Question

If regression lines are $3x+12y-19=0$ and $9x+3y-46=0$, then find the coefficient of correlation.

Answer 1

Given lines of regression are:
$3x+12y-19=0$ .........(1)
$9x+3y-46=0$ ..........(2)
From equation (1), the regression line of y on x is.
$12y=-3x+19$
$\Rightarrow y=-\frac{3}{12}x+\frac{19}{12}$
$\Rightarrow y=-\frac{1}{4}x+\frac{19}{12}$
$\therefore b_{yx}=-\frac{1}{4}$
From equation (2), the regression line of x on y is.
$9x=-3y+46$
$\Rightarrow x=-\frac{3}{9}y+\frac{46}{9}$
$\Rightarrow x=-\frac{1}{3}y+\frac{46}{9}$
$\therefore b_{xy}=-\frac{1}{3}$
Now, $r=\sqrt{b_{yx}\cdot b_{xy}}$
$r=\sqrt{(-\frac{1}{4})\times (-\frac{1}{3})}$
$r=\sqrt{\frac{1}{12}}$
$r=0.288$.