MATH 376 -- Probability and Statistics II

First Regression Example

April 16 - 19, 2004

>	read "/home/fac/little/public_html/ProbStat/MaplePackage/MSP.map";

Warning, the name changecoords has been redefined

The lists of   x   and   y   values.  As in problem 11.3 in the text,

      represents the year i = 1  is 1972, etc.

      represents the median sales price (in 1000's of $) for

       a single family house in the US in year

>	Xlist:=[1.,2,3,4,5,6,7,8];

>	Xbar:=Mean(Xlist);

>	Ylist:=[27.6,32.5,35.9,39.3,44.2,48.8,55.7,62.9];

We want to find the least-squares estimators for the coefficients

  in a linear model   Y  =   +

Using the formulas      and  

presented in class:

>	Ybar:=Mean(Ylist);

>	Sxy:=add((Xlist[i]-Xbar)*(Ylist[i]-Ybar),i=1..nops(Xlist));

>	Sxx:=add((Xlist[i]-Xbar)^2,i=1..nops(Xlist));

>	beta[1]:=Sxy/Sxx;

>	beta[0]:=Ybar-beta[1]*Xbar;

To display the regression line together with the observed data

points ( ),  first we ``zip together'' the two lists to

make a single list of pairs:

>	points:=[seq([Xlist[i],Ylist[i]],i=1..nops(Xlist))];

Then we load the plots package, create separate plots for the data

points, and the line (note the colons at the end of the plot commands --

see what happens if you use the normal semicolon to display the output;

also note that we are assigning the output from the plot commands

to names  PP (for ``point plot'')  and  LP  (for ``line plot'') --

these have no intrinsic meaning for Maple, but they should be

suggestive of what's actually going on here!

>	with(plots):

>	PP:=plot(points,style=point,color=blue):

>	LP:=plot(beta[0]+beta[1]*x,x=0..9,color=red):

Finally, to plot the data points and the regression line together on the

same coordinate axes, we use the   display   command from the plots

package:

>	display(PP,LP);

So, we can conclude that over this time frame, the median sales price

was increasing  roughly linearly, at a rate of approx.    = $4,842  dollars

per year.