Question

If there exists a linear statistical relationship between $2$ variables $x$ and $y$, then the regression coefficient of $y$ on $x$ is where ${\sigma }_x,{\sigma }_y$ are standard deviations of $x$ and $y$ respectively.

• A. $\frac{cov(x,y)}{{\sigma }_x{\sigma }_y}$
• B. $\frac{cov(x,y)}{{\sigma }_{y^2}}$
• C. $\frac{cov(x,y)}{{\sigma }_{x^2}}$
• D. $\frac{cov(x,y)}{{\sigma }_x}$

Answer 1

The correct option is C

$\frac{cov(x,y)}{{\sigma }_{x^2}}$

Find the regression coefficient of $y$ on $x$:

$\Rightarrow$ $r=\frac{cov(x,y)}{{\sigma }_x{\sigma }_y}$

The regression coefficient of $y$ on $x$, $b_{yx}=\frac{r{\sigma }_y}{{\sigma }_x}$

$\Rightarrow$$b_{yx}=\frac{[cov(x,y)\sigma y]}{[\sigma x.\sigma y.\sigma x]}$

$\Rightarrow$ $b_{yx}=\frac{cov(x,y)}{\sigma x^2}$

Hence, the option (C) is correct.