Question

From the equation of the two regression lines, $4x+3y+7=0$ and $3x+4y+8=0$, find: Regression coefficients.

Answer 1

Let equation (i) be the regression equation of y on x and (ii) the regression equation of x on y then from (i)
$y=-\frac{4}{3}x-\frac{7}{4}$
$\Rightarrow b_{yx}=-\frac{4}{3}$
and from (ii) $x=-\frac{4}{3}y-\frac{8}{3}$
$\Rightarrow b_{xy}=-\frac{4}{3}$
but $r^2=b_{yx}.b_{xy}=(-\frac{4}{3})\times (-\frac{4}{3})>1$
Which is impossible. Hence, equation (i) is the regression line of x on y and equation (ii) is the regression equation of y on x. From (i)
$x=-\frac{3}{4}y-\frac{7}{4}$
$\therefore b_{xy}=-\frac{3}{4}$
From (ii), $y=-\frac{3}{4}x-2$
$\therefore b_{yx}=-\frac{3}{4}$