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1、Chapter9StatisticalAnalysisofCategoricalDataInGodwetrust;allothersbringdata.Statistician’smotto9.1OrientationOneofthemostfamouscategoricaldatasetsinbiologyistheplantbreedingexperimentoftheAustrianmonkandbotanistG.Mendel(1822-1884)whoclassi?edpeasaccordingtotheirshape(round(r)orwrink

2、led(w))andcolor(yellow(y)orgreen(g)).Eachseedwasclassi?edintooneoffourcategories:(ry)=(round,yellow),(rg)=(round,green),(wy)=(wrinkled,yellow),and(wg)=(wrinkled,green).AccordingtoMendel’stheorythefrequencycountsofseedsofeachtypeproducedfromthisexperimentoccurinthefollowingratios:9:3

3、:3:1.Thus,theratioofthenumberof(ry)peasto(rg)peasshouldbeequalto9:3,andsimilarlyfortheotherratios.Mendel’sexperimentraisesthefollowingfundamentalscienti?cquestion:AreMendel’sdata,displayedinTable9.1,consistentwiththepredictionsofhisscienti?cmodel?Toanswertheseandmorecomplexquestions

4、arisinginthestudyofcategoricaldata,statisticiansusethechi-squaretest.OrganizationofChapter1.Section9.2:Chi-SquareTests2.Section9.3:ContingencyTables3.Section9.4:ChapterSummary9.2Chi-SquareTestsThechi-squaretestisusedtotesthypothesesabouttheparametersofoneormoremulti-nomialdistributi

5、ons.Recallthatamultinomialexperimentisthemathematicalmodelofnrepetitionsofanexperimentwithkmutuallyexclusiveandexhaustiveoutcomes,denotedC1,...,Ck,withprobabilitiespi=P(Ci),whereipi=1.(ThemultinomialdistributionisdiscussedinSection5.3.)TheoutcomesCiarecalledcategoriesorcells;theirp

6、robabil-itiespi=P(Ci)arecalledcellprobabilities.Inthesimplestcasethecellprobabilitiesarecompletelyspeci?ed,sothenullhypothesisisgivenby:H0:P(Ci)=pi,i=1,...,k.(9.1)Example9.1373374IntroductiontoProbabilityandStatisticsforScience,Engineering,andFinanceOneofthemostfamousmultinomialexpe

7、rimentsinthehistoryofsciencewasperformedbyG.Mendel,whoclassi?edn=556peasaccordingtotwotraits:shape(roundorwrinkled)andcolor(yelloworgreen).Eachseedwasputintooneofk=4categories:C1=ry,C2=rg,C3=wy,C4=wg,wherer,w,g,andydenoteround,wrinkled,green,andyellow,respectively.TheresultsofMendel

8、’sbreedingexperimen