Question

If two lines of regressions are $x+3y=5$ & $4x+3y=8$, then coefficient of correlation between x & y is

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $-\frac{1}{4}$
• D. $-\frac{1}{2}$

Answer 1

The correct option is D $-\frac{1}{2}$

Lines of regressions are $x+3y=5$ (i)
and $4x+3y=8$ (ii)
Suppose line (i) is line of regression x on y
$\therefore$ $b_{xy}=-3$
And line (ii) is line of regression y on x
$\therefore$ $b_{yx}=-\frac{4}{3}$
Now $b_{xy}{b}_{yx}=4>1$, so our supposition is wrong & line (i) is line of regression y on x & line (ii) is line of regression x on y
$\therefore$ $b_{yx}=-\frac{1}{3}$ & $b_{xy}=-\frac{3}{4}$
$\therefore$ $r^2=\frac{1}{4}$
We know that $r=\pm \sqrt{b_{yx}{b}_{xy}}$ and $r,b_{yx}$, $b_{xy}$ have same sign
$\therefore$ $r=-\sqrt{\frac{1}{4}}=-\frac{1}{2}$