Question

Given the 2 regression lines 3x+12y=19,3y+9x=46,find the coefficient of correlation between x and y?​

Answer 1

Question:

Given the 2 regression lines 3x+12y=19 , 3y+9x=46 , Find the coefficient of correlation between x and y

Answer:

Coefficient of correlation between x and y : 0.288

Given:

Two regression lines are :

3x+12y=19 and 3y+9x=46

To Find:

Coefficient of correlation between x and y

Solution:

Given lines of regression are:

$\sf 3x + 12y = 19$

$\sf i.e.\:\:\:3x + 12y - 19 = 0 \: .....................(1)$

$\sf 3y + 9x = 46$

$\sf i.e \: \:\: 9x + 3y - 46 = 0 \: ......................(2)$

From equation (1), the regression line of y on x is.

$\sf 12y-3x+19$

$\sf y=-\dfrac{3}{12}x + \dfrac{19}{12}$

$\sf y=-\dfrac{3}{12}x+\dfrac{19}{12}$

$\sf y =-\dfrac{1}{4} x+\dfrac{19}{12}$

The regression line of y on x is :

$\rm\large\bold{b_{yx} =-\dfrac{1}{4}}$

From equation (2), the regression line of x on y is.

$\sf 9x=-3y+46$

$\sf x=-\dfrac{3}{9}y+\dfrac{46}{9}$

$\sf x=-\dfrac{1}{3}y+\dfrac{46}{9}$

The regression line of x on y is : $\rm\large\bold{b_{xy}=-\dfrac{1}{3}}$

We know ,

${\boxed{\sf{r = \sqrt{ b_{yx} \times b_{xy}}}}}$

$\sf {r =\sqrt{-\: \dfrac{1}{4}\times -\dfrac{1}{3} } }$

$\sf r = \sqrt{ \dfrac{1}{12} }$

$\sf r = 0.288$

Coefficient of correlation between x and y : 0.288

Answer 2

$\sf{Given\:lines\:of\:regression\:are:}$

$\longrightarrow \sf\: 3x+12y-19=0........(1) \\ \\ \longrightarrow \sf\: 9x+3y-46=0........(2)$

$\bigstar$From equation (1), the regression line of y on x is,

$\longrightarrow \sf\:12y = -3x + 19 \\ \\ \longrightarrow \sf\:y= \frac{-3}{12}x + \frac{19}{12} \\ \\ \longrightarrow \sf\:y= \frac{-1}{4} + \frac{19}{12} \\ \\ \implies \sf\: {b}_{yx} = \frac{-1}{4}$

$\bigstar$From equation (2), the regression line of x on y is,

$\longrightarrow \sf\:9x = -3y + 19 \\ \\ \longrightarrow \sf\:x= \frac{-3}{9}x + \frac{46}{9} \\ \\ \longrightarrow \sf\:y= \frac{-1}{3} + \frac{46}{9} \\ \\ \implies \sf\: {b}_{xy} = \frac{-1}{3}$

${\boxed{\sf{r = \sqrt{ b_{yx} \times b_{xy}}}}}$

$\sf {r =\sqrt{-\: \dfrac{1}{4}\times -\dfrac{1}{3} } }$

$\sf r = \sqrt{ \dfrac{1}{12} }$

$\boxed{r=0.288}$