Question

For the given lines of regression, $3x-2y=5$ and $x-4y=7$, find:
coefficient of correlation $r(x,y)$

Answer 1

The two regression lines are
$3x-2y=5....\left(1\right)$
$x-4y=7....\left(2\right)$
From equation $\left(1\right)3x-2y=5$
$3x=2y+5$
$x=\frac{2}{3}y+\frac{5}{3}$
$\therefore b_{xy}=\frac{2}{3}$ (Regression of $x$ on $y$)
From equation $\left(2\right)x-4y=7$
$4y=x-7$
$y=\frac{x}{4}-\frac{7}{4}$
$\therefore b_{yx}=\frac{1}{4}$ (Regression of $y$ and $x$)

The two regression lines are
Coefficient of correlation $=|r|$
$=\sqrt{b_{xy}\cdot b_{yx}}$
$=\sqrt{\frac{1}{4}\times \frac{2}{3}}$
$=\sqrt{\frac{1}{6}}=\frac{\sqrt{6}}{6}$