Question

If the lines of regression of $y$ on $x$ and $x$ on $y$ make angles $30°$ and $60°$ respectively with the positive direction of the $x-$axis, then the correlation coefficient between $x$ and $y$ is

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{1}{2}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\frac{1}{3}$

Answer 1

The correct option is C

$\frac{1}{\sqrt{3}}$

Explanation for the correct option:

Step 1: Find the slope of the given regression lines.

In the question, it is given that the angle made by the lines of regression of $y$ on $x$ and $x$ on $y$ make angles $30°$ and $60°$ respectively with the positive direction of the $x-$axis.

We know that the slope $m$ of a line is given by $m=\tan \theta$, Where $\theta$ is the angle the given lines make with the positive direction of the $x-$axis.

Assume that the slope of the first regression line is $m_1$ and the slope of the second regression line is $m_2$.

So, the slope of the regression of $y$ on $x$ is given by:

$$\begin{gathered} m_1=\tan \left(30°\right) \\ \Rightarrow m_1=\frac{1}{\sqrt{3}} \end{gathered}$$

So, the slope of the regression of $x$ on $y$ is given by:

$$\begin{gathered} m_2=\tan \left(60°\right) \\ \Rightarrow m_2=\sqrt{3} \end{gathered}$$.

Step 2: Find the value of the coefficient of correlation.

According to the definition of regression lines, regression lines are the two best-fit lines for the any given regression one is the lines of regression of $y$ on $x$ and the other is the lines of regression of $x$ on $y$, and the relation between the slopes of the regression lines and the correlation is as follows:

$r·\frac{{\sigma }_Y}{{\sigma }_X}=m_1$ and $\frac{1}{r}·\frac{{\sigma }_Y}{{\sigma }_X}=m_2$

Where, $r$ is the coefficient of correlation.

Since the values of $m_1$ and $m_2$ are $\frac{1}{\sqrt{3}}$ and $\sqrt{3}$ respectively.

So, $r·\frac{{\sigma }_Y}{{\sigma }_X}=\frac{1}{\sqrt{3}}...1$ and $\frac{1}{r}·\frac{{\sigma }_Y}{{\sigma }_X}=\sqrt{3}...2$

Divide equation $1$ by equation $2$.

$$\begin{gathered} \frac{r·\frac{{\sigma }_Y}{{\sigma }_X}}{{\displaystyle \frac{1}{r}}·\frac{{\sigma }_Y}{{\sigma }_X}}=\frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} \\ \Rightarrow r^2=\frac{1}{{\left(\sqrt{3}\right)}^2} \\ \Rightarrow r=\pm \frac{1}{\sqrt{3}} \end{gathered}$$

Since the slope of lines of regression is greater than zero. So, the coefficient of correlation will be greater than zero.

That is, $r=\frac{1}{\sqrt{3}}$

Therefore, the value of the coefficient of correlation is $\frac{1}{\sqrt{3}}$.

Hence, option (C) is the correct answer.