试题英文数理统计

填空（一）各章节的introductionContinuousvariablesorintervaldatacanassumeanyvalueinsomeintervalofrealnumbers.连续变量或间隔数据可以假设在某个实数间隔中的任意值。（measurement）Discretevariablesassumeonlyisolatedvalues.离散变量只假定孤立的值。(counting)Thelowerorfirstquartileisthe25thpercentileandtheupperorthirdquartileisthe75thpercentile.ThefistqurtileQ1isthemedianoftheobservationsfallingbelowthemedianoftheentiresampleandthethirdquartileQ3isthemedianoftheobservationsfallingabovethemedianoftheentiresample.TheinterquartilerangeisdefinedasIQR=Q3-Q1.第一个四分位数Q1是低于整个样本中位数的观测值的中位数，第三个四分位数Q3是高于整个样本中位数的观测值的中位数。四分位数范围定义为IQR=Q3-Q1。Statisticsappliedtothelifesciencesinoftencalledbiostatisticsorbiometry.统计学应用于生命科学，通常称为生物统计学或生物计量学。Adescriptivemeasureassociatedwitharandomvariablewhenitisconsideredovertheentirepopulationiscalledaparameter.当在整个总体中考虑一个随机变量时，与它相关的描述性度量称为参数Oneisforcedtoexamineasubsetorsampleofthepopulationandmakeinferencesabouttheentirevariableofasampleiscalledastatistic.人们被迫检查总体中的一个子集或样本，并对样本中的整个变量做出推断，这被称为统计量。Thesummarydescriptivecharacteristicsofapopulationofobjectsarecalledpopulationparametersorjustparameters.对一个物体总体的概括描述特征称为总体参数或简称参数。These（population）parametersareusuallydenotedbyGreeklettersanddonotvarywithinapopulation.这些(人口)参数通常用希腊字母表示，在人口中不发生变化。thesummarydescriptivecharacteristicsofasampleofobjects,isasubsetofthepopulationarecalledstatistics.6.概括描述一个样本对象的特征，是总体的一个子集，称为统计量。Measuresofcentraltendency:meanmedianandmode.Thepopulationmeanisthesumofthevaluesofthevariableunderstudydividedbythetotalnumberofobjectsinthepopulation.8.总体平均值是被研究变量的值除以总体对象总数的总和。Themostwidelyutilizedmeasureofcentraltendencyisthearithmeticmeanoraverage.9.使用最广泛的集中趋势度量是算术平均值或平均。10、Thedepthofavalueisitspositionrelativetothenearestextremewhenthedataarelistedinorderfromsmallesttolargest.值的深度是当数据按从小到大的顺序排列时，它相对于最近的极值的位置。Themodeisdefinedasthemostfrequentlyoccurringvalueinadataset.模态定义为数据集中出现频率最高的值。Thereareseveralcommonlyusedmeasurestodescribethelocationorcenterofapopulationorsample，theseincludemean、medianandmode，themeasuresofdispersionandvariability:variance，standarddeviationandrange.Thequantityisthesumofthesesquareddeviatesandisreferredtoasthecorrectedsumsquares,denotedbyCSS.Ifafixedamount（c）isaddedorsubtractedfromeachobservationinadataset,thesamplemeanwillbeincreasedordecreasedbythatamount（c），butthevariancewillbeunchanged（equal）.Multiplicativecodinginvovlesmultiplyingordividingeachobservationinadatasetbyaconstant.乘法编码是将数据集中的每个观测值乘以或除以一个常数acompactwaytoreportthedescritiveinformationinvovlingthequartilesandtherangiswithafive-summaryofthedata.Itconsistsofthemedian,thetwoquartilesandtwoextremes.Thevisualcounterparttoafive-summaryisaboxplot.五摘要的视觉对应物是盒状图。Theprobabilityoftheeventisanumericalmeasureofthelikeihoodordegreeofpredictabilitythattheeventwilloccur.TheempiricalprobabilityofaneventAisdefinedasH0:nullhypothesis；Ha:alternativeorresearchhypothesisWehavegeneratedtwomutuallyexclusiveandall-inclusivepossibilities.21H0:nullhypothesis;Ha:alternativeorresearchhypothesis我们已经产生了两种相互排斥又包罗万象的可能性。thevaluestodeterminesignificantdifferencesfromexpectationwehavecutoffvalues,usuallycalledcriticalvalues.22.我们用截断值来确定与期望有显著差异的值，通常称为临界值。Thepowerofthetestdependsonboththelevelofsignificance(α)andthevalueofμ0-μ1,thetruedifferencebetweenmeans.检验的威力取决于显著性水平(α)和平均值之间的真实差值μ0-μ1Thespreadofthedistributionisdeterminedbythestandarderror,sandn.Quantiativevariablesfallintotwomajorcategoriescontinuousvariables（orintervaldata）？anddiscretevariables.EventA1andA2areindependentifandonlyifP（A1∩A2）=P（A1）·P（A2）Arandomvariableandisavariablewhoseactualvalueisdeterminedbychanceoperations一个随机变量，它的实际值由随机操作决定LetXbeanormalrandomvariableZwithmeanμ，andstandarddeviationσ，thrtransformationZ=（x-μ）/σ，expressXasthestandardnormalrandomvariablewithμ=0andσ=128设X为平均μ的正态随机变量Z，标准差σ，变换Z=(X-μ)/σ，将X表示为μ=0，σ=1的标准正态随机变量LetXbebinomialrandomvariablewithparametersnandp.Theμ=E（X）=npandσx²=VarX=np（1-p）.LetXbeaPoissonrandomvariablewithparameterμ.TheE（X）=μandσx²=VarX=μ.（z不要）is（ida）anormaldistributionwithμ=0andσ=1，while（不要t）is（ida）atdistributionwithμ=0andσdependingonthesamplesize，Therandomvariable（n-1）s²/σ²isachi-square/（X²）distribution.withμ=E（X²）=v（自由度）/n-1，andVarX²=2v/2（n-1）Forlargevalueofn，thebinomialrandomvariable（X）isappromatelynormalwithmeanμ=npandσ=√np（1-p）.Asimpleruleofthumbisthatthisapproximationisacceptableforvaluesnandpsuchthatnp（1-p）＞3Inhypithesistest，H0isusuallycallednullhypothesis，Haiscalledthealternativeorresearchhypothesis.TherelationshipbetweenH0andHaismutuallyexclusiveandall-inclusivepossibilities.IfwerejectH0，wemightmakeaTypeIerror（rejectingatrueHo），andtheprobabilityisα，IfrejectHawemightmakeTypeIIerror（acceptingafalseHo），andtheprobabilityisdenotedbyβ，1-βiscalledthepowerofthetest如果我们拒绝H0，我们可能犯I型错误(拒绝一个真Ho)，概率是α，如果我们拒绝Ha，我们可能犯II型错误(接受一个假Ho)，概率用β表示，1-β称为测试的幂Thequantity∑（Xi-X拔）²isthesumofthesquareddeviatesandisreferredtoasthecorrectedsumsquaresdenotedbyCSS.数量(Xi-X拔)偏离平方的总和,称为纠正平方和用CSS。Thepatternofbehaviorofadiscreterandomvariableisdescribedbyamathematicalfunctioncalledadensityfunctionorprobabilitydistribution离散随机变量的行为模式是由称为密度函数或概率分布的数学函数来描述的Themodernstudyofthelifesciencesincludesexperimentationdategatheringandinterpretation.生命科学的现代研究包括实验、数据收集和解释。Bonferronitalower（smaller）αlevel37Bonferroni是较低(较小)的α能级Regressionanalysisxcanmeasurewithouterror.Correlationanalysisbothvariablesmeasuredwitheroor.回归分析可以无误差地测量。相关分析两个变量测量的误差。Linearregression：SSTotal=SSR+SSE（=SSTreat+SSError）？RandomizedcompleteblockdesignANOVA.SSTotal：SSTreat（k-1）SSBlock（b-1）、SSError（k-1）（b-1）.meanseparationtechniquesormultiplecomparisons.approaches：Bonferronittest.Dubcan'smultiperangetest（DMRT）Afivenumbersummary：thetwoquartiles、twoextremes、median.Arandomvariableisavariablewhoseactualvaluesisdeterminedbychanceoperations.TheemiricalprobabilityofaneventAisdefinedasP（A）=nA/n=numberoftimesAoccurred？/numberoftrialsrun.Thenumbernfactoralisdenotedbyn！，ThenumberofwaysofchoosingKobjectsfromnwithoutregardtoorderis（nk）=nCk（括号里n在上k在下）n的阶乘用n表示!，从n个对象中任意顺序选择K个对象的方法个数为ANOVA：AnalysisofVarianceModelIANOVA：analyzeasafixeseffectsModelIIANOVA：analyzeasarandom-effectsParameter参数characteristics特征Population总体：Nμσ²；Sample样本：n、x拔、s²；statistics统计数；variable变量；mean（x拔、μ）平均数；Median中位数；Mode中数；range极差；variance方差（σ²、s²）；StandardDevration标准差（σ、s）；biostatistics/biometry生物统计学；Quartiles四分位数IQR=Q3-Q1；a平方和与自由度的拆分RandomizedcopleteblockdesignANOVATheanalysisofvariancetableforarandomizedcompleteblockdesignSourceofvarianceSSdfMSFc.v.TreatmentsSSTreatk-1SeeTableC.6BlocksSSBlocksb-1ErrorSSError(k-1)(b-1)TotalSSTotalkb-1FactorialDesignTwo-wayANOVASourceofvarianceSSdfCellsSSCellsab-1AfactorsSSAa-1BfactorsSSBb-1A×BinteractionSSA×B(a-1)(b-1)ErrorSSErrorab(n-1)TotalSSTotalabn-1各种分布的特征:概率分布泊松分布标准正态分布t分布分布F分布半期考试填空问答题考试内容：第五章ExplainwhymostresearchersaremorecomfortablerejectingH0thanacceptingit.Useaprobabilityargumentinyourexplanation.DiscusstherelationshipbetweentheCentralLimitTheoremandthesamplingdistributionfor.TheCentralLimitTheoremdescribesthesamplingdistributionofasnormalorapproximatelynormalwithastandarddeviationequaltothestandarderrorofthemean.theCentralLimitTheoremOutlinethefactorsaffectingTypeⅡerrorandthepowerofatestofthehypothesis.Explainwhyitisinappropriatetousetheterm“proof”whenperformingtestsofhypothesis.InstandardEnglish,toprovesomethingmeanstoestablishconclusivelyitstruthorvalidity.Withtestsofhypothesiswealwayshavesomeprobabilityorpossibilityofbeingincorrect.AcceptinganH0TypeⅠerror(acceptingafalseH0)andhasaprobabilityofα.RejectingWhyareH0andHaalwayswrittenasmutuallyexclusiveandall-inclusivepredictions?计算置信区间均值之间的比较（独立样本t检验不考）提出假设H0：Ha：确定检验统计量若总体方差已知，使用Z分布若总体方差已知，使用t分布做出统计判断P125P151方差分析（答题模式同半期一样，11.3中2×2列联表）One-wayANOVAThehypothesesareH0:Ha:atleastonepairofμ’sarenotequal.TheanalysisofvariancetableSourceofvariationSunofsquaresdfMsFc.v.TreatmentsSSTreatk-1SeeTableC.6ErrorSSErrorN-kTotalSSTotalN-12×2ContingencyTablesn11n12n1.n21n22n2.n.1n.2n..ThehypothesesareH0:twotitlesareindependent.Ha:twotitlesaredependent.Thescientificmethod:ObservationofaParticularEventStatementoftheProblemFormulationofaHypothesisMakingaPredictionDesignoftheExperimentExplainwhymostresearchersaremorecomfortablerejectingH0thanacceptingit.Useaprobabilityargumentinyourexplanation.Solution:BecauseifwerejectH0,wemightmakeaTypeⅠerrorandtheprobabilityisα.IfweacceptHα,wemightmakeaTypeⅡerrorandtheprobabilityisβ.Butascertainingtheprobabilityofthiskindoferrorisimpossibleinmostexperimentalsituations.SomostresearchersaremorecomfortablerejectingH0thanacceptingit.DiscusstherelationshipbetweentheCentralLimitTheoremandthesamplingdistributionfor.Solution:TheCentralLimitTheoremdescribesthesamplingdistributionofX-barasnormalorapproximatelynormalwithastandarddeviationequaltothestandarderrorofthemean.OutlinethefactorsaffectingTypeⅡerrorandthepowerofatestofthehypothesis.Solution:TheprobabilityofaTypeⅡerrorisdecreasedandthepowerofatestsubsequentlyincreasedwhenthesamplingdistributionsunderH0andHαoverlapless.Thespreadofthesedistributionsisdecreasedbydecreasingσorincreasingn.Explainwhyitisinappropriatetousetheterm“proof”whenperformingtestsofhypothesis.Solution:InstandardEnglish,toprovesomethingmeanstoestablishconclusivelyitstruthorvalidity.Withtestsofhypothesiswealwayshavesomeprobabilityorpossibilityofbeingincorrect.AcceptinganH0couldleadtoaTypeⅠerror(acceptingafalseH0)andhasaprobabilityofα.RejectinganH0couldleadtoaTypeⅡerror(rejectingatrueH0)andhasaprobabilityofβ.Whileαorβissometimesquitesmalleachisneverzero.WesayH0issupportedornotsupportedbythestatisticaltest,neverthatthetestprovesH0tobetrueorfalse.WhyareH0andHαalwayswrittenasmutuallyexclusiveandallinclusivepredictions?Solution:Accordingtothephilosophyofscienceonlyoneofthehypothesescanbetrue.IfH0istrue,thenHαmustbefalseandviceversa.ThisrequiresthatH0andHαbewrittenasmutuallyexclusiveandall-inclusivepredictions.Exactlyoneofthesepredictionswillbesupportedbythestatisticaltest.Todothistheymustbeinclusiveofallpossibleoutcomes.Canyouthinkofbiologicalexampleswherethiskindoflogicmightleadyoutochooseverysmallalphalevels?Solution:Supposeanewtreatmentforaparticularcanceristhoughttoprolonglifebysixmonths.Ifthistreatmentisverycostlyorpainful,youwouldchooseasmallalphaleveltobeveryconfidentthatitactuallyworksbeforeitismarketedasaneffectivetreatment.Itwouldbetragictosubmitpatientstopainfulorcostlytreatmentthatarenoteffective.ExplaintheCentralLimitTheorem.Solution:WhensamplingfromanonnormallydistributedpopulationwithmeanμXandvarianceσ2x,thedistributionofthesamplemean(samplingdistribution)willhavethisproperty:ThedistributionofX-barswillbeapproximatelynormalitybecomingbetterasthesamplesizeincreases.Generally,thesamplesizerequiredforthesamplingdistributionofX-bartoapproachnormalitydependsontheshapeoftheoriginaldistribution.Samplesof30ormoregiveverygoodnormalapproximationsforthissamplingdistributionofX-barinnearlyallsituations.Listthe9typicalstepsinastatisticaltestofhypothesis.Solution:(1)Statetheproblem.Formulatethenullandalternativehypotheses:H0:μ=10ozHα:μ≠10ozChoosethelevelofsignificance.Determinetheappropriateteststatistic.Calculatetheappropriateteststatistic.(a)Determinethecriticalvaluesforthesamplingdistributionandappropriatelevelofsignificance.(b)DeterminethePvalueoftheteststatistic(7)(a)Comparetheteststatistictothecriticalvalues(b)ComparethePvaluetoyourchosenlevelofsignificance(8)Basedonthecomparisoninstep7,acceptorrejectH0(9)Stateyourconclusionandanswerthequestionposedinstep1.Whenregressionanalysisisappropriate,wedothefollowing:GraphdatatoascertainthatalinearrelationshipisapparentCalculatetheregressionequationusingtheleastsquaresmethodTestthesignificanceofthisequationwithanalysisofvarianceIftheH0isrejected,plottheequationwithanalysisofvarianceFinally,calculateanyrequiredconfidenceintervalsinsupportoflinearrelationshipBoxPlotsDrawahorizontalorverticalreferencescalebasedontherangeofthedatasetCalculatethemedian,Q1,Q3,andtheIQRDeterminethefencesf1andf3usingtheformulasbelow.Anypointslyingoutsidethesefenceswillbeconsideredoutliersandmaywarrantfurtherinvestigation.f1=Q1-1.5(IQR)f3=Q3+1.5(IQR)Locatethetwo“adjacentvalues”,therearethesmallestandlargestdatavaluesinsidethefencesLightlymarkthemedian,quartilesandadjacentvaluesonthescale.Chooseascaletospreadthesepointsoutsufficiently.Besidethescale,constructaboxwithendsatthequartilesandwithadashedinteriorlinedrawnatthemedian.Generallythiswillnotbeatthemiddleofthebox.Drawa“whisker”(linesegment)fromthequartilestotheadjacentvaluesthataremarkedwithcrosses“X”.Markanyoutliersbeyondthefences(equivalently,beyondtheadjacentvalues)withopencircles”○”.Five-numbersummary:Median:XmedianQuartiles:Q1Q3Extremes：LargestnumberSmallestnumberModelANOVASolution:（1）ThehypothesesareH0:μ1=μ2=μ3=…,μkHa:Atleastofoneoftheμi’sarenotequalAccordingtotheformulationtocomputetheSSTotal=SSTreat+SSErrorThesummarydatatotestH0appearinthetablebelowSourceofvariationSumofsquaresdfMSFc.vTreatmentSSTreatk-1SSTreat/k-1MSTreat/MSEseeTableC.6ErrorSSErrorN-kSSError/N-kTotalSSTotalN-1ThecriticalvaluefromTableC.6witha=0.05andv1=k-1andv2=b-1,isC.V.SincewhenC.V.＜F,werejectH0,acceptHa.Itindicatesthatthetreatmentmeansaresignificantlydifferent.AndthenifC.V.＞F,weacceptH0,rejectHa.Itindicatesthatthemeansquaresdietsisnotsignificantlydifferent.If.C.V.＜F,wefindatleasttwoofthesemeansaresignificantlydifferent.weusetheDMRTtofindwhichmeanhavethesignificantlydifferent.firstthemeansarealreadyrank-ordered.WecalculatetheSSRp,ifcalculatedvalue＞SSRp,itissignificantlydifferent.①rpisatablevalueofTableC.7(2)MSEistheerrormeansquarefromtheANOVAtable(3)nisthecommonsamplesizeWeusedifferentsuperscriptletterstosaytheyaresignificantdifferences.Ifcalculatedvalue＜SSRp,itisnotsignificantlydifferent.Weusesamesuperscriptletterstosaytheyarenotsignificantdifferences.Forexample`X1a,`X2a,`X3bWecangettheconclusionthatthemeanfortheX1andX2isnotsignificantlydifferent,themeanfortheX2andX3issignificantlydifferent,themeanfortheX1andX3issignificantlydifferent.RandomizedCompleteBlockDesignANOVASolution:ThehypothesesareH0:μ1=μ2=μ3=…,μkHa:Atleastofoneoftheμi’sarenotequalAccordingtotheformulationtocomputetheSSTotal=SSTreat+SSErrorThesummarydatatotestH0appearinthetablebelowSourceofvariationSumofsquaresdfMSFc.v.TreatmentSSTreatk-1SSTreat/k-1MSTreat/MSEseeTableC.6BlocksSSBlocksb-1SSBlocks/b-1ErrorSSError(k-1)(b-1)SSError/(k-1)(b-1)TotalSSTotalkb-1ThecriticalvaluefromTableC.6witha=0.05andv1=k-1andv2=(k-1)(b-1),isC.V.SincewhenC.V.＜F,werejectH0,acceptHa.Itindicatesthatthetreatmentmeansaresignificantlydifferent.AndthenifC.V.＞F,weacceptH0,rejectHa.Itindicatesthatthemeansquaresdietsisnotsignificantlydifferent.If.C.V.＜F,wefindatleasttwoofthesemeansaresignificantlydifferent.WeusetheDMRTtofindwhichmeanhavethesignificantlydifferent.firstthemeansarealreadyrank-ordered.WecalculatetheSSRp,ifcalculatedvalue＞SSRp,itissignificantlydifferent.rpisatablevalueofTableC.7MSEistheerrormeansquarefromtheANOVAtablenisthecommonsamplesizeWeusedifferentsuperscriptletterstosaytheyaresignificantdifferences.Ifcalculatedvalue＜SSRp,itisnotsignificantlydifferent.Weusesamesuperscriptletterstosaytheyarenotsignificantdifferences.Forexample`X1a,`X2a,`X3bWecangettheconclusionthatthemeanfortheX1andX2isnotsignificantlydifferent,themeanfortheX2andX3issignificantlydifferent,themeanfortheX1andX3issignificantlydifferent.FactorialDesignTwo-WayANOVASolution:HeanalysisofvariancetableSourceofvariationSSdfMSE(MS)CellsSScellsab-1MScellsAfactorsSSBa-1MSABfactorsSSAb-1MSBA*BinteractionSSA*B(a-1)(b-1)MSA*BErrorSSErrorab(n-1)MSETotalSSTotalabn-1(1)TestforinteractionbetweenfactorsFirsttestforinteractionbetweenthetwotypeoffacterbeinginvestigated.ThehypothesesareH0：(αβ）ij=0foralli,jHa：(αβ）ij≠0forsomei,j.TheappropriatetestforthisnullhypothesisisFA*B=MSA*B/MSE,withdegreesoffreedom,ν1=(a-1)(b-1),ν2=ab(n-1)fromTableC.6witha=0.05andv1=(a-1)(b-1)andv2=ab(n-1),isC.V.WhenFA*B＜C.V.,weacceptH0.Therearenointeractions.Sotheanalysisiscontinuedcarryingouttwotests.(2)Iftherearenointeractions(a)TestwhethertherearedifferencesinmeansforA-factortreatmentSThehypothesesareH0：μi..，sallequal(αi=0foralli)Ha：atleastonepairofμi..，snotequalUsingtheteststatisticFA=MSA/MSE,Withdegreesoffreedomν1=a-1,ν2=ab(n-1)whenC.V.＜FA,werejectH0,acceptHa.Itindicatesthatatleastonepairofμi..，snotequal.AndthenifC.V.＞FA,weacceptH0,rejectHa.ItindicatesthatthemeanforfactorAlevelareallequal.If.C.V.＜FAwefindatleastsomeofthesemeansaresignificantlydifferent.weusetheDMRTtofindwhichmeanshavethesignificantlydifferent.WecalculatetheSSRp,ifcalculatedvalue＞SSRp,itissignificantlydifferent.Weusedifferentsuperscriptletterstosaytheyaresignificantdifferences.Ifcalculatedvalue＜SSRp,itisnotsignificantlydifferent.Weusesamesuperscriptletterstosaytheyarenotsignificantdifferences.Forexample`X1a,`X2a,`X3bWecangettheconclusionthatthemeanfortheX1andX2isnotsignificantlydifferent,themeanfortheX2andX3issignificantlydifferent,themeanfortheX1andX3issignificantlydifferent.(b)Similary,testwhethertherearedifferencesinmeansforB-factortreatmentsThehypothesesareH0：μi..，sallequal(αi=0foralli)Ha：atleastonepairofμi..，snotequalUsingtheteststatisticFB=MSB/MSE,Withdegreesoffreedomν1=b-1,ν2=ab(n-1)whenC.V.＜FBwerejectH0,acceptHa.Itindicatesthatatleastonepairofμi..，snotequal.AndthenifC.V.＞FB,weacceptH0,rejectHa.ItindicatesthatthemeanforfactorAlevelareallequal.If.C.V.＜FBwefindatleastsomeofthesemeansaresignificantlydifferent.weusetheDMRTtofindwhichmeanshavethesignificantlydifferent.WecalculatetheSSRp,ifcalculatedvalue＞SSRp,itissignificantlydifferent.Weusedifferentsuperscriptletterstosaytheyaresignificantdifferences.Ifcalculatedvalue＜SSRp,itisnotsignificantlydifferent.Weusesamesuperscriptletterstosaytheyare