Question

If the regression coefficient of $Y$ on $X$ is $\frac{8}{5}$and that of $X$ on $Y$ is $\frac{2}{5}$.

• A. $\frac{9}{25}$
• B. $\frac{9}{2\sqrt{5}}$
• C. $\frac{3}{25}$
• D. $\frac{9}{50}$

Answer 1

The correct option is D

$\frac{9}{50}$

Step 1: Find line slopes from the regression coefficient.

Let $b_{yx}and{b}_{xy}$ are regression coefficient.

Given

$$\begin{gathered} b_{yx}=\frac{8}{5} \\ {b}_{xy}=\frac{2}{5} \end{gathered}$$

Let $m_{1}and{m}_2$ be slopes of line.

$$\begin{gathered} m_1=b_{yx}=\frac{8}{5} \\ {m}_2=\frac{1}{b_{xy}}=\frac{1}{{\displaystyle \frac{2}{5}}} \\ =\frac{5}{2} \end{gathered}$$

Step 2: Find the angle between the lines by formula.

Let angle between the lines = $\theta$

Then $\tan \left(\theta \right)=\pm \frac{m_1-m_2}{1+m_1{m}_2}$

Put the value of $m_{1}and{m}_2$ in the above equation.

$m_{1}and{m}_2$ both are positive and hence $\theta$ between the line $\in$$(0,\frac{\pi }{2})$ so $\tan \theta >0$

$$\begin{gathered} \tan \theta =\pm \frac{\left(\frac{8}{5}\right)-\left(\frac{5}{2}\right)}{1+\left({\displaystyle \frac{8}{5}}\right)\left({\displaystyle \frac{5}{2}}\right)} \\ =-\frac{\left({\displaystyle \frac{16-25}{10}}\right)}{{\displaystyle \frac{(10+40)}{10}}}[Takingnegativesign] \\ =-\frac{{\displaystyle \frac{-9}{10}}}{{\displaystyle \frac{50}{10}}}=\frac{9}{10}\times \frac{10}{50} \\ =\frac{9}{50} \end{gathered}$$

Therefore, option (4) is correct.