Question

If the linear equations $3x=2y-1$ and $2x+3y+1=0$ are two lines of regression, then the coefficient of correlation will be:
1). $0$
2). $-1$
3). $1$
4). None of these

Answer 1

First of all we will assume both the equations as equation $(1)$ and $(2)$ then multiply equation $(1)$ with $3$ and $(2)$ with $2$ then add both the equation to get the value of $x$ then put the value of $x$ in equation $(1)$ to get the value of $y$after that find out the regression of $x$ on $y$ and $y$ on $x$ to check the correct option.

Complete step-by-step solution:
The measure of the extent of relationship between two variables is shown by the correlation coefficient. The range of this coefficient lies between $-1$ to $+1$ .
Regression is a technique used for the modeling and analysis of numerical data. Regression can be used for prediction, estimation, hypothesis testing and many more things.
The linear regression line equation is written as: $Y=a+bX$ where $X$ is plotted along $x-axis$ and is an independent variable and $Y$ is plotted along $y-axis$ which is a dependent variable.
Dependent variable means the variable we wish to predict and independent variable means the variable that is used to explain the dependent variable.
Linear regression shows the linear relationship between two variables.
Correlation analysis is used to measure strength of the association of linear relationships between two variables.
Now according to the question:
We have given two lines of regression:
$3x=2y-1$
$\Rightarrow 3x-2y+1=0$ assume it as equation $(1)$
$\Rightarrow 2x+3y+1=0$ assume it as equation $(2)$
Now multiply equation $(1)$ with $3$ and equation $(2)$ with $2$ we will get:
$\Rightarrow 9x-6y+3=0$
$\Rightarrow 4x+6y+2=0$
On adding both the equations we will get:
$\Rightarrow 9x-6y+3+4x+6y+2=0$
$\Rightarrow 13x+5=0$
$\Rightarrow 13x=-5$
$\Rightarrow x=\dfrac{-5}{13}$
Put the value of $x$ in equation $(1)$we will get the value of y:
$\Rightarrow 3x-2y+1=0$
$\Rightarrow 3\times (\dfrac{-5}{13})-2y+1=0$
$\Rightarrow \dfrac{-15}{13}-2y+1=0$
$\Rightarrow \dfrac{-15-26y+13}{13}=0$
$\Rightarrow -2-26y=0$
$\Rightarrow -2=26y$
$\Rightarrow y=-\dfrac{1}{13}$
Hence $\overline{x}=\dfrac{-5}{13}$ and $\overline{y}=-\dfrac{1}{13}$
Now let us suppose that the equation $3x=2y-1$ is regression equation of $y$ on $x$
$\Rightarrow 3x-2y+1=0$
$\Rightarrow -2y=-3x-1$
$\Rightarrow y=\dfrac{-3}{-2}x-\dfrac{1}{-2}$
$\Rightarrow y=\dfrac{1}{2}+\dfrac{3}{2}x$
Compare the equation from $y=a+bx$ we will get $a=\dfrac{1}{2}$ and $b=\dfrac{3}{2}$
As the equation is regression equation of $y$ on $x$ therefore ${{b}_{yx}}=\dfrac{3}{2}$
Now let us suppose the equation $2x+3y+1=0$ is regression equation of $x$ on $y$
$\Rightarrow 2x+3y+1=0$
$\Rightarrow 2x=-3y-1$
$\Rightarrow x=-\dfrac{3}{2}y-\dfrac{1}{2}$
Compare the equation from $x=a+by$ we will get $a=-\dfrac{1}{2}$ and $b=-\dfrac{3}{2}$
As the equation is regression equation of $x$ on $y$ therefore ${{b}_{xy}}=-\dfrac{3}{2}$
Here the values of ${{b}_{xy}}$ and ${{b}_{yx}}$ are of opposite sign hence the regression is not possible.
Hence option $(4)$ is correct as the regression is not possible.

Note: Students must know that the regression coefficient is the slope of the regression line which is equal to the average change in the dependent variable for a unit change in the independent variable. The strength of the linear relationship increases as $r$ moves away from $0$ .