The equations given of the two regression lines are 2x + 3y - 6 = 0 and 5x + 7y - 12 = 0. Find: (a) Correlation coefficient (b) σxσy - Mathematics and Statistics

Sum

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

Solution

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).

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
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