{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "MULTIVARIATE REGRESSION:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "with(stats): with (linalg): with(ListTools):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, \+ the protected names norm and trace have been redefined and unprotected \n" }}{PARA 7 "" 1 "" {TEXT -1 58 "Warning, the assigned name Group no w has a global binding\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Gener ating data (note the 'model' used):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "x1:=[$1..27];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G7=\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"* \"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\" #F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x2:=Flatten([[$1..9]$ 3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G7=\"\"\"\"\"#\"\"$\"\"% \"\"&\"\"'\"\"(\"\")\"\"*F&F'F(F)F*F+F,F-F.F&F'F(F)F*F+F,F-F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x3:=Flatten([[$1..3]$9]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G7=\"\"\"\"\"#\"\"$F&F'F(F&F'F(F &F'F(F&F'F(F&F'F(F&F'F(F&F'F(F&F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "y:=evalm(3*x1-7*x2+11*x3+20+6*[random[normald](27)]); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"yG-%'vectorG6#7=$\"+TP]0M!\")$ \"+A:)>1$F+$\"+(RO7C%F+$\"*D)pWjF+$\"+gA[_:F+$\"+77z')GF+$!+=q;^7F+$\" *Ks.M(F+$\"+66.)4\"F+$\"+d\"yKQ&F+$\"+:pu:qF+$\"+Fw#pV'F+$\"+ZZU)H%F+$ \"+T[\"=H&F+$\"+EuOv_F+$\"+.b,\"G%F+$\"+IVl5QF+$\"+UB-ISF+$\"+R9yIyF+$ \"+JWH%H*F+$\"+=)z7l*F+$\"+r!)4:qF+$\"+F.z'4)F+$\"+@(RYD(F+$\"+_i%QP'F +$\"+TGm.jF+$\"+!eUk;'F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Fitti ng the 'best' linear equation to this data:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "n:=27: k:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "X:=matrix(n,1,1.):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "X:=augment(X,x1,x2,x3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'matrixG6#7=7&$\"\"\"\"\"!F+F+F+7&F* \"\"#F.F.7&F*\"\"$F0F07&F*\"\"%F2F+7&F*\"\"&F4F.7&F*\"\"'F6F07&F*\"\"( F8F+7&F*\"\")F:F.7&F*\"\"*FF+F+7&F*\"#?F.F.7&F*\"#@F0F07&F*\"#AF2F+7&F*\"#BF4F.7&F*\"#CF6F0 7&F*\"#DF8F+7&F*\"#EF:F.7&F*\"#FF " 0 "" {MPLTEXT 1 0 30 "C:=inverse(transpose(X) &* X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'matrixG6#7&7&$\"+/Pq.P!#5$!+1&RG<'!#7$!+,zcM7! #6$!+f#f#f#*F27&F-$\"+i0reo!#8$!+i0reoF8$\"\"!F<7&F0F9$F7F/F-7&F3F;F-$ \"+1&RG<'F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Regression-coeffic ient point estimates:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "betaHat:=evalm(C &* transpose(X) &* y);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(betaHatG-%'vectorG6#7&$\"+_z!>K#!\")$\"+,Rj'= $!\"*$!+t9]uxF.$\"+l \+ " 0 "" {MPLTEXT 1 0 26 "yHat:=evalm(X &* betaHat);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%%yHatG-%'vectorG6#7=$\"+s$od#G!\")$\"+!zG'HLF+$\"+5 #*[LQF+$\"++\"3%\\9F+$\"+>&oK&>F+$\"+R*GrX#F+$\"*&Gy/t!\"*$\"*[#3pdF+$ \"+n'o23\"F+$\"+#))QPp&F+$\"+,$*f(>'F+$\"+@(f9q'F+$\"+5'ytJ%F+$\"+I!R7 #[F+$\"+]%*4D`F+$\"+S$=5%HF+$\"+f(y[W$F+$\"+y\"R([RF+$\"+%R4)F+$\"+Y)))*3eF +$\"+v#\\GJ'F+$\"+%o4n\"oF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Th e corresponding residuals:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "res:=evalm(y-yHat);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$resG-%'vectorG6#7=$\"*p`tz&!\")$!*oskn#F+$\"*(=ZxSF+ $!*v#Q\\\")F+$!*fiy+%F+$\"*tAmH%F+$!+Y[@C8F+$\"*%)*Gr:F+$\")WCEYsG#F+$\")'e\\=)F+$!*^5Dq\"F+$\" *'y!e2%F+$!*R-VQ*F+$\"*1u&[cF+$!(Mk=*F+$!*/rE]'F+" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 32 "Residual (error) sum of squares:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "SSe:=sum(res[i]^2,i=1. .n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SSeG$\"+A.!*o()!\"(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Mean square error:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "MSe:=SSe/(n-k-1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MSeG$\"+d`c7Q!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Point estimate of sigma:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sigmaHat:=sqrt(MSe); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)sigmaHatG$\"+Uufuh!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The coresponding CI (for sigma):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "sqrt(S Se/statevalf[icdf,chisquare[n-k-1]](.975)),\nsqrt(SSe/statevalf[icdf,c hisquare[n-k-1]](.025));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+K5)*)z %!\"*$\"+(3#[h')F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The standar d errors of regression coefficients:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for i to k+1 do sqrt(C[i,i]*MSe) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+O.udP!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+rR2<;!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%*oj8^!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+o54M:!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The corresponding CIs:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "tcrit:=statevalf[icdf,studentst[n-k -1]](.975):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for i to k+1 do evalm(betaHat[i]+[-1,1]*sqrt(C[i,i]*MSe)*tcrit) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+Q,cW:!\")$\"+mdD*4$F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+jm6_G!\"*$\"+R6:@NF) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$!+hyLK))!\"*$!+&3lm r'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$\"+.E'HX'!\"*$ \"+$4)**z7!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "ANALYSIS OF VAR IANCE" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "ONE-WAY ANOVA:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart : with(stats): with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been redefined and unprotecte d\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k:=3: n:=25:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Generating 3 samples of size 25 ea ch:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "X :=matrix(k,n,[random[normald[30,5]](n),random[normald[33,5]](n),random [normald[28,5]](n)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"XG-%'matr ixG6#7%7;$\"+%y>ze$!\")$\"+NzJ=FF,$\"+)*ppH^nSF,$\"+U>@#4$F, $\"+&G&o\"p#F,$\"+*>^cx#F,$\"+E?\">T$F,$\"+:l1EJF,$\"+f]\"f4$F,$\"+R>* RT$F,$\"+Mk')G@F,$\"+V&Q:c$F,7;$\"+$[N?_#F,$\"+=/k4HF,$\"+%>m1/$F,$\"+ ;v'3<$F,$\"+#*\\\">`#F,$\"+IGQIF,$\"+ds.GLF,$ \"+k()ygSF,$\"+%3SaS#F,$\"+dzi&f#F,$\"+\\5%=T#F,$\"+@dqEGF,$\"+a8aIF,7;$\"+G))pEIF,$\"+?!48[ #F,$\"+[+[EBF,$\"+&[u')f$F,$\"+xf[EIF,$\"+*[dl.#F,$\"+FF(f(HF,$\"+M-x; JF,$\"+bATOEF,$\"+aRHEBF,$\"+CYhNEF,$\"+l1)HMF,$\"+/%H2x#F,$\"+&Q#*QY$F,$\"+7%[J:#F,$\"+C](3^$ F,$\"+'**>f4$F,$\"+LL@tMF,$\"+lj)pL#F,$\"+`zEz=F,$\"+4RjzCF," }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Computing the thess sample means: " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eval m(X &* [1$n] /n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%$\" +S\\WjH!\")$\"+QHwiJF)$\"+<(4&zFF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Total sum of squares:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 75 "SSt:=sum(sum(X[i,j]^2,j=1..n),i=1..k)-sum(sum( X[i,j],j=1..n),i=1..k)^2/n/k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SS tG$\"*yLB7#!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Between (sampl es) sum of squares:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "SSb:=sum(sum(X[i,j],j=1..n)^2,i=1..k)/n-sum(sum(X[i,j ],j=1..n),i=1..k)^2/n/k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SSbG$\" )L-P=!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Within (samples) sum of squares:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "SSw:=SSt-SSb;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$SSwG$\"*XJ 'Q>!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Mean square between:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "MSb:=S Sb/(k-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MSbG$\"++l6&=*!\")" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Mean square within:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "MSw:=SSw/k/(n-1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$MSwG$\"+!oVDp#!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The resulting value of test statistic:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "T:=MSb/ MSw;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"+*o:8T$!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The corresponding critical value: " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "stat evalf[icdf,fratio[k-1,k*(n-1)]](.95);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\\u!R7$!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "We thus do have a statistically significance evidence (just barely) to rehect H0 . At least one of the 3 populations has a mean distinct from the rest. " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The corres ponding P-value:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "1-statevalf[cdf,fratio[k-1,k*(n-1)]](T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*z7=%Q!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "A point estimate of sigma:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sqrt(MSw);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:E(*)=&!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Would yo u know how to construct the corresponding CI for sigma?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Do you still remember ho w to construct a CI for each of the 3 population means?" }{MPLTEXT 1 0 0 "" }}}}{MARK "61 0 0" 47 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }