Question

The equations given of the two regression lines are 2x + 3y - 6 = 0 and 5x + 7y - 12 = 0.

Find:

(a) Correlation coefficient

(b) `sigma_x/sigma_y`

Answer 1

We assume that 2x + 3y - 6 = 0 to be the line of regression of y on x. 

2x + 3y - 6 = 0

⇒ `x = - 3/2y + 3`

⇒ `"bxy" = - 3/2`

5x + 7y - 12 = 0 to be the line of regression of x on y.

5x + 7y - 12 = 0

⇒ `y = - 5/7x + 12/7`

⇒  `"byx" = - 5/7`

Now,

r = `sqrt("bxy.byx") = sqrt(15/14)`

byx = `(rσ_y)/(σ_x) = - 5/7, "bxy" = (rσ_x)/(σ_y) = - 3/2`

⇒ `(σ_x^2)/(σ_y^2) =  (3/2)/(5/7)`

⇒ `(σ_x^2)/(σ_y^2) = 21/10`

⇒ `(σ_x)/(σ_y) = sqrt(21/10)`.