Question

If $3x=2y-1$ and $2x+3y+1=0$ are two lines of regression, then the coefficient of correlation will be

• A. $0$
• B. $-1$
• C. $1$
• D. None of these

Answer 1

The correct option is A

$0$

Explanation for the correct option:

Step 1. Finding the regression coefficient of correlation:

given,

The two regression lines are:
$\begin{array}{rcl}3x-2y+1& =& 0...\left(i\right)\\ 2x+3y+1& =& 0...\left(ii\right)\end{array}$

Step 2. From equation (i), we get

$\begin{array}{rcl}3x-2y& =& -1\end{array}$

$\Rightarrow$ $\begin{array}{rcl}3x& =& 2y-1\end{array}$

$\Rightarrow$ $\begin{array}{rcl}x& =& \frac{2y}{3}-\frac{1}{3}\end{array}$

$\begin{array}{rcl}\therefore b_{xy}& =& \frac{2}{3}\end{array}$ (Regression of $x$ on $y$)

Step 3. From equation (ii), we get

$\begin{array}{rcl}2x+3y& =& -1\end{array}$

$\Rightarrow$ $\begin{array}{rcl}3y& =& -x-1\end{array}$

$\Rightarrow$ $\begin{array}{rcl}y& =& -\frac{x}{3}-\frac{1}{3}\end{array}$

$\begin{array}{rcl}\therefore b_{yx}& =& -\frac{1}{3}\end{array}$ (Regression of $y$ on $x$)

$\because$The coefficients of regression have different signs which is not possible (also as the lines are perpendicular to each other)

$\therefore r=0$

Hence, the correct option is (A).