Homework(6(–(Solutions((1. Effectiveness,of,grasses,in,preventing,erosion,,,(a) There,is,one,sample,of,11,freeway,banks.,,At,each,location,,two,test,plots,are,established.,,The,observations,from,these,test,plots,are,paired,,so,we,will,want,to,analyze,the,differences.,,The,“group”,variable,is,the,type,of,grass.,,Since,we,have,control,over,the,assignment,of,grass,type,to,test,plots,at,a,freeway,bank,and,the,investigators,assign,grass,type,randomly,to,test,plots,,this,is,a,randomized,experiment.,,The,study,isn’t,based,on,a,random,sample,,but,we,hope,that,the,set,of,freeway,banks,are,representative,of,all,freeway,banks.,(b) Note,that,the,data,are,paired,,so,we,make,a,stem,and,leaf,plot,of,the,differences,Freeway Bank 1 2 3 4 5 6 7 8 9 10 11 Grass A 32 11 47 7 26 18 28 39 36 55 34 Grass B 19 13 21 8 20 15 21 30 42 25 29 Differences 13 -2 26 -1 6 3 7 9 -6 30 5 Stem,and,leaf,plot,…,30,|,0,20,|,6,10,|,3,,,0,|,35679,P0, |,126,The,data,appear,to,be,rightPskewed,with,a,couple,large,values.,,With,the,small,sample,size,,the,tPtest,might,not,be,robust.,(c) Since,we,have,a,paired,sample,,we,use,the,signedPrank,test.,,Note,that,we,don't,have,two,independent,samples,,so,we,would,NOT,use,the,rankPsum,test.,,,H0,:,M,=,0,vs.,Ha:,M,≠,0,,for,M,=,median,of,the,difference,in,erosion,for,A,and,B,(A,–,B).,,,,There,are,no,zero,differences,to,drop,,giving,n,=,11,nonPzero,differences.,,Order,the,absolute,differences,and,assign,ranks:,Sign - - + + + - + + + + + Abs diff 1 2 3 5 6 6 7 9 13 26 30 Order 1 2 3 4 5 6 7 8 9 10 11 Rank 1 2 3 4 5.5 5.5 7 8 9 10 11 S,=,sum,of,ranks,for,pairs,with,positive,difference,=,3,+,4,+,5.5,+,7,+,8,+,9,+,10,+,11,=,57.5,Mean(S),=,n#(n#+,1),/,4,=,11(12)/4,=,33, , ,SD(S),=,SD(S),=, ! ! + 1 (2! + 1)/24!=, 11 12 (23)/24!= 126.5,=,11.24722,Z,=,(S#P,Mean(S)),/,SD(S),=,(57.5,P,33),/,11.24722,=,2.18,twoPsided,pPvalue,=,2,x,(%,of,tPratios,with,infinite,degrees,of,freedom,larger,than,|Z|),, , ,,,,,,,=,2,x,(%,of,tPratios,with,infinite,degrees,of,freedom,larger,than,2.18),, , , is,between,2,x,0.02,=,0.04,and,2,x,0.01,=,0.02,0.02,<,p,<,0.04,There,is,moderate,evidence,of,a,difference,between,the,two,grasses,in,their,ability,to,prevent,soil,erosion.,,The,median,runPoff,for,Grass,A,is,larger,than,the,median,runPoff,for,Grass,B.,,,,,,,(c),See,SAS,program,on,web,(freeway.sas),,which,calculates,the,difference,and,then,runs,UNIVARIATE,on,the,difference,variable.,,Note,that,the,value,of,the,statistic,cited,by,SAS,is,actually,our,S,in,part,(b),minus,Mean(S),=,57.5,–,33,=,23.5.,,Also,,note,that,the,pPvalue,calculated,by,SAS,is,in,the,interval,from,our,pPvalue,in,(b).,,2. Approximate,onePsided,pPvalue,for,the,rank,sum,test,,H0,:,distributions,for,population,A,and,population,B,are,the,same,,Ha,:,distribution,for,population,B,tends,to,have,larger,values,than,the,distribution,for,population,A,,First,,we,combine,all,data,values,and,determine,the,ranks.,,Group,A(A(B,B,A(B,B,B,B,A(B,B,B,B,B,B,B,B,Data(1(1(1,1,2(2,2,3,4,5(7,15,32,41,77,107,299,976,Order,1(2(3,4,5(6,7,8,9,10(11,12,13,14,15,16,17,18,Rank,2.5(2.5(2.5,2.5,6(6,6,8,9,10(11,12,13,14,15,16,17,18,For,the,ties,,average,the,orders,for,the,tied,observations,to,get,the,rank,for,these,observations.,,,For,the,set,of,1’s,:,(1,+,2,+,3,+,4),/,4,=,2.5,,,,,,,,,,,,,For,the,set,of,2’s,:,(5,+,6,+,7),/,3,=,6,,},T,=,sum,of,the,ranks,corresponding,to,values,from,sample,A,=,2.5,+,2.5,+,6,+,10,=,21,Mean(T),=,n1,(n1,+,n2,+,1),/,2,=,4(19)/2,=,38,The,standard,deviation,of,all,the,ranks:,2.5,,2.5,,2.5,,2.5,,6,,6,,6,,8,,9,,10,,11,,12,,13,,14,,15,,16,,17,,18,is,sR,=,5.300,,SD(T),=,sRn!n!/(n!+n!)!=,5.300 4 14 /18!= 126.5,=,9.34832,,,(Note:,,this,formula,for,SD(T),is,not,really,appropriate,with,this,many,ties,–,it,overstates,the,SD.),Z,=,(21,–,38),/,9.34832,=,P1.819,The,onePsided,pPvalue,we,want,is,the,proportion,of,area,under,the,Z,distribution,curve,that,are,less,than,the,value,of,our,ZPstatistics,,P1.819.,,We,use,the,lower,tail,because,we,want,to,see,if,Population,B,has,larger,values,than,Population,A,(i.e.,,Population,A,values,are,smaller,that,Population,B,values,and,thus,the,ranks,are,lower,since,data,are,ordered,from,smallest,to,largest).,,We,can,use,the,tPtable,by,using,the,row,with,infinite,degrees,of,freedom,and,we,can,use,the,symmetry,of,the,distribution,to,change,the,problem,to,determining,the,proportion,of,values,larger,than,1.819.,,Using,this,method,,we,get,a,pPvalue,of,0.025,<,p,<,0.05.,,Thus,there,is,evidence,that,the,distribution,for,population,B,is,shifted,to,the,right,of,population,A,(i.e.,,has,larger,values).,,3. CA,population,estimate,data,,,,Recall,that,transformations,did,not,simultaneously,resolve,problems,with,normality,and,homogeneous,variances.,Thus,the,tPtest,might,not,be,robust,,and,a,rank,sum,test,would,be,a,good,alternative.,H0,:,CA,population,estimate,distributions,for,‘7’,and,‘70’,anchor,groups,have,the,same,shape,,Ha,:,The,estimates,for,‘7’,group,tend,to,be,smaller,or,larger,relative,to,the,‘70’,anchor,group,See,SAS,program,and,output,below.,SAS,will,use,the,‘70’,group,ranks,because,the,sample,size,is,smaller.,Z,=,[T,for,‘70’,group],–,Mean(T)],/,SD(T),=,4.337,SAS,uses,a,normal,approximation,to,generate,a,pPvalue,<,0.001,,There,is,convincing,evidence,that,the,anchoring,treatment,has,an,effect,on,the,distributions.,,The,median,CA,population,estimates,for,the,group,given,the,7,million,anchor,appear,to,be,smaller,than,the,estimates,for,the,group,anchored,with,70,million,residents.,proc npar1way wilcoxon data=a ; var guess ; class question ; title2 'Rank-sum test (called Wilcoxon test in SAS output)' ; run ; CA population estimates (guess) for each priming treatment (question) Rank-sum test (called Wilcoxon test in SAS output) The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable guess Classified by Variable question Sum of Expected Std Dev Mean question N Scores Under H0 Under H0 Score 7 21 261.50 409.50 34.008356 12.452381 70 17