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Foundations Of Statistics For Data Scientists With R And Python 1st Edition Alan Agresti - Solutions
Considerthestandarderror»π(1 − π)~n of asampleproportion ˆπ of successes in n binary trials.(a) Reportthestandarderrorwhen π = 0 or π = 1. Whywouldthisvaluemakesensetoyou evenifyoudidnotknowthestandarderrorformula?(b) Forfixed n, showthatthestandarderrorisgreatestwhen π = 0.50.
Whensampledata wereusedtorankstatesbybraincancerrates,Ellenberg(2014)noted that thehighestrankingstate(SouthDakota)andthenearlylowestrankingstate(North Dakota)hadrelativelysmallsamplesizes.Also,whenschoolsinNorthCarolinawereranked
Simulatewhatwouldhappenifeveryoneinacollegewith1000studentsflippedafaircoin100 times andobservedtheproportionofheads.Whatdoyougetforthemeanandthestandard deviation ofthe1000proportions?Whataretheirtheoreticalvalues?
Construct a populationdistributionthatisplausiblefor Y = numberofalcoholicdrinksinthe past day.(a) Simulateasinglerandomsampleofsize n = 1000 from thispopulationtoreflectresults of atypicalsamplesurvey.Summarizehowthesamplemeanandstandarddeviation
Usingsoftware,simulatetakingsimplerandomsamplesfromabimodalpopulationdistribu-tion. (Alternatively,youcanuseanappsuchasthe Sampling DistributionfortheSample Mean(ContinuousPopulation) app at www.artofstat.com/web-apps and selectthebimodal populationdistributionwithshape1.)(a)
Using software,simulateformingsampleproportionsforsimplerandomsamplesofsize n = 100 when π = 0.50. (Alternatively,youcanuseanapp,suchasthe Sampling Distributionforthe Sample Proportion app at www.artofstat.com/web-apps.)(a) Simulateonceandreportthecountsandtheproportionsforthetwocategories.Didyou
Inyourschool,supposethatGPAhasanapproximatenormaldistributionwith μ = 3.0, σ = 0.40.Not knowing μ, yourandomlysample n = 25 studentstoestimateit.Usingsimulationfor this application,illustratethedifferencebetweenasampledatadistributionandthesampling distribution of Y .
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 sampleof100residentsofSunshineCityhas y = 70 and s = 11.(a) Describethecenter
Atauniversity,60%ofthe7400studentsarefemale.Thestudentnewspaperreportsresultsof a surveyofasimplerandomsampleof50students,18femalesand32males,tostudyalcohol abuse, suchasbingedrinking.(a) Setupavariable Y to representgender,andidentifyitsprobabilitydistributionforthe
andastandarddeviationof1.5.SupposetheCensusBureauinsteadhadestimatedthis mean usingarandomsampleof225homes,andthatsamplehadameanof2.4andstandard deviation of1.4.Describethecenterandspreadofthe(a) populationdistribution,(b) sample data distribution,(c) samplingdistributionofthesamplemeanfor225homes.
AccordingtotheU.S.CensusBureau,thenumberofpeopleinahouseholdhasameanof
AccordingtoaGeneralSocialSurvey,intheUnitedStatesthepopulationdistributionof Y =numberofgoodfriends(notincludingfamilymembers)hasameanofabout5.5andastandard deviation ofabout3.9.(a) Isitplausiblethatthispopulationdistributionisnormal?Explain.(b)
Oneachbetinasequenceofbets,youwin$1withprobability0.50andlose$1(i.e.,win−$1) withprobability0.50.Let Y denote thetotalofyourwinningsandlosingsafter100bets.Giving yourreasoning,statetheapproximatedistributionof Y .
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 when n is large?(If youlike,conductasimulationtoinvestigate.)Thisillustratesthatnoteverystatistic has
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) pairs.Treatingthemasequallylikely,constructthe sampling distributionfortheirsamplemean.(b)
Constructthesamplingdistributionofthesampleproportionofheads,forflippingabalanced coin (a) once; (b) twice; (c) three times; (d) four times.Describehowtheshapechanges as thenumberofflips n increases. Whatwouldhappenif n keptgrowing?Why?
Inflippingabalancedcoin n times, areyoumorelikelytohave(i)between40and60heads in 100flips,or(ii)between490and510headsin1000flips?As n increases, explainwhythe proportion of headsconvergestoward1/2(becausethestandarderrorofasample proportion decreases) butthe number of headsneednotbecloseto n~2
ForthePresidentialelectionin2020,ofanexitpollof909votersinthestateofNewYork,64%votedforBidenand36%votedforTrump.Inresponsetothequestion“Isclimatechangea serious problem?”71%ofthosewhovotedforBidenresponded yes and 28%ofthosewhovoted for Trumpresponded yes.
Theexamplein Section 3.1.4 simulatedsamplingdistributionsofthesamplemeantodetermine howprecise Y for n = 25 mayestimateapopulation mean μ.(a) Findthetheoreticalstandarderrorof Y for thescenariovaluesof σ = 5 and 8. Howdo they
TheU.S.JusticeDepartmentandothergroupshavestudiedpossibleabusebypoliceofficers in theirtreatmentofminorities.Onestudy,conductedbytheAmericanCivilLibertiesUnion, analyzed whetherAfrican-Americandriversweremorelikelythanothersinthepopulation to
The49studentsinaclassattheUniversityofFloridamadeblindedevaluationsofpairsof cola drinks.Forthe49comparisonsofCokeandPepsi,Cokewaspreferred29times.Inthe populationthatthissamplerepresents,isthisstrongevidencethatamajorityprefersCoke?Use asimulationofasamplingdistributiontoanswer.
Inanexitpollof1648votersinthe2020SenatorialelectioninArizona,51.5%saidtheyvoted for MarkKellyand48.5%saidtheyvotedforMarthaMcSally.(a) Supposethatactually 50% ofthepopulationvotedforKelly.Ifthisexitpollhadthe propertiesofasimplerandomsample,findthestandarderrorofthesampleproportion votingforhim.(b)
In anexitpollof2123votersinthe2018SenatorialelectioninMinnesota,61.0%saidtheyvoted for theDemocraticcandidateAmyKlobucharinherraceagainsttheRepublicancandidate Jim Newberger.Basedonthisinformation,ifyoucouldtreatthisexitpolllikeasimplerandom sample,
Likethegammadistribution,thelog-normaldistribution(Exercise2.71),theWeibulldistri-bution (Exercise2.72),andtheParetodistibution(Exercise2.73),anotherdistributionfor skewed-rightvariablesisthe Gumbeldistribution, alsocalledthe typeIextreme-valuedis-tribution.
The Paretodistribution, introducedbytheItalianeconomistWilfredoParetoin1909to describe(onappropriatescales)incomeandwealth,isahighlypositively-skeweddistribution that has pdf f(y;α) = α~yα+1 for y ≥ 1 and aparameter α > 0.(a) Showthat f is alegitimate pdf (i.e.,
Likethegammaandlog-normaldistributions,the Weibulldistribution is positivelyskewed overthepositiverealline.Withshapeparameter k > 0 and scaleparameter λ > 0, its cdf is F(y; λ, k) = 1 − e−(y~λ)k for y > 0.(a) Findthe pdf.(b) Showthatthemedian= λ[log(2)]1~k.
When Y has positivelyskeweddistributionoverthepositiverealline,statisticalanalysesoften treat X = log(Y ) as havinga N(μ, σ2) distribution. Then Y is saidtohavethe log-normal distribution.(a) Deriveanexpressionforthe cdf G of Y in termsofthe cdf F of X, andtakethederivative to obtainthe pdf g of
The betadistribution is aprobabilitydistributionover(0,1)thatisoftenusedinapplications for whichtherandomvariableisaproportion.Thebeta pdf isfor parameters α and β, where Γ(⋅) denotes thegammafunction.(a) Showthattheuniformdistributionisthespecialcase α = β = 1.(b) Showthat μ = E(Y ) =
Forasequenceofindependent,identicalbinarytrials,explainwhytheprobabilitydistribution for Y = the numberofsuccessesbeforefailurenumber k occurshasprobabilityfunctionThis distribution,studiedfurtherin Section 7.5.2 for analyzingcountdata,iscalledthe neg-ative binomialdistribution. (y;k, ) = ( + k 1
Apopulationhas F females and M males. Forarandomsampleofsize n without replacement, explain whythe pmf for Y = numberoffemalesinthesampleisThis is called the hypergeometricdistribution. Itisanalternativetothebinomialforwhich
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 {yi} are arandomsamplefromanormaldistribution,theplotofthepoints(q1, y(1)), ..., (qn, y(n)) should
Momentsofadistributioncanbederivedbydifferentiatingthe moment generatingfunction(mgf ), m(t) = EetY .This functionprovidesanalternativewaytospecifyadistribution.(a) Showthatthe kth derivative m(k)(t) = EY ketY , andhence m′(0) = E(Y ) and m′′(0) =E(Y 2).(b) Showthatthe mgf is m(t) = 1
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 = intelligence(e.g.,IQ), V = time studyingmath, W = time studyingverbal.Explain howcorr(X,Y )
Boundsforthecorrelation:(a) Considerrandom variables X and Y and theirstandardizedvariables Zx and Zy. Using the equationsfromthepreviousexerciseandtherelationbetweenthecorrelationand covariance,showthatvar(Zx +Zy) ≥ 0 implies thatcorr(X,Y ) ≥ −1 and var(Zx −Zy) ≥ 0 implies28 corr(X,Y )
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 uncorrelated.
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, simulatethisMarkovchainafewtimes,eachtimeshowingaplotof(y1, y2, ...,y100) and reporting pn.(b)
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 whetherbothflipshadthesameresult,with z = 1 for yes and z = 0 for no.Showthat X and Z are independent.(b)
Let (Y1, Y2, ...,Yc) denote independentPoissonrandomvariables,withparameters(μ1,μ2, ...,μc).(a) Explainwhythejointprobabilitymassfunctionfor {Yi} is cΠi=1[exp(−μi)μyi i ~yi!]for allnonnegativeintegervalues (y1, y2, ...,yc).(b) Section 3.2.6 explains thatthesum n = Σi Yi also
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 independentrandomvariables?Whyorwhynot?
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 ). (We’lllearntherelevanceofthiswhen Chapter 5 introduces P-values.)
If Y is astandardnormalrandomvariable,with cdf Φ, whatistheprobabilitydistribution of X = Φ(Y )? Illustratebyrandomlygeneratingamillionstandardnormalrandomvariables, applying the cdf function Φ() to each,andplottinghistogramsofthe(a) y values,(b) x values.
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) From(a),showthat f(y) = (1~σ)ϕ[(y − μ)~σ], andshowthisisequation(2.8).
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 positively-valuedrandomvariable Y .
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) = tP (Y ≥ t).(b) If X is anyrandomvariablewithmean μ and variance σ2, applytheMarkovinequality
Section 2.4.5 showedthatforabinom(n, π) randomvariable,thesampleproportion ˆπ has standard deviation»π(1 − π)~n. Usethistoexplainwhy ˆπ tends tobecloserto π as n increases.Thus,intherainsimulationin Section 2.1.1, therelativefrequencyforaparticularoutcome convergesas n increases
Aprobabilitydistributionhasa scaleparameter θ if, whenyoumultiply θ byaconstantc, all valuesinthedistributionmultiplybyc. Ithasa locationparameter θ if, whenyouincrease θbyaconstantc, allvaluesinthedistributionincreaseby c.(a) Forascaleparameter θ, thedistributionof Y ~θ doesnotdependon θ.
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 memoryless property,ifaneventhasnotoccurredbytime u, the additional
Reparameterizingthegamma distribution byreplacing λ by k~μ, showthat f(y; k,μ) =(k~μ)kΓ(k) e−ky~μyk−1, y ≥ 0 (2.17)For this parameterization,showthattheexpressionsin(2.11)simplifyto E(Y ) = μ and σ =μ~ºk. So,forfixed k, asthe mean grows,sodoesthestandarddeviation.Thisisthecasein
Forthegamma pdf (2.10), find E(Y 2) and useittogetherwith μ = k~λ to findvar(Y ).
Considertheexponential pdf f(y; λ) = λe−λy and cdf F(y; λ) = 1 − e−λy, for y ≥ 0.(a) Findthemedian.(b) Findthelowerquartileandtheupperquartile.(c) Find μ byshowingthatitequals 1~λ times theintegralofagamma pdf. Explainwhy μis greaterthanthemedian.(d) Find σ byfinding E(Y 2) using
Showthatthebinomialprobabilitymassfunction(2.6)convergestothePoissonwhenyou substitute π = μ~n and let n → ∞ with μ fixed.
Useformula(2.8)forthenormal pdf to showthatthe pdf is symmetric.
Let Y beyourwaitingtimeinalineatagrocerystore.Let Y1, Y2, ...bethewaitingtimesof other people.Let N = the numberofpeoplethatmustbeinlinesatthestoreuntilsomeonehas to waitlongerthanyou.Ifthewaitingtimesareindependentandhavethesamecontinuous probabilitydistribution,explainwhy(a) P(N > n) = P(Y >
ForthePoissondistribution,showthat E[Y (Y −1)] = μ2. Usethistoshowthat E(Y 2) = μ+μ2 and thusvar(Y ) = μ.
Forageometricrandomvariablewithprobabilityfunction(2.1),showthat(a) the cdf is F(y) = 1 − (1 − π)y for y = 1, 2, ...; (b) E(Y ) = 1~π.
Forindependentobservationswithprobabilityofsuccess π on each,specifytheprobability mass functionof Y = numberoffailuresuntilthefirstsuccess.27
Let X beauniformdistributionover [L,U] with L < U.(a) Specifythe pdf of X and find E(X).(b) From Section 2.3.3, auniformrandomvariable Y over[0,1]hasmean1/2andstandard deviation 1~º12. Express X as alinearfunction of Y and usethisrelationandresultsin Section 2.3.5
Althoughanobservationofacontinuousrandomvariableisaparticularvalue,explainwhy eachpossiblevaluehasprobability0.Justifythisinthecontextoftherelativefrequency interpretationofprobability.
Ifevents A and B are independent,thenare A and Bc independent,ordependent?Showwhich is thecase.
Fordiscreterandomvariables X and Y , suppose P(Y = y S X = x) = P(Y = y) for allpossible values x of X and y of Y . Showthat P(X = x S Y = y) = P(X = x) for allthosevalues.
Onamultiple-choiceexam,with k possibleresponsesforeachquestion,astudentknowsthe answerwithprobability π and hastoguesstheanswerrandomlywithprobability (1−π). Given that astudentcorrectlyanswersaquestion,findtheprobabilitytheytrulyknewtheanswer.Evaluatetheexpressionyouderivewhen k = 5 and π is
For continuousrandomvariables,formulateaversionofBayes’Theoremtoobtain f(x S y)from thefunctions f(y S x) and f1(x).
Fordiscreterandomvariables X and Y , derivethefollowinggeneralizationofBayes’Theorem(Section 2.1.5):P(X = x S Y = y) = P(Y = y S X = x)P(X = x)Σa P(Y = y S X = a)P(X =a) , where thedenominatorsumisover allthepossiblevalues a for X. Statethecorresponding result for P(Bj S A) for anevent A and
De Morgan’slaw states, forthecaseoftwoevents, (A∪B)c = AcBc. ShowthiswithaVenn diagram, andexplainhowthelawgeneralizesto p events A1, ...,Ap.
Showthe rule oftotalprobability: Ifasamplespace S partitions intodisjointevents B1, ...,Bc (the unionofwhichis S), then P(A) = P(A S B1)P(B1) + ⋯+ P(A S Bc)P(Bc).
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 and plot1000observationsfromthisapproximatebivariatenormaldistribution.Approximate the marginalmeansandstandarddeviationsfor X and
Referto Table2.4 crossclassifyinghappinesswithfamilyincome.(a) Findandinterpretthecorrelationusingscores(i)(1,2,3)foreachvariable,(ii)(1,2,3)for familyincomeand(1,4,5)forhappiness.(b) Constructthejointdistributionthathasthesemarginaldistributionsandexhibitsinde-pendenceof X and Y .
Fortheexamplein Section 2.7.4 in whichamidtermexamscore X has a N(70, 102) distribution and theconditionaldistributionofthefinalexamscore Y given X = x is N(70+0.60(x−70), 62), use theformula E(Y ) = E[E(Y S X)] to find E(Y ).
Considerthejurylistexamplein Section 2.6.4, butwith (π1, π2, π3) = (0.25, 0.25,0.50)for(Hispanic, African-American,White).(a) With n = 12, theexpectedcountsare(3,3,6).Usethemultinomialdistributiontofind the probabilityofthisresult.(b)
The Afterlife data fileatthebook’s websitecontainsdatafromthe2018GeneralSocial Surveyonpostlife = beliefintheafterlife(1 = yes,2 = no) andreligion(1 = Protestant,2 =Catholic, 3 = Jewish, othercategoriesexcluded).Usingthesedata,formacontingencytable and
Considerthemammogramdiagnosticexamplein Section 2.1.4.(a) Showthatthejointprobabilitydistributionofdiagnosisanddiseasestatusisasshownin Table2.6. Giventhatadiagnostictestresultispositive,explainhowthisjointdistribution showsthatthe12%ofincorrectdiagnosesforthe99%ofwomennothavingbreastcancer
Plotthegammadistributionbyfixingtheshapeparameter k = 3 and settingthescaleparameter= 0.5, 1,2,3,4,5.Whatistheeffectofincreasingthescaleparameter?(SeealsoExercise2.48.)
Createadatafilewiththeincomevaluesinthe Income data fileatthetextwebsite.(a) Constructahistogramorasmooth-curveapproximationforthe pdf of incomeinthe correspondingpopulationbyplottingresultsusingthe density function in R (explained in Exercise1.18).(b)
LakeWobegonJuniorCollegeadmitsstudentsonlyiftheyscoreabove400onastandardized achievementtest.ApplicantsfromgroupAhaveameanof500andastandarddeviationof 100 onthistest,andapplicantsfromgroupBhaveameanof450andastandarddeviationof 100.
(iii) 0.05.(b) Findthe z-valuesuchthatforanormaldistributiontheintervalfrom μ − zσ to μ + zσcontainsprobability (i) 0.90, (ii) 0.95, (iii) 0.99.(c) Findthe z-valuessuchthat μ + zσ is the (i) 75th, (ii) 95th, (iii) 99th percentileofa normal distribution.(d) Showthattheupperquartileequals μ
(ii)
Normalprobabilitiesandpercentiles:(a) Findthe z-valueforwhichtheprobabilitythatanormalvariableexceeds μ + zσ equals(i)
AninstructorgivesacoursegradeofBtostudentswhohavetotalscoreonexamsandhome-worksbetween800and900,wherethemaximumpossibleis1000.Ifthetotalscoreshave approximatelyanormaldistributionwithmean830andstandarddeviation50,aboutwhat proportionofthestudentsreceiveaB?
Eachdayahospitalrecordsthenumberofpeoplewhocometotheemergencyroomfortreat-ment.(a) Inthe first week,theobservationsfromSundaytoSaturdayare10,8,14,7,21,44,60.Do youthinkthatthePoissondistributionmightdescribetherandomvariabilityofthis phenomenon adequately.Whyorwhynot?(b)
Eachweekaninsurancecompanyrecords Y = numberofpaymentsbecauseofahomeburning down.Stateconditionsunderwhichwewouldexpect Y to approximatelyhaveaPoisson distribution.
ToassessthepopularityoftheprimeministerinItaly,eachofseveralsamplesurveystakesa simple randomsampleof1000peoplefromthepopulationof40millionadultsinItaly.(a) With10surveys,findtheprobabilitythatnonesampleaparticularperson,Vincenzoin the villageofFerrazzanointheregionofMolise.(b)
Inhisautobiography A SortofLife, BritishauthorGrahamGreenedescribedaperiodofsevere mentaldepressionduringwhichheplayedRussianroulette—puttingabulletinoneofthesix chambersofapistol,spinningthechamberstoselectoneatrandom,andthenfiringthepistol once athishead.(a)
+ 0.05? (ii) number of successfulfreethrowsisbetween 0.80n − 5 and 0.80n + 5? Explainyouranswers.
and
−
ofmaking(i)thatmanyfreethrows,(ii)thatproportionoffreethrows.Stateany assumptions youmake.(b) Whentheplayerattempts n free throws,as n increases, wouldyouexpecttheprobability to increase,ortodecrease,thatthe (i) proportion of successfulfreethrowsisbetween
Abasketballplayerhasprobability0.80ofmakinganyparticularfreethrow(astandardized shot taken15feetfromthebasket).(a) Foraseasonwith200freethrowattempts,usethemeanandstandarddeviationofa binomial distributiontostateanintervalwithinwhichtheplayerhasprobabilityabout
(and thusclosetoasymmetric,bellshape)requires n > 25.
Showthatforthebinomialdistributiontohaveabsolutevalueoftheskewnesscoefficient < c for anyparticular c > 0 requires n > (1 − 2π)2~c2π(1 − π). Showthatwhen π = 0.20, having skewness
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