Question

The equations of the two regression line are $2x+3y-6=0$ and $5x+7y-12=0$

Find

$\left(a\right)$ Correlation coefficient

$\left(b\right)$ $\frac{{\sigma }_x}{{\sigma }_y}$

Answer 1

Let $2x+3y-6=0$ be $x$ on $y$ and $5x+7y-12=0$ be $y$ on $x$

$2x+3y-6=0⟹x=-\frac{3}{2}y+3⟹bxy=-\frac{3}{2}$

$5x+7y-12=0⟹y=-\frac{5}{7}x+\frac{12}{7}⟹byx=-\frac{5}{7}$

$r=\sqrt{bxybyx}=\sqrt{\frac{15}{14}}$

$byx=\frac{r{\sigma }_y}{{\sigma }_x}=-\frac{5}{7},bxy=\frac{r{\sigma }_x}{{\sigma }_y}=-\frac{3}{2}⟹\frac{{\sigma }_x^2}{{\sigma }_y^2}=\frac{\frac{3}{2}}{\frac{5}{7}}=\frac{21}{10}⟹\frac{{\sigma }_x}{{\sigma }_y}=\sqrt{\frac{21}{10}}$