 for Italianemploymentinthe Employment data file,usinglogitandidentitylinkfunctions.Ineachcase,explainhowtointerpretthe estimated gendereffect.
Forfirst-degreemurderconvictionsinEastBatonRougeParish,Louisiana,between1990and 2008, thedeathpenaltywasgivenin3of25casesinwhichaWhitekilledaWhite,in0of3 cases inwhichaWhitekilledaBlack,in9of30casesinwhichaBlackkilledaWhite,and in 11of132casesinwhichaBlackkilledaBlack.25 Construct adatafile,letting
ForindictmentsincaseswithmultiplemurdersinFloridaduringa12-yearperiod,thedeath penaltywasgivenin53of467casesinwhichaWhitekilledaWhite,in0of16casesinwhich a WhitekilledaBlack,in11of48casesinwhichaBlackkilledaWhite,andin4of143cases in whichaBlackkilledaBlack.24(a)
The SoreThroat data fileatthebook’swebsitecontainsdatafromfromastudy23 about Y =whether apatienthavingsurgeryhadasorethroatonwaking(1 = yes,0 = no) asafunction of D = duration ofthesurgery(inminutes)and T = typeofdeviceusedtosecuretheairway(1 = trachealtube,0 = laryngeal maskairway).(a)
The Soybeans data fileatthebook’swebsiteshowsresultsofanexperiment22 to comparethe proportionsofsoybeanseedsthathavehealthyradicleofatleast2 mm after beingwrapped with wetgerminatingpaperundertreatmentAof25degreesCelsiusfor24hoursandunder
The Afterlife data fileatthebook’swebsiteshowsdatafromthe2018GeneralSocialSurvey on beliefintheafterlife(postlife = 1, yes;postlife = 2, no),religion(1 = Protestant,2 = Catholic, 3 = Jewish, othercategoriesexcluded),andgender(1 = male, 2 = female). Analyzethesedata
Usethefollowingtoydatatoillustrateeffectsofgroupedversusungroupedbinarydataonthe estimates andthedevianceforlogisticregressionmodeling:x Numberoftrials Numberofsuccesses 0 4 1 1 4 2 2 4 4 Denote by M0 the null modelandby M1 the modellogit(πi) = β0 + β1xi.(a)
AstudyofsheepmentionedinExercise1.27analyzedwhetherthesheepsurvivedforayear from theoriginalobservationtime(1 = yes,0 = no) asafunctionoftheirweight(kg) atthe original observation.(a) Doesthesurvivalofthesheepseemtodependontheirweight?Ifso,howdoestheweight of
Analogouslytothepreviousexercise,randomlysample30 X observationsfromauniformin the interval(-4,4)andconditionalon X = x, 30normalobservationswith E(Y ) = 3.5x3 −20x2 + 0.5x + 20 and σ = 30. FitpolynomialnormalGLMsoflowerandhigherorderthan that
Refertothesimulationin Section 7.1.6. Randomlysample6observationsfromthesameuni-form/normal structure.Fitthenullmodel,thestraight-linemodel,andthefifth-degreepoly-nomial. Foreachmodelfit,findSSEandfindthesumofsquarederrorsaroundthetruemeans.Explain whatthisshows.
FortheScottishhillracesdata(Section 6.1.4) inthe ScotsRaces data file,itisplausiblethat the recordtimesaremorevariableathighervaluesofdistanceandclimb.(a) Usingtheidentitylinkfunction,fitthegammaGLMtothewomen’srecordtimeswith distance
For the Houses data filedescribedin Section 7.1.3, consider Y = selling price, x1 = tax bill(in dollars), and x2 = whether thehouseisnew.(a) Formthescatterplotof y and x1. DoesthenormalGLMstructureofconstantvariability in y seem appropriate?Ifnot,howdoesitseemstobeviolated?(b)
Withquantitative x and y, itcanbeinformativetoplotthedataandshowasmoothingofthe relationship withoutassuminglinearityorsomeotherfunctionalform.WithanInternetsearch, read aboutthe lowess (locallyweightedscatterplotsmoothing)methodforthis.Explainthe method,withanexample,inareportofatmost300words.
Usingformula6.13,explainheuristicallywhyvar( ˆβj) tends todecreaseas n increases.
Showthemodelmatrixfor12observationsandthreeexplanatoryvariables,thefirstofwhichis quantitative,thesecondisbinary,andthethirdhasthreecategories,suchthateachcombination of thecategoricalvariableshastwoobservations.
Tosummarizetheinteractionbetweenquantitative x and binary z, supposewecenter x at its mean andexpressthemodelas E(Y ) = β0 + β1(x − μx) + β2z + β3(x − μx) × z.Explain howtointerpret β2. Howdoesthisdifferfromtheinterpretationforthemodelwithout centering?
Thelinearmodelfortheone-waylayoutgeneralizesfortwoormorefactors.Forthetwo-way r × c layoutwithfactors A and B:(a) Writethelinearmodelformulathatassumesalackofinteraction.Explainhowtocon-struct an F statistic fortestingthatthe A main effectsarezero.(b)
Derivethe F test in Section 6.5.6 for comparingnestednormallinearmodelsasalikelihood-ratio test,usinganargumentliketheonein Section 6.4.6 for theglobal F test.
Inconductingalargenumberofsignificancetests,the false discoveryrate (FDR) isthe expectedproportionoftherejectednullhypothesesthatareerroneouslyrejected.Asimple algorithm canensureFDR ≤ α when appliedwith t independenttestsandisespeciallyuseful when
ToshowBoole’sinequality(Section 6.5.4), let B1 = E1, B2 = Ec 1 ∩ E2, B3 = Ec 1 ∩ Ec 2 ∩ E3, ...Explain why P(∪jEj) = P(∪jBj) = Σj P(Bj) ≤ Σt j=1 P(Ej).
Illustratehowbetween-groupsandwithin-groupsvariabilityaffecttheresultoftheANOVA F test of H0: μ1 = μ2 = μ3 byconstructingtwoscenarios,eachwith n1 = n2 = n3 = 4 and y2 = 10, y2 = 12, y2 = 14, suchthatonescenario has arelativelylarge P-valueandonehasarelatively small P-value.36
Supposethatthepopulationcorrelation ρ is closeto1.(a) Whatshapewouldyouexpectthesamplingdistributionof r to have?Why?(b) In1915,R.A.Fishershowedthatwithsimplerandomsampling,has anapproximate N[T(ρ), 1~(n − 3)] distribution. Usingthis,explainhowtoconstruct a confidenceintervalfor ρ. = T(r)
(a)Explainintuitivelywhyapredictionintervalfor Y is muchwiderthanaconfidenceinterval for E(Y ).(b) Explainhowpredictionintervalsfor Y at variousvaluesof x wouldlikelybeinsubstantial error if(i)thevariabilityin Y tends toincreaseas x increases, (ii) Y is highlydiscrete, suchas Y =
Considertheconfidenceinterval(6.8)for E(Y ) in thebivariatemodel.(a) Showthatat x0 = x, theintervalis the sameastheinterval y ± tα~2,n−1(s~ºn) in the formula(4.10)forthe marginal distributionof y, exceptforthe t quantilescore.(b) At x0 ≠ x, explainwhyyou can estimate E(Y ) more preciselywhen
Forindependentrandomsamplesofsizes n1 and n2 from normalpopulationswithcommon variance σ2, using Section 4.4.6 and thedefinitionofthe F distribution in Section 6.4.1, explain why S2 1 ~S2 2 has an F distribution with df 1 = n1 − 1 and df 2 = n2 − 1.
Usingthedefinitionsof T and F random variables, showthatif T has a t distribution, then T2 has an F distribution with df = 1 and df2 equal df for the t distribution.
F statistics havealternateexpressionsintermsof R2 values.(a) Showthatfortesting H0: β1 = ⋯ = βp = 0,Explain whylargervaluesof R2 yield largervalues of F.(b) Showthatfor comparing nestedlinearmodels, F = (TSS-SSE)/p SSE/[n-(p+1)] R2/p is equivalently - (1 R2)/[n (p+1)]*
Usetheequationsin Section 6.2.6 that aresolvedtofindtheleastsquaresestimatestoshow that theleastsquaresfitofamultipleregressionmodelsatisfies 1 - 5) - . - 5) + . - .).
From Section 6.1.5 and Exercise6.31,thecorrelationisa standardizedslope. Whenwefita multipleregressionmodeltostandardizedversionsof y and theexplanatoryvariables,the estimated regressionparametersarecalled standardizedregressioncoefficients.(a) Explainhowtointerpretthestandardizedcoefficientfor xj
Usinghow r2 for thebivariatemodelisbasedonthemarginalvariabilityof y and theconditional variabilityof y given x, explainwhyforaparticulartrend,we’dexpect r2 (and SrS) tobelarger when therangeof x-valuessampledislarger.
Explainwhyconditionalindependencebetween Y and X1, given X2, doesnotimplymarginal independencebetween Y and X1. Toillustratethis,sketchahypotheticalscatterplotbetween y and x1 valuesatdifferentvaluesof x2.
Foruncorrelatedrandomvariables U, V , and W, let X = U + V and Y = U + W. (a) What happenstothecorrelationbetween X and Y when weconditionon U bykeepingitfixed;(b)Explain howaspuriousassociationcanoccur.
ForrecentU.S.presidentialelections,ineachstatewealthiervoterstendtobemorelikely to voteRepublican,yetstatesthatarewealthierinanaggregatesensetendtohavesmaller proportionsvotingfortheRepublican.35 Sketchaplotofstatewidevaluesof x = median income against y =
Astudyfoundthatchildrenwhoregularlyeatbreakfastgetbettermathgradesthanthose who donoteatbreakfast.Thisresultwasbasedontheassociationbetween X = whether eat breakfast (yes,no)and Y = grade inlastmathcoursetaken.Suggestapossiblelurkingvariable and
Aresearcharticlestatesthatsubjectswhoexerciseregularlyreportedonlyhalfasmanyserious illnesses peryear,ontheaverage,asthosewhodonotexerciseregularly.Theresultssection states, “Wenextanalyzedwhetheragewasaconfoundingvariableaffectingthisassociation.”(a)
Aten-yearstudy34 of elderlypeoplefoundthatthosewhoplayedatleastoneroundofgolf everyweekweremorelikelytobealiveadecadelater.Explainhowthiscouldbespuriousby describing potentialassociationsbetweenanothervariableandbothmortalityandfrequency of playinggolf.
Youcanfitthequadraticequation E(Y ) = β0+β1x+β2x2 byfittingamultipleregressionmodel with x1 = x and x2 = x2.(a) Simulate100independentobservationsfromthemodel Y = 40.0−5.0x+0.5x2+ϵ, where X has auniformdistributionover[0,10]and ϵ ∼ N(0, 1). Plotthedataandfitthequadratic
Forthemodel E(Yi) = β0 + β1xi, considertheexpressionfor ˆβ1 in equation(6.1).(a) Specifythesamplingdistributionof ˆβ1 and showthat E( ˆβ1) = β1 and var( ˆβ1) =σ2~[Σi(xi − x)2].(b) Fromthevariance formula,explainwhymorepreciseestimationof β1 is possiblewhenthe data
FortheScottishhillracesdatawith y = women’srecordtime,wecouldimposethenatural constraintthatwhen x = distance =0, E(Y ) = 0.(a) Forthemodel E(Yi) = βxi, derivetheleastsquaresestimateof β.(b) Fitthemodeltothedata.(In R, youcanuse lm(timeW ∼ -1
Section 6.1.5 showedthatasingleoutlyingScottishraceobservationinfluenceswhetherthe correlation betweenwomen’srecordtimeandclimbis0.69or0.85. Section 5.8.3 showedthat in comparinggroups,itcanbemorerobusttouseranks.Transformthetimeandclimbvalues to
Thedistributionof X = heights(cm) ofwomenintheU.K.isapproximately N(162, 72).Conditional on X = x, suppose Y = weight(kg) hasa N(3.0 + 0.40x, 82) distribution. Simulate a millionobservationsfromthisdistribution.Approximatethecorrelation.Approximatethe regression equationforpredicting X using y.
For Y = income (thousandsofdollars)and x = education (years),randomlygenerate n = 100 observationswhen X has auniformdistributionovertheintegers10to17and,conditionalon X = x, Y has a N(−25.0 + 6.0x, 152) distribution. Usesoftwaretofindtheleastsquaresfitof the
Let β1 denote theeffectof x in theregressionequationusing x to predict y and let β∗1 denote the effectof y in theregressionequationusing y to predict x. Showthatthecorrelation r is the geometricmean(Exercise1.44)of ˆβ1 and ˆβ∗1 .
Inbivariatemodeling,supposewetreat X also asarandomvariable,with (X,Y ) havinga bivariatenormaldistributionwithcorrelation ρ. Regressionanalysisestimatestheequation E(Y S X = x) = μY + ρ(σY ~σX)x − μX).(a) Statetheequationestimatedifweinsteaduse Y to predict X.(b) If E(Y ) = β0 + β1x
Forthe bivariate normal distribution (Section 2.7.4) withcorrelation ρ, in1886FrancisGalton showedthattheconditionaldistributionof (Y S X = x) is N[μy+ρ(σY ~σX)(x−μx), σ2 Y (1−ρ2)].Showthatand useittoexplain theconceptofregressiontowardthemean. E(YX = x)-Hy x-x 0x
Ahealthysystolicbloodpressureis120 mm Hg or less.Astudytookavolunteersampleof subjectswithhigh-bloodpressureandaskedthemtowalkfor30minutesatlunchtimeeach day.Afteronemonth,theirmeansystolicbloodpressurefellfrom150to140.Explainhowthis could merelyreflectregressiontowardthemean.
and sx = sy.Showthat (ˆμi − y) = 0.70(xi − x). Explainhowa score onthefinalexamispredictedto regress towarditsmeanrelativetothedistancebetweenthemidtermexamscoreandits mean.
Withleastsquaresforthebivariatelinearmodel,equation(6.1)statesthat ˆβ0 = y − ˆβ1x.(a) Usingthisformula,show thatanalternativeexpressionforthefittedlineis (ˆμi − y) =ˆβ1(xi − x).(b) Fordataon y = final exam score and x = midterm score,suppose r =
Whenobservationsonboth x and y are standardized,showthatthebivariatepredictionequa-tion hastheform ˆμi = rxi, where r is thesamplecorrelation.
Whenthevaluesof y are multipliedbyaconstantc, fromtheirformulas,showthat sy and ˆβ1 in thebivariatelinearmodelarealsothenmultipliedbyc. Thus,showthat r = ˆβ1(sx~sy) does not dependontheunitsofmeasurement.
Analternativeparameterizationofthebivariateregressionmodelcenters x around itsmean and expressesthemodelas E(Y ) = β0 + β1(x − μx).(a) Forthisparameterization,explainhowtointerpret β0. Howdoesthisdifferfromthe interpretationforthemodelwithoutcentering?(b) Ifyoureplace μx in thisexpressionby
Showthatthecorrelation betweenavariableanditselfisnecessarily1.0.
Showthatforthenullmodel E(Yi) = β0, leastsquaresyields ˆβ0 = y.
ThestatisticianGeorgeBox,whohadanillustriousacademiccareerattheUniversityofWis-consin, isoftenquotedassaying,“Allmodelsarewrong,butsomemodelsareuseful.”Whydo youthinkthat,inpractice, (a) all modelsarewrong, (b) some modelsare not useful.
TheInternetsite www.artofstat.com/web-apps has a Guess theCorrelation app. Playthe CorrelationGame 10 times.Showthetableofguessesandactualvaluesandfindthecorrelation betweenthem.
Constructascatterplotwith5pointsthathaveacorrelationcloseto +1, andthenaddasingle pointthatchangesthecorrelationtoastrongnegativevalue.33 Explain whythissingleoutlying observationhassomuchinfluence.
The Hare data fileatthebook’swebsite32 has datafor550haresonbodymass(grams)and hind footlength(mm), bygenderofthehare.Useregressionmodelswithbodymassasthe responsevariable,interpretingcarefullyanyeffectsorinteractions.Prepareareportofatmost 300
for thepriordistributionfor β0. Showthatshrinkingβ0 far fromtheleastsquaresestimateof28.2toward0alsoforcesdramaticchangesin the otherBayesianposteriormeanestimates.(c) With τ = 10 and similarresultsaswithleastsquares,explaindifferencesbetweenthe interpretationsoftheBayesianposterior P(β1 ≤ 0)
for theeffectsoflife eventsandSESandcompareresultswithhighlydispersenormalpriorshaving τ = 10.0.Explain howtheposteriormeansfor β1 and β2 dependon τ .(b) In(a),supposeyoualsoused τ =
for β1 and β2 (and thusprecision 1~(0.20)2 = 25) yethaveahighlydispersepriorfor β0 (e.g., precision = 10−10), wecan specifya3×3 matrix A to usefortheprecision B0, wherethevaluesonthemaindiagonal are theprecisionsfor (β0, β1, β2):> A=diag(c(10^(-10),25,25))> fit.bayes
Refertothementalimpairmentexamplein Sections 6.4.2 and 6.6.2.(a) Theexamplein Section 6.6.2 used highlydispersepriors.Let’scompareresultstoin-formativepriorsbasedonsubjectivebeliefsthat β1 and β2 are notvarylarge.With the MCMCregress function in R,
FortheScottishhillsracesdata,modeltherecordtimesimultaneouslyformenandwomenby using distance,climb,andgenderasexplanatoryvariables.Prepareareportofatmost300 words,includinghighlyeditedsoftwareoutputinanappendix.Forthefinalmodelchosen,in-terpret parameterestimates,findandinterpret R2,
UselinearmodelingtoanalyzetherecordtimesformenintheScottishhillraces,withdistance and climbasexplanatoryvariables,(a) withleastsquares,(b) withBayesianmethods.Prepare a report,includinghighlyeditedsoftwareoutputinanappendix.
ConductBayesianfittingofthelinearmodelfortheScottishhillraces,usingclimbanddis-tance predictorsofwomen’srecordtimeswithouttheoutlyingHighlandFlingraceobservation.Compare theposteriormeanestimatesoftheparameterstotheleastsquaresestimates,asyou varythespreadofthenormalpriordistributions.
Apharmaceuticalclinicaltrial31 randomly assigned24patientstothreetreatmentgroups(drug A, drugC,placebo)andcomparedthemonameasureofrespiratoryability(FEV1 = forced expiratory volumein1second,inliters)after1houroftreatment.Forthe FEV data fileat the book’swebsite,fitthelinearmodelfor fev with
For72young girls sufferingfromanorexia,the Anorexia.dat file atthebook’swebsiteshows their weightsbeforeandafteranexperimentalperiod.Thegirlswererandomlyassignedto receivecognitivebehavioraltherapy(cb). familytherapy(f ), orbeinacontrolgroup(c).(a)
Refertothepreviousexample.Thedataexhibitastrongpositivecorrelationof0.84between selling priceandthetaxbill.Mightthisbeaspuriousassociation,withthesizeofthehome causally affectingbothofthem?Sketchafiguretoreflectthepotentialspuriousrelationship.Analyze
912 11732 1 no 3 365.55165430764 2 no(a) Summarizethedatawithascatterplot between y and x1, usingseparatesymbolsforthe twocategoriesof x2.(b) Fitthemodel E(Yi) = β0 + β1xi1 + β2xi2. Interprettheestimatedeffects.(c) Reportandinterpret R2 for summarizingtheglobalpredictivepower.(d)
Table6.5 showsobservationsonhomesalesinGainesville,Florida,fromthe Houses data file at thebook’swebsitefor100homes.Variableslistedaresellingprice(thousandsofdollars), size ofhouse(squarefeet),annualpropertytaxbill(dollars),numberofbedrooms,numberof
1136.3607
0.03411.651 Anova(lm(SpermTotal~CW+factor(Color) + CW:factor(Color),data=Crabs2))Anova Table(TypeIItests)Sum SqDfFvaluePr(>F)CW
cm), whichisameasureofitssize,and color (1 = dark, 2 = medium, 3 = light),whichisa measure ofadultage,darkeronesbeingolder.> summary(lm(SpermTotal~CW+factor(Color), data=Crabs2))Estimate Std.ErrortvaluePr(>|t|)(Intercept) 11.3660.63817.822
and s = 2.0. Thetwo explanatory variablesusedin the R output arethehorseshoecrab’s carapacewidth (CW, mean18.6 cm, standarddeviation
Thedataset30 Crabs2 at thebook’swebsitecomesfromastudyoffactorsthataffectsperm traits ofmalehorseshoecrabs.Aresponsevariable, SpermTotal, isthelogofthetotalnumber of sperminanejaculate.Ithas y =
Forthe Polid data filesummarizedin Table5.2, conductanANOVAtoanalyzewhethermean politicalideologyvariesbyrace.Useafollow-upmultiplecomparisonmethodwithoverall confidence level0.95toestimatedifferencesofmeansbetweenpairsofracesonpoliticalideology.Interpretresults.
Forthe UN data fileatthebook’swebsite(seeExercise1.24),constructamultipleregression modelpredictingInternetusingalltheothervariables.Usetheconceptofmulticollinearity to explainwhyadjusted R2 is notdramaticallygreaterthanwhenGDPisthesolepredictor.Compare
levelifweusetheBonferroniapproachto test thefamilyofthreeindividualeffects?Explain.(d) Aretheeffectsof tv and sport significant?Proposeanalternativemodel.
RefertothemodelfittedinthepreviousexercisetopredictcollegeGPA.(a) Test H0: β1 = β2 = β3 = 0. Reportthe P-valueandinterpret.(b) Showhowtoconductasignificancetestabouttheindividualeffectof hsgpa, adjusting for tv and sport, using α = 0.05. Interpret.(c) Istheeffectin(b)significantatthe α =
The Students data fileshowsresponsesonvariablessummarizedinExercise1.2.(a) Fitthelinearmodelusing hsgpa = high schoolGPA, tv = weeklyhourswatchingTV,and sport = weeklyhoursparticipatinginsportsaspredictorsof cogpa = college GPA.Report the predictionequation.Whatdothe P-valuessuggest?(b)
Exercise1.49gavealinkto2020U.S.statewidedataon x = percentageofpeoplewearing masks inpublicand y = percentageofpeoplewhoknowsomeonewithCovid-19symptoms.Interpretthevalueof r2 for thedata,whichis0.724,andreportthecorrelation.
Refertotheexample in Section 6.2.5 of thecrimerateinFloridacounties.(a) Explainwhatitmeanswhenwesaythesedataexhibit Simpson’sparadox. Whatcould cause thischangeinthedirectionoftheassociationbetweencrimerateandeducation when weadjustforurbanization?(b) Usingthe Florida data
Forthemodelpermittinginteractionbetweendistanceandclimbintheireffectonwomen’s record timesfortheScottishhillraces,analyzewhetheranyobservationisinfluentialinthe least squaresfit.Howdoestheparameterestimatefortheinteractiontermchangewhenyou analyze thedatawithoutit?
Refertothestudyofmentalimpairmentin Section 6.4.2.(a) Sketchafiguretoillustrateanassociationbetweenimpairmentandlifeeventsthatis spurious, explainbySES.Fittwomodelstoinvestigatewhetherthisisthecase,and summarize.(b) Section 6.4.2
Forthe Covid19 data fileatthetextwebsite:(a) Constructthetwoscatterplotsshownin Figure 6.3.(b) Findandinterpretthecorrelationbetweentimeand(i)cases,(ii) log(cases).(c) Fitthelinearmodelforthelog-transformedcountsandreportthepredictionequation.29 Explain whythepredictedcountatday x+1 equals
Inthe2000PresidentialelectionintheU.S.withDemocraticcandidateAlGoreandRepublican candidate GeorgeW.Bush,somepoliticalanalyststhoughtthatmostofthevotesinPalm BeachCounty,Florida,fortheReformpartycandidate,PatBuchanan,mayhaveactually
The Firearms2 data fileatthetextwebsiteshowsU.S.statewidedataon x = percentage of peoplewhoreportowningagunand y = firearm deathrate(annualnumberofdeaths per100,000population),from www.cdc.gov. Identifyapotentiallyinfluentialobservationfrom the
Foradvancedindustrializednations,the Firearms data fileatthetextwebsiteshowsannual homicide rates(permillionpopulation)andthenumberoffirearms(per100people),withdata takenfromWikipediaand smallarmssurvey.org.(a) Constructascatterplotandhighlightanyobservationsthatfallapartfromthegeneral trend.(b)
For theScottishhillracesdata,alinearmodelcanpredictmen’srecordtimesfromwomen’s record times.(a) Showthescatterplotandreportthepredictionequation.Predictthemen’srecordtime for theHighlandFling,forwhichtimeW=490.05minutes.(b) Findandinterpretthecorrelation.(c)
Afamilyofdistributions f(y; θ) is saidtohave monotone likelihoodratio (MLR) ifastatistic T(y) exists suchthatwhenever θ′ < θ, ℓ(θ)~ℓ(θ′) is anondecreasingfunctionof T. Forany suchfamily,themostpowerfultestof H0: θ = θ0 against Ha: θ > θ0 forms P-valuesfromvalues of T at
Foradiscretedistributionandateststatistic T with observedvalue tobs and one-sided Ha suchthatlarge T contradicts H0, mid P-value = 1 2P(T = tobs) + P(T > tobs).(a) Supposethat P(T = tj) = πj , j = 1, . ..c. Showthat E(mid P-value) = 0.50. (Hint: Show that Σj πj[(πj~2) + πj+1 + ⋯+ πc] =
find the P-valueforeachsample.Plotahistogramordensityestimate using the1,000,000 P-values.Relateittotheresultin(a).
is true.Generate1,000,000randomsamples,eachofsize n = 1500, and for Ha: π >
From Section 2.5.7, if T is acontinuousrandomvariablewith cdf F, then F(T) has theuniform distribution over[0,1].(a) Insignificancetestingwithateststatistic T, explainwhy F(T) and 1−F(T) correspond to one-sided P-values.Explainwhy 1 − F(T) also hasauniformdistributionunder
Explainwhatismeantby censored data. Giveanexampleofasituationinwhichsomeobser-vationswouldbe(a) right-censored,(b) left-censored.
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