> restart: read `C:\\Tanis-Hogg\\Maple Examples\\stat.m`: with(plots): randomize(): read `C:\\Tanis-Hogg\\Maple Examples\\HistogramFill.txt`: read `C:\\Tanis-Hogg\\Maple Examples\\ScatPlotCirc.txt`: read `C:\\Tanis-Hogg\\Maple Examples\\ScatPlotPoint.txt`: read `C:\\Tanis-Hogg\\Maple Examples\\BoxPlot.txt`:

```Warning, the name changecoords has been redefined

```

Elliot Tanis
April 16, 2001
T-distribution -- Equal Variances--  (Tev.mws)

and see if T  actually has a T -distribution with n  + m  - 2 degrees of freedom.  Also calculate the value of   " Z" ,

.

How do the sizes of n  and m  affect the distribution of T ? How important is it for the distribution variances to be equal for T  to actually have a t-distribution with n  + m  - 2 degrees of freedom?

For this simulation the variances are equal.

 > randomize():

 > N := 500; n := 6; m := 18; VarX := 16; VarY := 16;

 > for k from 1 to N do X := NormalS(0, VarX, n): Y := NormalS(0, VarY, m): TT[k] := evalf((Mean(X) - Mean(Y))/sqrt(((n-1)*Variance(X) + (m-1)*Variance(Y))/(n + m - 2)*(1/n + 1/m))): ZZ[k] := evalf((Mean(X) - Mean(Y))/sqrt(Variance(X)/n + Variance(Y)/m)): od: T := [seq(TT[j], j = 1 .. N)]: Z := [seq(ZZ[j], j = 1 .. N)]:

 > MedianofTs := Median(T); MedianofZs := Median(Z); MeanofTs := Mean(T); MeanofZs := Mean(Z); SDofTs := StDev(T); SDofZs := StDev(Z); Min(T), Max(T); Min(Z), Max(Z);

 > display({BoxPlot(T, Z)}, title = `Box Plots for Z (above) and T (below)`, titlefont=[TIMES,BOLD,16]);

 > xtics := [seq(-3.5 + k*0.5, k = 0 .. 14)]: ytics := [seq(0.1*k, k = 1 .. 11)]:

 > P1 := HistogramFill(T, -3.5 .. 3.5, 28): P2 := plot(TPDF(n+m-2, x), x = -3.5 .. 3.5, y = 0 .. 0.45, color=black, thickness=2, xtickmarks=xtics, ytickmarks=ytics, labels=[``,``]): display({P1, P2}, title = `Histogram of T Observations\nwith T(n+m-2) Superimposed`, titlefont=[TIMES,BOLD,16]);

 > n+m-2;

 > Percentile(T, 0.05), Percentile(T, 0.95); TP(n+m-2,0.05), TP(n+m-2,0.95);

 > P3 := HistogramFill(Z, -3.5 .. 3.5, 28): P4 := plot(NormalPDF(0, 1, x), x = -3.5 .. 3.5, y = 0 .. 0.45, color=black, thickness=2, xtickmarks=xtics, ytickmarks=ytics, labels=[``,``]): display({P3, P4}, title = `Histogram of Z Observations\nwith N(0, 1) p.d.f. Superimposed`, titlefont=[TIMES,BOLD,16]);

 > Percentile(Z, 0.05), Percentile(Z, 0.95); NormalP(0, 1, 0.05), NormalP(0, 1, 0.95);

 > TQuants := [seq(TP(n+m-2, k/(N+1)), k = 1 .. N)]:

 > T := sort(T): P5 := ScatPlotCirc(T, TQuants): P6 := plot([[-3.5, -3.5], [3.5, 3.5]], x = -3.5 .. 3.5, y = -3.5 .. 3.5, color=black, thickness=2, xtickmarks=xtics, ytickmarks=xtics, labels=[``,``]): display({P5, P6}, title=`q-q Plot of T(n+m-2) Quantiles\nVersus the T Order Statistics`, titlefont=[TIMES,BOLD,16]);

 > n+m-2;

 > ZQuants := [seq(NormalP(0, 1, k/(N+1)), k = 1 .. N)]:

 > Z := sort(Z): P7 := ScatPlotCirc(Z, ZQuants): P6 := plot([[-3.5, -3.5], [3.5, 3.5]], x = -3.5 .. 3.5, y = -3.5 .. 3.5, color=black, thickness=2, xtickmarks=xtics, ytickmarks=xtics, labels=[``,``]): display({P7, P6}, title=`q-q Plot of N(0, 1) Quantiles\nVersus the Z Order Statistics`, titlefont=[TIMES,BOLD,16]);

 >

Welch showed that U = Z,  where

,

really has an approximate T  distribution with the number of degrees of freedom given by the greatest integer (or the floor) in the following expression:

We shall now fit a t -distribution to the Z  data using as the number of degrees of freedom Welch's suggestion but with distribution variances rather than sample variances.

 > v := floor((VarX/n + VarY/m)^2/((VarX/n)^2/(n-1) + (VarY/m)^2/(m-1)));

 > P1 := HistogramFill(Z, -3.5 .. 3.5, 28): P2 := plot(TPDF(v, x), x = -3.5 .. 3.5, y = 0 .. 0.45, color=black, thickness=2, xtickmarks=xtics, ytickmarks=ytics, labels=[``,``]): display({P1, P2}, title=`Histogram of U Observations\nwith T(v) Superimposed`, titlefont=[TIMES,BOLD,16]);

 > v;

 > Percentile(Z, 0.05), Percentile(Z, 0.95); TP(v, 0.05), TP(v, 0.95);

 > TvQuants := [seq(TP(v, k/(N+1)), k = 1 .. N)]:

 > Z := sort(Z): P5 := ScatPlotCirc(Z, TvQuants): P6 := plot([[-3.5, -3.5], [3.5, 3.5]], x = -3.5 .. 3.5, y = -3.5 .. 3.5, color=black, thickness=2, xtickmarks=xtics, ytickmarks=xtics, labels=[``,``]): display({P5, P6}, title=`q-q Plot of T(v) Quantiles\nVersus the U Order Statistics`, titlefont=[TIMES,BOLD,16]);

 > v;

 >