Question

The two lines of regressions are $4x+2y-3=0$ and $3x+6y+5=0$. Find the correlation coefficient between $x$ and $y$.

Answer 1

Let $4x+2y-3=0$ be the regression equation of $y$ on $x$ and

$3x+6y+5=0$ the regression equation of $x$ on $y$ then

$2y=-4x+3$
$y=-2x+\frac{3}{2}$
$\therefore b_{yx}=-2$
and,

$3x=-6y-5$
$x=-\frac{6}{3}y-\frac{5}{3}$
$x=-2y-\frac{5}{3}$
$\therefore b_{xy}=-2$

$r^2=b_{yx}\times b_{xy}$
$=-2\times -2=4>1$ which is impossible.

So regression equation of $y$ on $x$ is
$3x+6y+5=0$ ..........(i)

and regression equation of $x$ on $y$ is
$4x+2y-3=0$ ...........(ii)

From (i), we get
$6y=-3x-5$
$y=-\frac{1}{2}x-\frac{5}{6}$
$\therefore b_{yx}=-\frac{1}{2}$

From (ii), we get
$4x=-2y+3$
$x=-\frac{1}{2}y+\frac{3}{4}$
$\therefore b_{xy}=-\frac{1}{2}$

$\therefore$ Correlation coefficient is given by
$|r|=\sqrt{b_{yx}.b_{xy}}$

$=\sqrt{-\frac{1}{2}\times -\frac{1}{2}}=\frac{1}{2}$
$\Rightarrow r=\pm \frac{1}{2}$

But $r$ has the same sign as regression coefficient
$\therefore r=-\frac{1}{2}$.