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statistics informed decisions using data

Questions and Answers of Statistics Informed Decisions Using Data

Showthatforthenullmodel E(Yi) = β0, leastsquaresyields ˆβ0 = y.
ThestatisticianGeorgeBox,whohadanillustriousacademiccareerattheUniversityofWis-consin, isoftenquotedassaying,“Allmodelsarewrong,butsomemodelsareuseful.”Whydo youthinkthat,inpractice, (a) all
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
for thepriordistributionfor β0. Showthatshrinkingβ0 far fromtheleastsquaresestimateof28.2toward0alsoforcesdramaticchangesin the otherBayesianposteriormeanestimates.(c) With τ = 10 and
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,
Refertothementalimpairmentexamplein Sections 6.4.2 and 6.6.2.(a) Theexamplein Section 6.6.2 used highlydispersepriors.Let’scompareresultstoin-formativepriorsbasedonsubjectivebeliefsthat β1 and β2
FortheScottishhillsracesdata,modeltherecordtimesimultaneouslyformenandwomenby using distance,climb,andgenderasexplanatoryvariables.Prepareareportofatmost300
UselinearmodelingtoanalyzetherecordtimesformenintheScottishhillraces,withdistance and climbasexplanatoryvariables,(a) withleastsquares,(b) withBayesianmethods.Prepare a
ConductBayesianfittingofthelinearmodelfortheScottishhillraces,usingclimbanddis-tance predictorsofwomen’srecordtimeswithouttheoutlyingHighlandFlingraceobservation.Compare
Apharmaceuticalclinicaltrial31 randomly assigned24patientstothreetreatmentgroups(drug A, drugC,placebo)andcomparedthemonameasureofrespiratoryability(FEV1 = forced expiratory
For72young girls sufferingfromanorexia,the Anorexia.dat file atthebook’swebsiteshows their weightsbeforeandafteranexperimentalperiod.Thegirlswererandomlyassignedto
Refertothepreviousexample.Thedataexhibitastrongpositivecorrelationof0.84between selling priceandthetaxbill.Mightthisbeaspuriousassociation,withthesizeofthehome causally
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.
Table6.5 showsobservationsonhomesalesinGainesville,Florida,fromthe Houses data file at thebook’swebsitefor100homes.Variableslistedaresellingprice(thousandsofdollars), size
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
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
Forthe UN data fileatthebook’swebsite(seeExercise1.24),constructamultipleregression modelpredictingInternetusingalltheothervariables.Usetheconceptofmulticollinearity to explainwhyadjusted R2 is
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,
The Students data fileshowsresponsesonvariablessummarizedinExercise1.2.(a) Fitthelinearmodelusing hsgpa = high schoolGPA, tv = weeklyhourswatchingTV,and sport =
Exercise1.49gavealinkto2020U.S.statewidedataon x = percentageofpeoplewearing masks inpublicand y = percentageofpeoplewhoknowsomeonewithCovid-19symptoms.Interpretthevalueof r2 for
Refertotheexample in Section 6.2.5 of thecrimerateinFloridacounties.(a) Explainwhatitmeanswhenwesaythesedataexhibit Simpson’sparadox. Whatcould cause
Forthemodelpermittinginteractionbetweendistanceandclimbintheireffectonwomen’s record timesfortheScottishhillraces,analyzewhetheranyobservationisinfluentialinthe least
Refertothestudyofmentalimpairmentin Section 6.4.2.(a) Sketchafiguretoillustrateanassociationbetweenimpairmentandlifeeventsthatis spurious,
Forthe Covid19 data fileatthetextwebsite:(a) Constructthetwoscatterplotsshownin Figure 6.3.(b) Findandinterpretthecorrelationbetweentimeand(i)cases,(ii) log(cases).(c)
Inthe2000PresidentialelectionintheU.S.withDemocraticcandidateAlGoreandRepublican candidate GeorgeW.Bush,somepoliticalanalyststhoughtthatmostofthevotesinPalm
The Firearms2 data fileatthetextwebsiteshowsU.S.statewidedataon x = percentage of peoplewhoreportowningagunand y = firearm deathrate(annualnumberofdeaths per100,000population),from www.cdc.gov.
Foradvancedindustrializednations,the Firearms data fileatthetextwebsiteshowsannual homicide rates(permillionpopulation)andthenumberoffirearms(per100people),withdata takenfromWikipediaand
For theScottishhillracesdata,alinearmodelcanpredictmen’srecordtimesfromwomen’s record times.(a) Showthescatterplotandreportthepredictionequation.Predictthemen’srecordtime for
Afamilyofdistributions f(y; θ) is saidtohave monotone likelihoodratio (MLR) ifastatistic T(y) exists suchthatwhenever θ′ < θ, ℓ(θ)~ℓ(θ′) is anondecreasingfunctionof T. Forany
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 =
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
Explainwhatismeantby censored data. Giveanexampleofasituationinwhichsomeobser-vationswouldbe(a) right-censored,(b) left-censored.
Explainthelogicunderlyinginvertingpermutationteststoobtainaconfidenceintervalforthe difference betweentwopopulationmeans.Illustratethemethodbyfindingthe95%confidence
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 >
Explainthelogicunderlyingthepermutationtesttocomparetwodistributions.Compareits assumptions withthoseofthetwo-sample t test.
ExplainwhytheconfidenceintervalbasedontheWaldtestof H0: θ = θ0 is symmetricaroundˆθ (i.e., havingcenterexactlyequalto ˆθ. Thisisnottruefortheconfidenceintervalsbasedon the
Foralargenumber n of independentPoissonrandomvariables {Yi}, with μ = E(Yi), consider testing H0: μ = μ0.(a) Showthatthescoreteststatisticis Z =ºn( ¯ Y − μ0)~ºμ0.(b) Showthatthe
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}.
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
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)
Whenarticlesinthemassmediaaboutmedicalstudiesreportlargedangersofcertainagents(e.g., coffeedrinking),laterresearchoftensuggeststhattheeffectsaresmallerthanfirstbe-lieved,ormaynotevenexist.Explainwhy.
Somejournalspublishresearchresultsonlyiftheyachievestatisticalsignificanceatthe0.05α-level.Explain publicationbias and itsdangers.
Inanevaluationof32schoolsinacountyoverthepastfiveyearsaccordingtothemean score ofseniorstudentsonastandardizedachievementtest,onlyoneschoolperformedabove the
Aresearchstudyconducts40significancetests.Ofthese,onlytwoaresignificantatthe0.05 level.Theauthorswriteareportaboutthosetworesults,notmentioningtheother38tests.Explain whatismisleadingabouttheirreport.
Aresearcherconductsasignificancetesteverytimesheanalyzesanewdataset.Overtime, she conducts100significancetests,eachatthe0.05level.If H0 is trueineverycase,whatis the
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 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)
Criminaldefendantsareconvictedifthejuryfindsthemtobeguilty“beyondareasonable doubt.”Ajuryinterpretsthistomeanthatifthedefendantisinnocent,theprobabilityof
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) −
Refertothepreviousexerciseandthe P-valueof0.057.(a) Explainwhythe P-valueisthesmallest α-levelatwhich H0 can berejected.(b) Explainwhythe94.3%confidenceintervalisthenarrowestconfidenceintervalfor μ
level.
Arandomsampleofsize40has y = 120. The P-valuefortesting H0: μ = 100 against Ha:μ ≠ 100 is 0.057. Explainwhatisincorrectabouteachofthefollowinginterpretationsofthis
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
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,
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
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 ≠
Explainwhy the terminology“donotreject H0” ispreferableto“accept H0.”
Small P-valuesindicatestrongevidenceagainst H0, becausethedatawouldthenbeunusual if H0 weretrue.Whydoesitnotmakesensetodefinea P-valueastheprobabilitythatthe test statisticequalsthe observedresult
Abook44 on methodsformodelingsurvivaltimesdiscussedanexamplecomparingtimesof remission (inweeks)ofleukemiapatientstakingadrugorcontrol.Thedata,withcensored
Fortheexamplein Section 5.8.4, thesubjecttakingthedrugwhowascensoredafter4months is nowfoundtohavehadasurvivaltimeof11months.(a) Conductasignificancetestofidenticalsurvivaldistributions.Interpretthe
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)
Refertothepettingversuspraiseofdogsexamplein Section 5.8.2.(a) Forthe14timesobserved,showthepartitioningofthevaluestothetwogroupsforwhich the P-valuewouldbesmallest.Whatisthat P-value?(b)
Section 5.3.2 used a t test tocomparecognitivebehavioralandcontrolgroupsforanorexia patients.Usingsimulationwithsoftwareorwiththe Permutation Test app at www.artofstat.com/web-apps,
Foran α = 0.05-levellikelihood-ratiotestof H0: θ = θ0 using thelikelihoodfunctionvaluesℓ(ˆθ) and ℓ0 = ℓ(θ0), explainwhythecorresponding95%confidenceintervalfor θ is thesetofθ0 for which
Aremanymedical“discoveries”actuallyTypeIerrors?Inmedicalresearch,suppose43 that an actualpopulationeffectexistsonly10%ofthetimeandthatwhenaneffecttrulyexists, the
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
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
JonesandSmithseparatelyconductstudiestotest H0: π =
Section 5.5.5 mentionedastudyaboutwhetherastrologerscanpredictthecorrectpersonality chartforagivenhoroscopebetterthanbyrandomguessing.(a)
Astudy42 compared populationdynamicsofthethreatenedspeciesSootyFalcononFahal Island andtheDaymaniyatislandsintheSeaofOmanduring2007-2014.Theclutchsizeshad mean
The2018General SocialSurveyasked1136subjectswhethertheybelieveinheavenandwhether they believeinhell.Ofthem,804said yes to both,209said no to both,113said yes to heaven and no to hell,and10said no to
Astudyof100womensufferingfromexcessivemenstrualbleedingconsiderswhetheranew analgesic providesgreaterreliefthanthestandardanalgesic.Ofthewomen,40reportedgreater relief
The Afterlife data fileatthebook’swebsiteshowsdatafromthe2018GeneralSocialSurvey on postlife = beliefintheafterlife(1 = yes,2 = no), religion(1 = Protestant,2 = Catholic, 3 =Jewish,
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)
Withthedataintheexamplein Section 5.4.5, conductandinterpretthePearsonchi-squared test (a) comparingdivorced/separatedwithnevermarriedonhappiness;(b) comparingmarried with
Fortheexamplein Section 5.4.5, interpretthestandardizedresidualsforthe not toohappy category.
Usethe Happy data filefromthe2018GeneralSocialSurveyatthetextwebsitetoforma contingencytablethatcrossclassifieshappinesswithgender.For H0: independencebetween happiness andgender:(a)
The Substance data fileatthebook’swebsiteshowsacontingencytableformedfromasurvey that askedasampleofhighschoolstudentswhethertheyhaveeverusedalcohol,cigarettes, and marijuana.Findthe

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