Question

If the lines of regression of $y$ on $x$ and that of $x$ on $y$ are $y=kx+4$ and $x=4y+5$ respectively, then

• A. $k\le 0$
• B. $k\ge 0$
• C. $0\le k\le \frac{1}{4}$
• D. $0\le k\le 1$

Answer 1

The correct option is C

$0\le k\le \frac{1}{4}$

Explanation for the correct option:

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

In the question, the equation of two lines is given.

$$\begin{gathered} y=kx+4...1 \\ x=4y+5...2 \end{gathered}$$

we know that the slope-intercept form of a line is $y=mx+c...3$.

Where, $(x,y)$ are the general point of the given line, $m$ is the slope of the line and $c$ is the $y-$intercept.

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

On comparing equation $1$ and equation $3$, we get

$m_1=k$

Since the second equation is the line of regression of $x$ on $y$. So, to find the slope of this equation we treat $x$ as $y$ and $y$ as $x$.

So, the slope of the second regression line is $m_2=4$.

Step 2: Find the value of $k$.

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 is as follows:

$0\le m_1·m_2\le 1$

Since the values of $m_1$ and $m_2$ are $k$ and $4$ respectively.

So,

$$\begin{gathered} 0\le k·4\le 1 \\ \Rightarrow 0\le k\le \frac{1}{4} \end{gathered}$$

Therefore, the value of $k$ lies in the range of $0\le k\le \frac{1}{4}$.

Hence, option (C) is the correct answer.