Question

The regression equation of y on x is 4y = 9x + 15
The regression equation of x on y is 25x = 6y + 7
Find the mean values of x, y and r.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given regression lines are

The regression equation of y on x is 4y = 9x + 15

The regression equation of x on y is 25x = 6y + 7

The above lines can be rewritten as

The line y on x is 9x - 4y = - 15

The line x on y is 25x - 6y = 7

Calculation of mean value of x and y

We know,

• The point of intersection of regression line gives the mean .

So, let we solve these two linear equations using Cross Multiplication method.

We have equations

$\rm :\longmapsto\:9x - 4y = - 15 - - - (1)$

and

$\rm :\longmapsto\:25x - 6y = 7 - - - (2)$

Using Cross Multiplication method, we have

$\begin{gathered}\boxed{\begin{array}{c|c|c|c} \bf 2 & \bf 3 & \bf 1& \bf 2\\ \frac{\qquad}{} & \frac{\qquad}{}\frac{\qquad}{} &\frac{\qquad}{} & \frac{\qquad}{} &\\ \sf - 4 & \sf - 15 & \sf 9 & \sf - 4\\ \\ \sf - 6 & \sf 7 & \sf 25 & \sf - 6\\ \end{array}} \\ \end{gathered}$

So,

$\rm :\longmapsto\:\dfrac{x}{ - 28 - 90} = \dfrac{y}{ - 375 - 63} = \dfrac{ - 1}{ - 54 + 100}$

$\rm :\longmapsto\:\dfrac{x}{ - 118} = \dfrac{y}{ -438} = \dfrac{ - 1}{ 46}$

On Multiply by - 2, we get

$\rm :\longmapsto\:\dfrac{x}{59} = \dfrac{y}{219} = \dfrac{1}{23}$

$\bf\implies \:x = \dfrac{59}{23} \: \: and \: \: y = \dfrac{219}{23}$

Hence, Mean is given by

$\bf\implies \: \bar{x} = \dfrac{59}{23} \: \: and \: \: \bar{ y} = \dfrac{219}{23}$

Calculation of Correlation coefficient

We have

Regression equation of y on x is

$\rm :\longmapsto\:9x - 4y = - 15$

So, regression coefficient of y on x is

$\rm :\longmapsto\:b_{yx} = - \: \dfrac{coefficient \: of \: x}{coefficient \: of \: y}$

$\rm :\longmapsto\:b_{yx} = - \dfrac{9}{( - 4)}$

$\rm :\longmapsto\:b_{yx} = \: \dfrac{9}{4}$

Regression line of x on y is

$\rm :\longmapsto\:25x - 6y = 7$

$\rm :\longmapsto\:b_{xy} = - \: \dfrac{coefficient \: of \: y}{coefficient \: of \: x}$

$\rm :\longmapsto\:b_{xy} = - \: \dfrac{( - 6)}{25}$

$\rm :\longmapsto\:b_{xy} = \: \dfrac{6}{25}$

Now, we know that

Correlation Coefficient r, is evaluated as

$\rm :\longmapsto\:r = \pm \: \sqrt{b_{yx} \times b_{xy}}$

$\rm :\longmapsto\:r = \pm \: \sqrt{\dfrac{9}{4} \times \dfrac{6}{25} }$

$\rm :\longmapsto\:r = \pm \: \dfrac{3}{10} \sqrt{6 }$

As sign of regression coefficient is + ve, so r is + ve

$\rm :\longmapsto\:r = \: \dfrac{3}{10} \sqrt{6 }$

$\bf\implies \:r = 0.7348$