Consider the regression model z qx ry s for the following

Consider the regression model z = qx + ry + s for the following data, where the x-value is a scaled score (to have average value of 0) of high school grades, the y-value is a scaled score of SAT scores, and the z-value is a scaled score of college grades. Determine q, r, and s. Note that the x, y, and 1 vectors (in the regression matrix equation z = qx + ry + s1) are orthogonal.

Solution

considering the values of x,y,z we have the following equations

-4q+2r+s=3 ........................(1)

-2q-r+s=6.............................(2)

-2r+s=7................................(3)

2q-r+s=7...............................(4)

4q+2r+s=6.............................(5)

by equation (3) we have 2r=s-7 implies r=(s-7)/2

substituting the value of r in equation (4) we have2q-(s-7)/2+s=7

multiplying by 2 in the above equation we have

4q-2s+14+2s=14

hence q=0 substituting q=0 and r=(s-7)/2 in (1) we have

2(s-7)/2+s=3 implies s=10

now r=(s-7)/2 substituting the value of s we have r=3/2

hence q=0, r=3/2,s=10