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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`
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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)`.
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