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statistics informed decisions using data

Questions and Answers of Statistics Informed Decisions Using Data

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
Inthe2018GeneralSocialSurvey,whenaskedwhethertheybelievedinlifeafterdeath,1017 of 1178femalessaid yes, and703of945malessaid yes. Construct95%confidenceintervals for
Usingthe Students data file,forthecorrespondingpopulation,constructa95%confidencein-terval(a) forthemeanweeklynumberofhoursspentwatchingTV;(b) tocomparefemalesand males
Sections 4.4.3 and 4.5.3 analyzed datafromastudyaboutanorexia.The Anorexia data file at thetextwebsitecontainsresultsforthecognitivebehavioralandfamilytherapiesandthe
The Income data fileatthebook’swebsiteshowsannualincomesinthousandsofdollarsfor subjectsinthreeracial-ethnicgroupsintheU.S.(a) Useagraphicsuchasaside-by-sideboxplot(Section 1.4.5)
TheobservationsonnumberofhoursofdailyTVwatchingforthe10subjectsinthe2018GSS who identifiedthemselvesasIslamicwere0,0,1,1,1,2,2,3,3,4.(a)
ArecentGeneralSocialSurveyaskedmalerespondentshowmanyfemalepartnerstheyhave had sexwithsincetheir18thbirthday.Forthe131malesbetweentheagesof23and29,the median = 6 andmode = 1
Astudyinvestigatesthedistributionofannualincomeforheadsofhouseholdslivinginpublic housing inChicago.Forarandomsampleofsize30,theannualincomes(inthousandsof dollars) areinthe Chicago data
ToestimatetheproportionoftrafficdeathsinCalifornialastyearthatwerealcoholrelated,de-termine thenecessarysamplesizefortheestimatetobeaccuratetowithin0.04withprobability 0.90.
AsocialscientistwantedtoestimatetheproportionofschoolchildreninBostonwholivein a single-parentfamily.Shedecidedtouseasamplesizesuchthat,withprobability0.95,the error
Inarandomsampleof25people to estimatetheextentofvegetarianisminasociety,0peo-ple werevegetarian.Constructa99%Waldconfidenceintervalforthepopulationproportion of
TheGeneralSocialSurveyhasaskedrespondents,“Doyouthinktheuseofmarijuanashould bemadelegalornot?”Viewresultsatthemostrecentcumulativedatafileat sda.berkeley.edu/archive.htm
Forthe Students data file(Exercise1.2in Chapter 1) andcorrespondingpopulation,findthe ML estimateofthepopulationproportionbelievinginlifeafterdeath.ConstructaWald95%confidence
with y = 6 for n = 10 (e.g.,in R bytaking pi (π) asasequencebetween0 and 1andthentakingLasthelogof dbinom(6, 10,pi)). Identify ˆπ in theplotandexplain
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
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
RefertoExercise2.66and the momentgeneratingfunction m(t) = EetY , analternativeto the probabilityfunctionforcharacterizingaprobabilitydistribution.Forindependentrandom variables Y1, Y2, …, Yn,
and π2 = 0.10. Simulateamillionsample proportionsfromeachtreatmentwith n1 = n2 = 50. Constructhistogramsfor ˆπ1~ˆπ2 and for log(ˆπ1~ˆπ2).
Forindependent Y1 ∼ binom(n1, π1) and Y2 ∼ binom(n2, π2) random variables, ˆπ1~ˆπ2 is called the relativerisk or risk ratio. Thismeasureisoftenusedinmedicalresearchtocomparethe
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
Manypositively-valuedresponsevariableshaveaunimodaldistributionbutwithstandard deviation proportionaltothemean.Identifyatransformationforwhichthevarianceisapprox-imately
Forabinomialsample proportion ˆπ, showthattheapproximatevarianceof sin−1(ºˆπ) (with the anglebeingmeasured in radians)is1/4n, sothisisavariancestabilizingtransformation.(In
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
Explainthedifferencebetween convergenceinprobability and convergenceindistribution.Explain howonecanregardconvergenceinprobabilityasaspecialcaseofconvergencein distribution
is reasonableforgivingabellshape.Forthis M, showthat Y has anapproximatenormal sampling distributionwhen n ≥ 10S2. Forthis guideline, howlargearandomsampledoyouneedfromanexponentialdistributionto
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
Theformula σY= σ~ºn for thestandarderrorof Y treats thepopulation size as infinitely large relativetothesamplesize n. Withafinitepopulationsize N, separateobservationsare
Forindependentobservations,var(Σi Yi) = Σi var(Yi). Explainintuitivelywhy Σi Yi has a larger variancethanasingleobservation Yi.
Inthepreviousexercise,explainwhatisincorrectabouteachoptionthatyoudidnotchoose.
TheCentralLimitTheoremimpliesthat(a) Allvariableshaveapproximatelybell-shapedsampledatadistributionsifarandomsample containsatleastabout30observations.(b)
Inonemilliontossesofafaircoin,theprobabilityofgettingexactly500,000headsand500,000 tails (i.e.,asampleproportionofheadsexactly=1/2)is(a) verycloseto1.0,bythelawoflargenumbers.(b)
Thestandarderrorofastatisticdescribes(a) Thestandarddeviationofthesamplingdistributionofthatstatistic.(b) Thestandarddeviationofthesampledata.(c)
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
Considerthestandarderror»π(1 − π)~n of asampleproportion ˆπ of successes in n binary trials.(a) Reportthestandarderrorwhen π = 0 or π = 1. Whywouldthisvaluemakesensetoyou
Whensampledata wereusedtorankstatesbybraincancerrates,Ellenberg(2014)noted that thehighestrankingstate(SouthDakota)andthenearlylowestrankingstate(North
Simulatewhatwouldhappenifeveryoneinacollegewith1000studentsflippedafaircoin100 times andobservedtheproportionofheads.Whatdoyougetforthemeanandthestandard deviation
Construct a populationdistributionthatisplausiblefor Y = numberofalcoholicdrinksinthe past day.(a) Simulateasinglerandomsampleofsize n = 1000 from thispopulationtoreflectresults of
Usingsoftware,simulatetakingsimplerandomsamplesfromabimodalpopulationdistribu-tion. (Alternatively,youcanuseanappsuchasthe Sampling DistributionfortheSample Mean(ContinuousPopulation) app at
Using software,simulateformingsampleproportionsforsimplerandomsamplesofsize n = 100 when π = 0.50. (Alternatively,youcanuseanapp,suchasthe Sampling Distributionforthe Sample Proportion app at
Inyourschool,supposethatGPAhasanapproximatenormaldistributionwith μ = 3.0, σ = 0.40.Not knowing μ, yourandomlysample n = 25 studentstoestimateit.Usingsimulationfor this
Toapproximatethemean μ of aprobabilitydistributionthatdescribesarandomphenomenon, yousimulate n observationsfromthedistributionandfind y. Explainhowto assess howclose y is likelytobe to μ.
GenerateamillionindependentPoissonrandomvariableswithparameter(i) μ = 9, (ii) μ =100, (iii) μ = 100000. Showhowtoconstructtransformedvaluesthathaveaboutthesame variabilityineachcase.
SunshineCity,whichattractsprimarilyretiredpeople,has90,000residentswithameanage of 72yearsandastandarddeviationof12years.Theagedistributionisskewedtotheleft.A random
Atauniversity,60%ofthe7400studentsarefemale.Thestudentnewspaperreportsresultsof a surveyofasimplerandomsampleof50students,18femalesand32males,tostudyalcohol abuse, suchasbingedrinking.(a)
andastandarddeviationof1.5.SupposetheCensusBureauinsteadhadestimatedthis mean usingarandomsampleof225homes,andthatsamplehadameanof2.4andstandard deviation of1.4.Describethecenterandspreadofthe(a)
AccordingtotheU.S.CensusBureau,thenumberofpeopleinahouseholdhasameanof
AccordingtoaGeneralSocialSurvey,intheUnitedStatesthepopulationdistributionof Y =numberofgoodfriends(notincludingfamilymembers)hasameanofabout5.5andastandard deviation ofabout3.9.(a)
Oneachbetinasequenceofbets,youwin$1withprobability0.50andlose$1(i.e.,win−$1) withprobability0.50.Let Y denote thetotalofyourwinningsandlosingsafter100bets.Giving
Simulaterandomsamplingfromauniformpopulationdistributionwithseveral n valuesto illustrate theCentralLimitTheorem.
Simulaterandomsamplingfromanormalpopulationdistributionwithseveral n valuesto illustrate thelawoflargenumbers.
Simulatetakingarandomsampleofsize n from aPoissondistributionwith μ = 5. Find y for n = 10, n = 1000, n = 100, 000, and n = 10, 000, 000 to illustrate thelawoflargenumbers.
Refertothepreviousexercise.For n rolls ofthedice,let X = max(y1, y2, ...,yn).(a) Constructthe sampling distributionof X when n = 2.(b) Whatdoyouexpectfortheappearanceofthesamplingdistributionof X
speculatedabouttheshapeofthesamplingdistribution of Y for 10rolls.Useasimulation andconstructahistogramtoportraythesampling distribution inthatcase.
Theoutcomeofrollingabalanceddicehasprobability 1~6 for eachof {1, 2, 3, 4, 5, 6}. Let(y1, y2) denotetheoutcomesfortworolls.(a) Enumeratethe36possible(y1, y2)
Constructthesamplingdistributionofthesampleproportionofheads,forflippingabalanced coin (a) once; (b) twice; (c) three times; (d) four times.Describehowtheshapechanges as thenumberofflips n increases.
Inflippingabalancedcoin n times, areyoumorelikelytohave(i)between40and60heads in 100flips,or(ii)between490and510headsin1000flips?As n increases, explainwhythe proportion of
ForthePresidentialelectionin2020,ofanexitpollof909votersinthestateofNewYork,64%votedforBidenand36%votedforTrump.Inresponsetothequestion“Isclimatechangea serious
Theexamplein Section 3.1.4 simulatedsamplingdistributionsofthesamplemeantodetermine howprecise Y for n = 25 mayestimateapopulation mean μ.(a) Findthetheoreticalstandarderrorof Y for
TheU.S.JusticeDepartmentandothergroupshavestudiedpossibleabusebypoliceofficers in theirtreatmentofminorities.Onestudy,conductedbytheAmericanCivilLibertiesUnion, analyzed
The49studentsinaclassattheUniversityofFloridamadeblindedevaluationsofpairsof cola drinks.Forthe49comparisonsofCokeandPepsi,Cokewaspreferred29times.Inthe
Inanexitpollof1648votersinthe2020SenatorialelectioninArizona,51.5%saidtheyvoted for MarkKellyand48.5%saidtheyvotedforMarthaMcSally.(a) Supposethatactually 50%
In anexitpollof2123votersinthe2018SenatorialelectioninMinnesota,61.0%saidtheyvoted for theDemocraticcandidateAmyKlobucharinherraceagainsttheRepublicancandidate Jim
Likethegammadistribution,thelog-normaldistribution(Exercise2.71),theWeibulldistri-bution (Exercise2.72),andtheParetodistibution(Exercise2.73),anotherdistributionfor skewed-rightvariablesisthe
The Paretodistribution, introducedbytheItalianeconomistWilfredoParetoin1909to describe(onappropriatescales)incomeandwealth,isahighlypositively-skeweddistribution that has pdf f(y;α) = α~yα+1 for y
Likethegammaandlog-normaldistributions,the Weibulldistribution is positivelyskewed overthepositiverealline.Withshapeparameter k > 0 and scaleparameter λ > 0, its cdf is F(y; λ, k) = 1 −
When Y has positivelyskeweddistributionoverthepositiverealline,statisticalanalysesoften treat X = log(Y ) as havinga N(μ, σ2) distribution. Then Y is saidtohavethe log-normal distribution.(a)
The betadistribution is aprobabilitydistributionover(0,1)thatisoftenusedinapplications for whichtherandomvariableisaproportion.Thebeta pdf isfor parameters α and β, where Γ(⋅) denotes
Forasequenceofindependent,identicalbinarytrials,explainwhytheprobabilitydistribution for Y = the numberofsuccessesbeforefailurenumber k occurshasprobabilityfunctionThis distribution,studiedfurtherin
Apopulationhas F females and M males. Forarandomsampleofsize n without replacement, explain whythe pmf for Y = numberoffemalesinthesampleisThis is called the hypergeometricdistribution.
For n observations {yi}, let y(1) ≤ y(2) ≤ ⋯ ≤ y(n) denote theirorderedvalues,called order statistics. Let qi bethe i~(n + 1) quantileofthestandardnormaldistribution,for i = 1. ...,n. When
Momentsofadistributioncanbederivedbydifferentiatingthe moment generatingfunction(mgf ), m(t) = EetY .This functionprovidesanalternativewaytospecifyadistribution.(a) Showthatthe kth derivative
Foruncorrelatedrandomvariables U, V , and W, let X = U + V and Y = U +W.(a) Showthatcov(X,Y ) = var(U) and(b) Forsomescaling,suppose X = math achievementtestscore, Y = verbalachievementtest score, U
Boundsforthecorrelation:(a) Considerrandom variables X and Y and theirstandardizedvariables Zx and Zy. Using the equationsfromthepreviousexerciseandtherelationbetweenthecorrelationand
Considertworandomvariables X and Y :(a) Showthatvar(X + Y ) = var(X) + var(Y ) + 2cov(X,Y ).(b) Showthatvar(X − Y ) = var(X) + var(Y ) − 2cov(X,Y ).(c) Showhow(a)and(b)simplifywhen X and Y are
Forindependentbinom(1, π) random variables X and Y , let U = X + Y and V = X − Y . Find the jointprobabilitydistributionof U and V . Showthat U and V are uncorrelatedbutnot independent.
ContructanexampleofaMarkovchain,andusesimulationtorandomlygenerate100values from it.PlotthesequenceanddescribehowtheMarkovpropertyaffectstheplot.
For n coin-flip betsasdescribedin Section 2.6.7, let pn denote theproportionof t between1 and n for whichthetotalwinnings Yt at time t is positive.(a) With n = 100,
Explainhowaboardgameusingdice,suchas“SnakesandLadders,”hasasequenceofoutcomes that satisfiestheMarkovproperty.
Abalancedcoinisflippedtwice.Let X denote theoutcomeofthefirstflipand Y denote the outcomeofthesecondflip,representing head by1and tail by0.Supposetheflipsare independent.(a) Let Z indicate
Let (Y1, Y2, ...,Yc) denote independentPoissonrandomvariables,withparameters(μ1,μ2, ...,μc).(a) Explainwhythejointprobabilitymassfunctionfor {Yi} is cΠi=1[exp(−μi)μyi i ~yi!]for
Consider the multinomialdistribution(2.14)with c = 3 categories.(a) Explainwhythemarginaldistributionof Y1 is binomial.Basedonthis,report E(Y1) and var(Y1).(b) Are Y1 and Y2
Supposethatconditionalon λ, thedistributionof Y is Poissonwithmeanparameter λ, but λitself variesamongdifferentsegmentsofapopulation,with μ = E(λ). Usethelawofiterated expectationtofind E(Y ).
Reviewtheresultaboutthe“probabilityintegraltransformation”in Section 2.5.7. Foracontin-uous randomvariable Y with cdf F, findtheprobabilitydistributionoftheright-tailprobability X = 1 − F(Y ).
If Y is astandardnormalrandomvariable,with cdf Φ, whatistheprobabilitydistribution of X = Φ(Y )? Illustratebyrandomlygeneratingamillionstandardnormalrandomvariables, applying the cdf function Φ()
The pdf f of a N(μ, σ2) distribution canbederivedfromthestandardnormal pdf ϕ shownin equation (2.9).(a) Showthatthenormal cdf F relates tothestandardnormal cdf Φ by F(y) = Φ[(y−μ)~σ].(b)
By Jensen’sinequality, convexfunctionssatisfy E[g(Y )] ≥ g[E(Y )]. Usethistoprovethat for concavefunctions, E[g(Y )] ≤ g[E(Y )]. Applytheappropriatecasesto log(Y ) and 1~Y for a
The Markov inequality states thatwhen P(Y ≥ 0) = 1, then P(Y ≥ t) ≤ E(Y )~t.(a) When Y is discreteoverthenonnegativeintegers,provethisbyexplainingwhy E(Y ) ≥Σy≥t yf(y) ≥ Σy≥t tf(y) =
Section 2.4.5 showedthatforabinom(n, π) randomvariable,thesampleproportion ˆπ has standard deviation»π(1 − π)~n. Usethistoexplainwhy ˆπ tends tobecloserto π as n
Aprobabilitydistributionhasa scaleparameter θ if, whenyoumultiply θ byaconstantc, all valuesinthedistributionmultiplybyc. Ithasa locationparameter θ if, whenyouincrease θbyaconstantc,
Whenaprobabilityfunctionissymmetricanditsmomentsexist,explainwhy E(Y − μ)3 = 0, so theskewnesscoefficient=0.
Section 2.5.5 showedthatthewaitingtime T for thefirstoccurrenceofaPoissonprocesshas the exponentialdistributionwithparameter λ. Forthisdistribution,showthat P(T > u + t S T > u) = P(T > t). Bythis

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