Question

If there exists a linear statistical relationship between two variables x and y, then the regression coefficient of y on x is

• 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. $\frac{Cov(x,y)}{\sigma_{x}}$, where $\sigma_{x},\sigma_{y}$are standard deviations of x and y respectively.

Answer 1

Answer: (C)

$\frac{Cov(x,y)}{\sigma_{x}^{2}}$

Answer 2

We know that, correlation coefficient i.e., $r = \frac{Co ⥂ v(x,y)}{\sigma_{x}\sigma_{y}}$

and regression coefficient of y on x i.e., $b_{yx} = \frac{r\sigma_{y}}{\sigma_{x}}$.

∴ $b_{yx} = \frac{Co ⥂ v(x,y).\sigma_{y}}{\sigma_{x}.\sigma_{y}.\sigma_{x}} \Rightarrow b_{yx} = \frac{Co ⥂ v(x,y)}{\sigma_{x}^{2}}$.