Correlation and Regression

Concept:The arithmetic mean of the two regression coefficients $b_{xy}$ and $b_{yx}$ is always greater than or equal to the correlation coefficient $r$.Explanation:We know $b_{xy} = r \cdot \frac{\sigma_x}{\sigma_y}$ and $b_{yx} = r \cdot \frac{\sigma_y}{\sigma_x}$.Their arithmetic mean is $\frac{1}{2}(b_{xy} + b_{yx}) = \frac{r}{2}\left(\frac{\sigma_x}{\sigma_y} + \frac{\sigma_y}{\sigma_x}\right)$.Since $\frac{\sigma_x}{\sigma_y} + \frac{\sigma_y}{\sigma_x} \geq 2$ by AM–GM inequality, we have $\frac{1}{2}(b_{xy} + b_{yx}) \geq r$.Thus the mean is never less than $r$, and equals $r$ only when $\sigma_x = \sigma_y$.Option A (equality) is not always true.Option B (less than) is false.Option C (greater than) is not always true (equality possible).Therefore, none of the given options is universally correct.Answer: