Question

The regression equation of Y on X is y = `2/9` x and the regression equation of X on Y is `x=y/2+7/6` 

Find:

1.  The correlation coefficient between X and Y. 
2.  `σ_y^2 if σ _x^2=4`

Answer 1

The repression equation of y on x is y =`2/9x` 

Comparing with y − `barY = b_yx (x − barx)` 

Here

`b_yx=2/9`

Now the regression eqn. of x on y is x =`y/2+7/6` 

Comparing with` x-barx=bxy=1/2` 

(a) we have, correlation coefficient between x and y is 

`r=sqrt(b_(yx) b_(xy))` 

=`sqrt2/9xx1/2=sqrt1/9+-1/3` 

∴ `r=1/3 (∵b_(yx) and b_(xy) "are positive")` 

`therefore` r = 0.33

(b) `σ_x^2=4⇒ σ_x=2` 

We have `b_yx= r.σ_y/σ_x`  

∴ `2/9=1/3 σ_y/2` 

⇒ `σ_y=12/9` 

⇒ `σ_y=4/3`

`therefore ""σ_y^2= 16/9`