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statistics informed decisions using data
Foundations Of Statistics For Data Scientists With R And Python 1st Edition Alan Agresti - Solutions
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 = P(+ S Dc), andthe prevalence ρ = P(D). Relevanttoapatientwhohas receivedapositivediagnosisis P(D S
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 weaddtwosuccessesandtwofailurestothedataset.(b) Hassquareofthecoefficientof zα~2 in
Bias-variancetradeoff3: For n independentweightchangesinananorexiastudy,aresearcher believesitplausiblethat μ = 0 and toestimate μ decides tousetheestimator ˜μ = cY for a constant c with 0 ≤ c ≤ 1.(a) Show that as c decreases from1to0,|bias|increasesbutvar(˜μ) decreases.(b)
Bias-variancetradeoff2: Somestatisticalmethodsdiscussedin Chapters 7 and 8 use a“shrinkage” or“smoothing”oftheMLestimator,suchthatastheamountofshrinkagein-creases, thebiasincreasesbutthevariancedecreases.Toillustratehowthiscanhappen,for Y ∼ binom(n, π), let ˜π = (Y + c)~(n + 2c)
Bias-variance tradeoff : Forabinomialparameter π, considertheBayesianestimator (Y +1)~(n + 2) that occurswithauniformpriordistribution.(a) Deriveitsbiasanditsvariance.ComparethesewiththebiasandvarianceoftheML estimator ˆπ = Y ~n.(b) ShowthatitsMSE = [nπ(1−π)+(1−2π)2]~(n+2)2. For n = 10
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 and giveitsapproximatemeanforlarge n. (Thegammaisthe conjugate
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 hasapproximatevariance ˆπ(1 − ˆπ)~n.) Interpret,andrelatetoclassicalinference.(b)
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) Ifrandomsamplesof50recordswererepeatedlyselected,then95%ofthetimethesample mean
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 aratiooftwoindependentstandardnormalrandomvariables.(b)
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 , then X2 1 +X2 2 ∼ χ2 d1+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) Findthemaximumlikelihoodestimatorof λ and of E(Y ).(c) Onecanshowthat 2λ(Σi Yi) has
and8.0.(d) Ifrandomsamplesofsize1467wererepeatedlyselected,theninthelongrun95%ofthe confidence intervalsformedwouldcontainthetruevalueof μ.(e) Ifrandomsamplesofsize1467wererepeatedlyselected,theninthelongrun95%ofthe y valueswouldfall between6.8and8.0.
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 (are)correct,andindicatewhatiswrongwiththeothers.(a) Wecanbe95%confidentthat y is between
Selectthecorrectresponse:Otherthingsbeingequal,quadruplingthesamplesizecausesthe width ofaconfidenceintervalforapopulationmeanorproportionto(a) double,(b) halve,(c) beonequarteraswide.
Selectthecorrectresponse:Thereasonweusea zα~2 normal quantilescoreinconstructinga confidence intervalforaproportionisthat:(a) Forlargerandomsamples,thesamplingdistributionofthesampleproportionisapprox-imately normal.(b) Thepopulationdistributionisapproximatelynormal.(c)
and draw10,000randomsamplesof size 20each.Lookatthesimulatedsamplingdistributionof ˆπ, whichishighlydiscrete.Is it bell-shapedandsymmetric?UsethistohelpexplainwhytheWaldconfidenceinterval performspoorlyinthiscase.(c) When ˆπ = 0, showthattheWaldconfidenceintervalis(0,0).As π
Usesimulationoranapp,suchasthe ExploreCoverage app at www.artofstat.com/web-apps, to exploretheperformanceofconfidenceintervalsforaproportionwhen n is notverylarge and π maybenear0or1,byrepeatedlygeneratingrandomsamplesandconstructingthe confidence intervals.Takethepopulationproportion π = 0.06,
Explainthereasoningbehindthefollowingstatement:Studiesaboutmorediversepopula-tions requirelargersamplesizes.Illustrateforestimatingmeanincomeforallmedicaldoctors compared toestimatingmeanincomeforallentry-levelemployeesatMcDonald’srestaurants.
Forasimplerandomsampleof n subjects,explainwhyitisabout95%likelythatthesample proportionhaserrornomorethan 1~ºn in estimatingthepopulation proportion.(Hint: To showthis“1~ºn rule,” findtwo standard errorswhen π = 0.50, andexplainhowthiscompares to twostandarderrorsatothervaluesof π.)
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). Assumeamultinomialdistribution(Section 2.6.4) for n observationswithcounts(y1, y2, y3) of
Refertotheprevioustwoexercises.Considerthesellingprices(inthousandsofdollars)inthe Houses data filementionedinExercise4.31.(a) FitthenormaldistributiontothedatabyfindingtheMLestimatesof μ and σ for that distribution.(b)
Exercise2.71mentionedthatwhen Y has positivelyskeweddistributionoverthepositivereal line, statisticalanalysesoftentreat log(Y ) as havinga N(μ, σ2) distribution. Then Y has the log-normaldistribution, whichhas pdf for y > 0, f(y; μ, σ) = [1~y(º2πσ)] exp{−[log(y) − μ]2~2σ2}.(a) For n
Consider n independentobservationsfroma N(μ, σ2) distribution.(a) Focusingon μ, findthelikelihoodfunction.Showthatthelog-likelihoodfunctionisa concave,parabolicfunctionof μ. FindtheMLestimator ˆμ.(b) Considering σ2 to beknown,findtheinformationusing(i)equation(4.3),(ii)equation(4.4).
For n independentobservations {yi} from thegamma pdf (2.10), withshapeparameter k known(suchaswiththeexponentialdistributionspecialcase):(a) FindthelikelihoodfunctionandshowthattheMLestimatorof λ is ˆλ = k~Y .(b) WhatistheMLestimator of E(Y ) = k~λ?(c) Showthatthelarge-samplevarianceof ˆλ is
Exercise2.73introducedthe Paretodistribution, whichhas pdf f(y;α) = α~yα+1 for y ≥ 1 and a parameter α > 0. Astudyusesthisfamilytomodel n independentobservationsonincome.Find thelikelihoodfunctionandtheMLestimatorof α and itsasymptoticvariance.
if n = (i) 1,(ii)5,(iii) 10.Howdoestheshapeof L(λ) changeas n increases? Whataretheimplications of thenarrowinglog-likelihood?(Also,whenthelog-likelihoodisclosertoparabolic,the distribution oftheMLestimatorisclosertonormality.)(d) Findtheinformation I(λ) using
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) Suppose y = 10. Plot L(λ) − L(ˆλ) between λ =
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 = M. Usingformula(2.8),show that ˆM has asymptoticstandarderror»π~2(σ~ºn) (for π = 3.14...).(b)
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 MSE.(a) Showthat ˜σ2 is anasymptoticallyunbiasedestimatorof σ2.(b) Showthat ˆσ2 is
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 squarederrorsaroundthecommonpopulationmeanandmedianof0.Whichseemsto
The Afterlife data fileatthebook’swebsiteshowsdatafromthe2018GeneralSocialSurvey on postlife = beliefintheafterlife(1 = yes,2 = no) andreligion(1 = Protestant,2 = Catholic, 3 =Jewish, othercategoriesexcluded).Analyzethesedatawithmethodsofthischapter.Summarize results
The Houses data fileatthebook’swebsitelists,for100homesalesinGainesville,Florida, severalvariables,includingthesellingpriceinthousandsofdollarsandwhetherthehouse is new(1 = yes,0 = no). Prepareashortreportinwhich,statingallassumptionsincluding the
Whena2015PewResearchsurvey(www.pewresearch.org) asked Americans whetherthereis solid evidenceofglobalwarming,92%ofDemocratswhoidentifiedthemselvesasliberalsaid yes whereas 38%ofRepublicanswhoidentifiedthemselvesasconservativesaid yes. Suppose n = 200 for
Astudythat Section 7.3.2 discusses aboutendometrialcanceranalyzedhowahistologygrade responsevariable(HG = 0, low;HG = 1, high)relatestothreeriskfactors,includingNV =neovasculation(1 = present,0 = absent).Usingthe Endometrial data fileatthebook’swebsite, conduct
Refertothepreviousexercise.ConductBayesianmethodswithimproperpriorstoobtain a posteriorintervalforthepopulationmeanweightchangeforthefamilytherapyandthe difference betweenthepopulationmeansforthattherapyandthecontrolcondition.Forthe difference,
RefertoExercise4.13.ConsiderBayesianinferenceforthepopulationmeanweightchange μfor thefamilytherapy.(a) Selectdiffusepriordistributionsfor μ and σ2 and findtheposteriormeanestimateand a 95%posteriorintervalfor μ.(b) Constructtheclassical95%confidenceintervalfor μ.
Refertotheclinicaltrialexamplein Section 4.7.5. UsingtheJeffreyspriorfor π1 and for π2, simulateandplottheposteriordistributionof π1 −π2. FindtheHPDinterval.Whatdoesthat intervalreflectabouttheposteriordistribution?
Refertothepreviousexercise,regardingestimating π when y = 0 in n = 25 trials.(a) Withuniformpriordistribution,findthe95%highestposteriordensity(HPD)interval, and comparewiththe95%equal-tailposteriorinterval,usingthe2.5and97.5percentiles.Whyaretheydifferent?(b) When y = 0,
RefertothevegetariansurveyresultinExercise4.6,with n = 25 and novegetarians.(a) FindtheBayesianestimateof π using abetapriordistributionwith α = β equal (i) 0.5,(ii) 1.0,(iii) 10.0.Explainhowthechoiceofpriordistributionaffectstheposteriormean estimate.(b)
Youwanttoestimate the proportionofstudentsatyourschoolwhoanswer yes when asked whether governmentsshoulddomoretoaddressglobalwarming.Inarandomsampleof10 students,everystudentsays yes. Giveapointestimateoftheprobabilitythatthenextstudent interviewedwillanswer yes, ifyouuse(a) MLestimation,(b)
Ina2016PewResearchsurvey(www.pewresearch.org), the percentagewhofavoredallowing gaysandlesbianstomarrylegallywas84%forliberalDemocratsand24%forconservative Republicans. Suppose n = 300 for eachgroup.Usinguniformpriordistributions,(a) Construct95%posteriorintervalsforthepopulationproportions.(b)
The trimmedmean calculates thesamplemeanafterexcludingasmallpercentageofthe lowestandhighestdatapoints,tolessentheimpactofoutliers.Forexample,withthefunction mean(y, 0.05) applied toavariable y, R finds the 10% trimmedmean, excluding5%ateach end.
Toillustratehowsamplingdistributionofthecorrelationcanbehighlyskewedwhenthe the truecorrelationisnear −1 or+1,constructandplotthebootstrapdistributionforthe correlation between GDP and CO2 for the UN data. (Thisismerelyforillustration,because this datasetisnotarandomsampleofnations).
Withthedataonthenumberofyearssinceabookwascheckedout(variable C) inthe Library data fileatthetextwebsite,usethebootstraptofindandinterpretthe95% (a) percentile confidence intervalforthemedian, (b) percentileandbias-correctedconfidenceintervalsfor the standarddeviation.
Astandardizedvariable Y dealing withafinancialoutcomeisconsideredtotakeanunusually extreme valueif y > 3. Find P(Y > 3) if (a) Y is modeledashavingastandardnormaldistri-bution Y ∼ N(0, 1), (b) Y instead behaveslikearatioofstandardnormalrandomvariables, in whichcase Y has
Randomlygenerate10,000observationsfroma t distribution with df = 3 and constructa normal quantileplot(Exercise2.67).Howdoesthisplotrevealnon-normalityofthedataina waythatahistogramdoesnot?
The Substance data fileatthebook’swebsiteshowsacontingencytableformedfromasurvey that askedasampleofhighschoolstudentswhethertheyhaveeverusedalcohol,cigarettes, and marijuana.Constructa95%Waldconfidenceintervaltocomparethosewhohaveusedor not usedalcoholonwhethertheyhaveusedmarijuana,using(a)
Inthe2018GeneralSocialSurvey,whenaskedwhethertheybelievedinlifeafterdeath,1017 of 1178femalessaid yes, and703of945malessaid yes. Construct95%confidenceintervals for thepopulationproportionsoffemalesandmalesthatbelieveinlifeafterdeathandforthe difference betweenthem.Interpret.
Usingthe Students data file,forthecorrespondingpopulation,constructa95%confidencein-terval(a) forthemeanweeklynumberofhoursspentwatchingTV;(b) tocomparefemalesand males onthemeanweeklynumberofhoursspentwatchingTV.Ineachcase,stateassumptions, including
Sections 4.4.3 and 4.5.3 analyzed datafromastudyaboutanorexia.The Anorexia data file at thetextwebsitecontainsresultsforthecognitivebehavioralandfamilytherapiesandthe controlgroup.Usingdataforthe17girlswhoreceivedthefamilytherapy:(a)
The Income data fileatthebook’swebsiteshowsannualincomesinthousandsofdollarsfor subjectsinthreeracial-ethnicgroupsintheU.S.(a) Useagraphicsuchasaside-by-sideboxplot(Section 1.4.5) tocompareincomesofBlack, Hispanic, andWhitesubjects.(b)
TheobservationsonnumberofhoursofdailyTVwatchingforthe10subjectsinthe2018GSS who identifiedthemselvesasIslamicwere0,0,1,1,1,2,2,3,3,4.(a) Constructandinterpreta95%confidenceintervalforthepopulationmean.(b) Supposetheobservationof4wasincorrectlyrecordedas24.Whatwouldyouobtainfor the
ArecentGeneralSocialSurveyaskedmalerespondentshowmanyfemalepartnerstheyhave had sexwithsincetheir18thbirthday.Forthe131malesbetweentheagesof23and29,the median = 6 andmode = 1 (16.8%ofthesample).Softwaresummarizesotherresults:Variable nMeanStDevSEMean95.0%CI NUMWOMEN
Astudyinvestigatesthedistributionofannualincomeforheadsofhouseholdslivinginpublic housing inChicago.Forarandomsampleofsize30,theannualincomes(inthousandsof dollars) areinthe Chicago data fileatthetextwebsite.(a) Basedonadescriptivegraphic,describetheshapeofthesampledatadistribution.Find and
ToestimatetheproportionoftrafficdeathsinCalifornialastyearthatwerealcoholrelated,de-termine thenecessarysamplesizefortheestimatetobeaccuratetowithin0.04withprobability 0.90. BasedonresultsofstudiesreportedbytheNationalHighwayTrafficSafetyAdministra-tion (www.nhtsa.gov), we
AsocialscientistwantedtoestimatetheproportionofschoolchildreninBostonwholivein a single-parentfamily.Shedecidedtouseasamplesizesuchthat,withprobability0.95,the error wouldnotexceed0.05.Howlargeasamplesizeshouldsheuse,ifshehasnoideaofthe size ofthatproportion?
Inarandomsampleof25people to estimatetheextentofvegetarianisminasociety,0peo-ple werevegetarian.Constructa99%Waldconfidenceintervalforthepopulationproportion of vegetarians.Explaintheawkwardissueindoingthis,andfindandinterpretamorereli-able confidenceinterval.(As Section 4.3.3
TheGeneralSocialSurveyhasaskedrespondents,“Doyouthinktheuseofmarijuanashould bemadelegalornot?”Viewresultsatthemostrecentcumulativedatafileat sda.berkeley.edu/archive.htm byenteringthevariablesGRASSandYEAR.(a) Describeanytrendyouseesince1973inthepercentagefavoringlegalization.(b)
Forthe Students data file(Exercise1.2in Chapter 1) andcorrespondingpopulation,findthe ML estimateofthepopulationproportionbelievinginlifeafterdeath.ConstructaWald95%confidence interval,usingitsformula(4.8).Interpret.
with y = 6 for n = 10 (e.g.,in R bytaking pi (π) asasequencebetween0 and 1andthentakingLasthelogof dbinom(6, 10,pi)). Identify ˆπ in theplotandexplain whymaximizationoccursatthesamepointasintheplotof ℓ(π) itself.
Plotthelog-likelihoodfunction L(π) correspondingtothebinomiallikelihoodfunction ℓ(π)shownin Figure
Forasequenceofobservationsofabinaryrandomvariable,youobservethegeometricrandom variable(Section 2.2.2) outcomeofthefirstsuccessonobservationnumber y = 3. Findandplot the likelihoodfunction.
Forapoint estimate ofthemeanofapopulationthatisassumedtohaveanormaldistribution, a datascientistdecidestousetheaverageofthesamplelowerandupperquartilesforthe n = 100 observations,sinceunlikethesamplemean ¯ Y , thequartilesarenotaffectedbyoutliers.Evaluate the precisionofthisestimatorcomparedto ¯
RefertoExercise3.45andthe mgf m(t) = eμt+σ2t2~2 for anormaldistribution.For n indepen-dentobservations {Yi} from anarbitrarydistributionwithmean μ and variance σ2, let m(t)bethe mgf of thestandardizedrandomvariable Zi = (Yi − μ)~σ.(a) Explainwhythe mgf of ºn(Y − μ)~σ is [m(t~ºn)]n.(b)
RefertoExercise2.66and the momentgeneratingfunction m(t) = EetY , analternativeto the probabilityfunctionforcharacterizingaprobabilitydistribution.Forindependentrandom variables Y1, Y2, …, Yn, let T = Y1 + ⋯+ Yn.(a) Explainwhy T has mgf determined bytheseparateonesas m(t) =
and π2 = 0.10. Simulateamillionsample proportionsfromeachtreatmentwith n1 = n2 = 50. Constructhistogramsfor ˆπ1~ˆπ2 and for log(ˆπ1~ˆπ2). Whichseemstohavesamplingdistributionclosertonormality?(This showsthatsomestatisticsmayrequireaverylargesamplesizetohaveanapproximate normal
Forindependent Y1 ∼ binom(n1, π1) and Y2 ∼ binom(n2, π2) random variables, ˆπ1~ˆπ2 is called the relativerisk or risk ratio. Thismeasureisoftenusedinmedicalresearchtocomparethe probabilityofacertainoutcomefortwotreatments.(a) Arandomizedstudywith n1 = n2 = 50 compares
When T is astandard normal randomvariable,whydoesthedeltamethodnotimplythat T2 has anormaldistribution?(Hint: FormtheTaylor-seriesexpansionof g(t) = t2 around g(0).In Section 4.4.5 weshallseethat T2 has achi-squareddistribution.)
Manypositively-valuedresponsevariableshaveaunimodaldistributionbutwithstandard deviation proportionaltothemean.Identifyatransformationforwhichthevarianceisapprox-imately thesameforallvaluesofthemean.Identifyatleastonedistributionthathasstandard deviation proportionaltothemean.
Forabinomialsample proportion ˆπ, showthattheapproximatevarianceof sin−1(ºˆπ) (with the anglebeingmeasured in radians)is1/4n, sothisisavariancestabilizingtransformation.(In reality,simulationrevealsthatthevarianceofthis arcsinetransformation is notnearly constantwhen π is near0ornear1and n
Forabinomialparameter π, g(π) = log[π~(1 − π)] is calledthe logit. Itisusedforcategorical data, asshownin Chapter 7. Let T = ˆπ for n independentbinarytrials.Usethedeltamethod to showthat Vlog()()(0-1)). villog()-10g( N
Explainthedifferencebetween convergenceinprobability and convergenceindistribution.Explain howonecanregardconvergenceinprobabilityasaspecialcaseofconvergencein distribution forwhichthelimitingdistributionhasprobability1atasinglepoint.
is reasonableforgivingabellshape.Forthis M, showthat Y has anapproximatenormal sampling distributionwhen n ≥ 10S2. Forthis guideline, howlargearandomsampledoyouneedfromanexponentialdistributionto achieveclosetonormalityforthesamplingdistribution?
Consider n independentobservationsofarandomvariable Y that hasskewnesscoefficient S = E(Y − μ)3~σ3.(a) Showhow E(Y −μ)3 relates to E(Y −μ)3. Based on this,showthattheskewnesscoefficient for thesamplingdistributionof Y satisfies skewness(Y ) = S~ºn. Explainhowthis result relates
Theformula σY= σ~ºn for thestandarderrorof Y treats thepopulation size as infinitely large relativetothesamplesize n. Withafinitepopulationsize N, separateobservationsare veryslightlynegativelycorrelated,sovar(Σi Yi) < Σivar(Yi), andactuallyThe term »(N − n)~(N − 1) is calledthe finite
Forindependentobservations,var(Σi Yi) = Σi var(Yi). Explainintuitivelywhy Σi Yi has a larger variancethanasingleobservation Yi.
Inthepreviousexercise,explainwhatisincorrectabouteachoptionthatyoudidnotchoose.
TheCentralLimitTheoremimpliesthat(a) Allvariableshaveapproximatelybell-shapedsampledatadistributionsifarandomsample containsatleastabout30observations.(b) Populationdistributionsarenormalwheneverthepopulationsizeislarge.(c) Forlarge random samples,thesamplingdistributionof Y is
Inonemilliontossesofafaircoin,theprobabilityofgettingexactly500,000headsand500,000 tails (i.e.,asampleproportionofheadsexactly=1/2)is(a) verycloseto1.0,bythelawoflargenumbers.(b) verycloseto1/2,bythelawoflargenumbers.(c) verycloseto0,bythestandarddeviationofthebinomialdistributionanditsapproximate
Thestandarderrorofastatisticdescribes(a) Thestandarddeviationofthesamplingdistributionofthatstatistic.(b) Thestandarddeviationofthesampledata.(c) Howclosethatstatisticislikelytofalltotheparameterthatitestimates.(d) Thevariabilityinvaluesofthestatisticforrepeatedsimplerandomsamplesofsize n.(e)
Whichdistributiondoesthesampledatadistributiontendtoresemblemoreclosely—thesam-pling distributionofthesamplemeanorthepopulationdistribution?Explain.Illustrateyour answerforavariable Y that
and σ = 7.75. Plot the populationdistributionandthesimulatedsamplingdistributionofthe10,000 y values.Explain whatyour plotsillustrate.
bytaking10,000simulations of arandomsampleofsize n = 200 from agammadistributionwith μ =
MimictheappdisplayoftheCentralLimitTheoremin Figure
and amedianof0.69.As n increases, doesthesamplingdistributionofthesamplemedianseemtobeapproximately normal?
Simulatefor n = 10 and n = 100 the samplingdistributionsofthesamplemedianforsampling from theexponentialdistribution(2.12)with λ = 1.0, whichhas μ = σ =
Asurveyisplannedtoestimatethepopulationproportion π supportingmoregovernment action toaddressglobalwarming.Forasimplerandomsample,if π maybenear0.50,how large should n besothatthestandarderrorofthesampleproportionis0.04?
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