Question

The coefficient of correlation between the values denoted by $X$ and $Y$ is $0.5$. The mean of $X$ is $3$ and that of $Y$ is $5$. Their standard deviations are $5$ and $4$ respectively. Find:
The two lines of regression

Answer 1

$\overline{x}=3,\overline{y}=5,{\sigma }_x=5,{\sigma }_y=4,r=0.5$
$b_{yx}=r\cdot \frac{{\sigma }_y}{{\sigma }_x}=0.5\times \frac{4}{5}=0.4$,
$b_{xy}=r\cdot \frac{{\sigma }_x}{{\sigma }_y}=0.5\times \frac{5}{4}=0.625$
Regression equation,
$y-\overline{y}=b_{yx}(x-\overline{x})|x-\overline{x}=b_{xy}(y-\overline{y})$
$y-5=\left(0.4\right)(x-3)|x-3=\left(0.625\right)(y-5)$
$y=0.4x+3.8,x=0.625y-0.125$