if the tangent of the angle between the lines of regression of Y on X and X on Y is 0.6, and the S.D. of Y is twice that of X, find the correlation coefficient between X and Y.

Let $L_{yx}\ $be the line of regression of y on x.

Let the angle made by $L_{yz}\ $with X-axis be $\mathrm{\theta}$

$\mathrm{\therefore }$ Tangent of the angle made by $L_{yz}$ with X- axis = tan $\mathrm{\theta}$ = 0.6

But , slope of any line with X-axis = tan $\mathrm{\theta}$

However, slope of $L_{yx}=\ b_{yx}$

$\mathrm{\therefore }$ $b_{yx}$ = tan $\mathrm{\theta}$ = 0.6

But $b_{yx}=r\ \frac{{\sigma }_y}{{\sigma }_x}$

$\therefore 0.6 = r\ \frac{2{\sigma }_x}{{\sigma }_x}\ (given)$

$\therefore 0.6 = 2 r$

$So,\ r=0.3$

$\therefore$ The correlation coefficient between x and y(r) = 0.3. Ans.