Question

Find the coefficient of correlation from the regression lines: $x-2y+3=0$ and $4x-5y+1=0$.

Answer 1

The coefficient of correlation between two variables $x$ and $y$ is the simple geometric mean of the two regression coefficients.

The sign of the correlation coefficient would be the common sign of the two regression coefficient.

Let us assume that $x-2y+3-0$ represents the regression line of $y$ on $x$ and $4x-5y+1=0$ repersents the regression line of $x$ on $y:$

Now $x-2y+3=0\Rightarrow y=\frac{1}{2}x+\frac{3}{2}$

$\therefore b_{yx}=\frac{1}{2}$

Again $4x-5y+1=0$

$\Rightarrow x=\frac{5y}{4}-\frac{1}{4}$

$\therefore b_{xy}=\frac{5}{4}$

$\therefore r^2=b_{yx}{b}_{xy}=\frac{1}{2}\frac{5}{4}=\frac{5}{8}<1$

Now, $r^2\le 1$, our assumptions are correct.

So, $r=\sqrt{\frac{5}{8}}$

where $r=$coefficient of correlation and $b_{xy}=$ regression coefficient.