Question

The equation of the line of regression of Y on X is 3x + 2y = 26 and X on Y is 6x + y = 31. Find Var. (X) if Var. (Y) = 36.

Answer 1

The regression line of Yon Xis 3x + 2y = 26

`=> "y" = (-3"x" + 26)/2` 

`therefore "y" = -3/2 "x" + 26/2`

`therefore "b"_"yx" = "Coefficient of x" = (-3)/2`

The regression line of x on y is 6x + y = 31

⇒ 6x = -y + 31

⇒ `"x" = (-"y" + 31)/6`

`therefore "x" = -1/6 "y" + 31/6`

`therefore "b"_"xy" = "Coefficient of y" = (-1)/6`

Now `"b"_"yx" . "b"_"xy" = (-3/2)(-1/6) = 1/4 < 1`

`"r" = sqrt ("b"_"yx" . "b"_"xy") = sqrt (1/4) = +-  1/2`

Since b_yx and b_xy both are negative 

`therefore "r" = -1/2`

Given Var (Y) = 36

∴ σ²y = 36 

∴ σ y = 6

Again b_xy = r . `(sigma "x")/(sigma "y")`

`=> -1/6 = -1/2 xx (sigma "x")/6`

⇒ σ x = 2

∴ Var (X) = σ²x = (2)² = 4