Question

If $x_1$ and $x_2$ are regression coefficients and $r$ is the coefficient of correlation, then

• A. $x_1-x_2>r$
• B. $x_1+x_2<r$
• C. $x_1+x_2\ge 2r$
• D. None of the above

Answer 1

The correct option is C

$x_1+x_2\ge 2r$

Explanation for the correct answer:

Find the correct relation :

Given,

$$\begin{gathered} x_1=r\frac{{\sigma }_y}{{\sigma }_x}...\left(1\right) \\ {x}_2=r\frac{{\sigma }_x}{{\sigma }_y}...\left(2\right) \end{gathered}$$

By adding $\left(1\right)$ and $\left(2\right)$, we get

$x_1+x_2=r\left(\frac{{\sigma }_y}{{\sigma }_x}+\frac{{\sigma }_x}{{\sigma }_y}\right)$

We know ,

arithmetic mean $\ge$ geometric mean

so,

$$\begin{gathered} \frac{\left(\frac{{\sigma }_y}{{\sigma }_x}+\frac{{\sigma }_x}{{\sigma }_y}\right)}{2}\ge \sqrt{\left(\frac{{\sigma }_y}{{\sigma }_x}\times \frac{{\sigma }_x}{{\sigma }_y}\right)} \\ \therefore \left(\frac{{\sigma }_y}{{\sigma }_x}+\frac{{\sigma }_x}{{\sigma }_y}\right)\ge 2 \end{gathered}$$

Thus, $x_1+x_2\ge 2r$

Hence, the correct option is C.