Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap !-Nonparametric TestsChapter 16Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-2!"apter %oalsAfter completing this chapter, you should be able to:&ppl' an( interpret t"e )ilcoxon ran* s+m test for t"e (ifference ,et-een t-o me(iansPerform nonparametric anal'sis of .ariance +sing t"e /r+s*al-)allis ran* test for one--a' &N01&Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-2#$# Nonparametric or (istri,+tion-free met"o(sNonparametric tests are +se( -"en normalit' consi(eration isn3t nee(e(In fact, t"ese tests are (istri,+tion-free an( +se( -it" increasing fre4+enc' ,' (ata anal'stsStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-4Sign testStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-5Signe( 6an*TestStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-$Test t-o in(epen(ent pop+lation me(iansPop+lations nee( not ,e normall' (istri,+te(7istri,+tion free proce(+reUse( -"en onl' ran* (ata are a.aila,leM+st +se normal approximation if eit"er of t"e sample si8es is larger t"an #0)ilcoxon 6an*-S+m Test for 7ifferences in 2 Me(iansStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-9)ilcoxon 6an*-S+m Test: Small Samples!an +se -"en ,ot"n# , n2; #0&ssign ran*s to t"e com,ine(n# < n2 sample o,ser.ationsIf +ne4+al sample si8es, let n# refer to smaller-si8e( sampleSmallest .al+e ran* = #, largest .al+e ran* = n# < n2 &ssign a.erage ran* for tiesS+m t"e ran*s for eac" sample: T# an( T2 0,tain test statistic, T#>from smaller sample?Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-@!"ec*ing t"e 6an*ingsT"e s+m of t"e ran*ings m+st satisf' t"e form+la ,elo-!an +se t"is to .erif' t"e s+ms T# an( T22#? n>nT T2 #+ +-"eren = n# < n2Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-A)ilcoxon 6an*-S+m Test:H'pot"esis an( 7ecision 6+leH0: M# = M2H#: M# M2H0: M# M2H#: M# > M2H0: M# M2H#: M# < M2T-o-Tail Test Beft-Tail Test 6ig"t-Tail TestM# = me(ian of pop+lation #CM2 = me(ian of pop+lation 2 6eDectT1LT1U6eDect7o Not 6eDect6eDectT1L7o Not 6eDectT1U6eDect7o Not 6eDectTest statistic = T#>S+m of ran*s from smaller sample?6eDect H0 if T# E T#Bor if T# F T#U6eDect H0 if T# E T#B6eDect H0 if T# F T#UStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-#0Sample (ata is collecte( on t"e capacit' rates >G of capacit'? for t-o factories &re t"e me(ian operating rates for t-o factories t"e sameHIor factor' &, t"e rates are 9#, @2, 99, A4, @@Ior factor' J, t"e rates are @5, @2, A2, A9Test for e4+alit' of t"e sample me(ians at t"e 005 significance le.el )ilcoxon 6an*-S+m Test: Small Sample ExampleStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-##)ilcoxon 6an*-S+m Test: Small Sample ExampleCapacity Rankactory A actory ! actory A actory !"1 1"" #$# %&'$# %&'$' '$$ 6(# "() $(" (Rank *ums: #+&' #)&'Tie in 2r( an( 4t" placesRankedCapacity,alues:(continued)Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-#2)ilcoxon 6an*-S+m Test: Small Sample Example(continued)Iactor' J "as t"e smaller sample si8e, so t"e test statistic is t"e s+m of t"e Iactor' J ran*s:T# = 245T"e sample si8es are: n# = 4 >factor' J?n2 = 5 >factor' &?T"e le.el of significance is K = 05Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-#2n2n#0ne-Taile(T-o-Taile(4 54505 #0 #2, 2@ #A, 2$025 05 ##, 2A #9, 2@0# 02 #0, 20 #$, 2A005 0# --, -- #5, 40$)ilcoxon 6an*-S+m Test: Small Sample ExampleBo-er an( Upper !ritical 1al+es for T# from &ppen(ix ta,le E@:(continued)T1L - 11andT1U - #(Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-#4H0: M# = M2H#: M# M2T-o-Tail Test6eDectT1L-11 T1U-#(6eDect7o Not 6eDect6eDect H0 if T# E T#B=##or if T# F T#U=2A = 05n# = 4 ,n2 = 5 Test *tatistic >S+m of ran*s from smaller sample?:T# = 245 .ecision:Conclusion:7o not reDect at = 005T"ere is not eno+g" e.i(ence to pro.e t"at t"e me(ians are not e4+al)ilcoxon 6an*-S+m Test:Small Sample Sol+tion(continued)Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-#5)ilcoxon 6an*-S+m Test >Barge Sample?Ior large samples, t"e test statistic T#is approximatel' normal -it" meanan( stan(ar( (e.iation : M+st +se t"e normal approximation if eit"er n# or n2 F #0&ssign n# to ,e t"e smaller of t"e t-o sample si8es !an +se t"e normal approximation for small samples#TL#TM2? # n > nL#T#+#2? # n > n nM2 #T#+Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-#$)ilcoxon 6an*-S+m Test >Barge Sample?T"e Ntest statistic is )"ere N approximatel' follo-s a stan(ar(i8e( normal (istri,+tion##TT #ML TN(continued)Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-#9)ilcoxon 6an*-S+m Test: Normal &pproximation ExampleUse t"e setting of t"e prior example:T"e sample si8es -ere: n# = 4 >factor' J?n2 = 5 >factor' &?T"e le.el of significance -as K = 05T"e test statistic -as T# = 245Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-#@)ilcoxon 6an*-S+m Test: Normal &pproximation ExampleT"e test statistic is202? # A > 42? # n > nL#T#++92A2#2? # A > ? 5 > 4#2? # n > n nM2 #T#++(continued)$4#292A20 524ML TN##TT #N = #$4 is not greater t"an t"e critical N .al+e of #A$ >for K = 05? so -e (o not reDect H0 O t"ere is not s+fficient e.i(ence t"at t"e me(ians are not e4+alStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-#A/r+s*al-)allis 6an* TestTests t"e e4+alit' of more t"an 2 pop+lation me(iansUse -"en t"e normalit' ass+mption for one--a' &N01& is .iolate(&ss+mptions:T"e samples are ran(om an( in(epen(ent.aria,les "a.e a contin+o+s (istri,+tiont"e (ata can ,e ran*e(pop+lations "a.e t"e same .aria,ilit'pop+lations "a.e t"e same s"apeStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-20/r+s*al-)allis Test Proce(+re0,tain relati.e ran*ings for eac" .al+eIn e.ent of tie, eac" of t"e tie( .al+es gets t"e a.erage ran*S+m t"e ran*ings for (ata from eac" of t"e c gro+ps!omp+te t"e H test statisticStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-2#/r+s*al-)allis Test Proce(+reT"e /r+s*al-)allis H test statistic: >-it" c O # (egrees of free(om?) 1 ( 3) 1 (1212+ 11]1

+nnRn nHcjjj-"ere:n = s+m of sample si8es in all samplesc = N+m,er of samplesTD = S+m of ran*s in t"e Dt" samplenD = Si8e of t"e Dt" sample(continued)Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-227ecision r+le6eDect H+ if test statistic H F 2U0t"er-ise (o not reDect H+(continued)/r+s*al-)allis Test Proce(+re!omplete t"e test ,' comparing t"e calc+late( H .al+e to a critical 2 .al+e from t"e c"i-s4+are (istri,+tion -it" c O # (egrees of free(om22U0 6eDect H07o not reDect H0Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-227o (ifferent (epartments "a.e (ifferent class si8esH/r+s*al-)allis Example!lass si8e >Mat", M?!lass si8e >Englis", E?!lass si8e >Jiolog', J?#%)'')"$66''6+"#)'"+%+)+1$%)))Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-247o (ifferent (epartments "a.e (ifferent class si8esH/r+s*al-)allis Example!lass si8e >Mat", M?6an*ing!lass si8e >Englis", E?6an*ing!lass si8e >Jiolog', J?6an*ing#%)1')"$66#6(1'1#''6+"#)'"+1+111)$1%%+)+1$%)))%'1)" - )) - '6 - #+Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-25T"e H statistic is(continued)/r+s*al-)allis Example92$ ? # #5 > 252055$544? # #5 > #5#2? # n > 2n6? # n > n#2H2 2 2c# DD2D + 1]1

,_

+ +++ 11]1

+e4+al are Me(ians pop+lation all ot N : HMe(ian Me(ian Me(ian : H&H E M 0 Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-2$SinceH = $92 F reDect H0(continued)/r+s*al-)allis Example991 . 52 U!ompareH = $92to t"e critical .al+e from t"e c"i-s4+are (istri,+tion for2 O # = 2(egrees of free(om an( = 05:991 . 52 UT"ere is s+fficient e.i(ence to reDect t"at t"e pop+lation me(ians are all e4+alStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap #$-29!"apter S+mmar'Use( t"e )ilcoxon ran* s+m test for t-o pop+lation me(iansSmall SamplesBarge sample N approximation&pplie( t"e /r+s*al-)allis H-test for m+ltiple pop+lation me(ians