Question

The regression equation of Y on X is $y=\frac{2}{9}x$ and the regression equation of X on Y is $x=\frac{y}{2}+\frac{7}{6}$
Find: a. Correlation coefficient between X and Y.
b. ${\sigma }_y^2$ if ${\sigma }_x^2=4$.

Answer 1

The repression equation of $y$ on $x$ is $y=\frac{2}{9}x$
Comparing with $y-\overline{y}=b_{yx}(x-\overline{x})$
Here, $b_{yx}=\frac{2}{9}$
Now, the regression equation of $x$ on $y$ is $x=\frac{y}{2}+\frac{7}{6}$
Comparing with $x-\overline{x}=b_{xy}(y-\overline{y})$
Here, $b_{xy}=\frac{1}{2}$
Part $\left(a\right)$
We have, correlation coefficient between $x$ and $y$ is
$r=\sqrt{b_{yx}.b_{xy}}$
$r=\sqrt{\frac{2}{9}\times \frac{1}{2}}$
$r=\pm \frac{1}{3}$
Therefore,
$r=\frac{1}{3}$ (Since $b_{yx}$ and $b_{xy}$ are positive)
Part $\left(b\right)$
Since, ${\sigma }_x^2=4\Rightarrow {\sigma }_x=2$
We have,
$b_{yx}=r.\frac{{\sigma }_y}{{\sigma }_x}$
Therefore,
$\frac{2}{9}=\frac{1}{3}.\frac{{\sigma }_y}{2}$
${\sigma }_y=\frac{4}{3}$
Hence, this is the answer.