Given the regression lines X + 2Y - 5 = 0, 2X + 3Y - 8 = 0 and Var(X) = 12, the value of Var(Y) is

Question

Question

This question was previously asked in

SSC CGL Tier-II ( JSO ) 2019 Official Paper ( Held On : 17 Nov 2020 )

Answer 1

Given

Regression lines

x + 2y - 5 = 0

2x + 3y - 8 = 0

var(x)= σ_x = 12

Calculation

x + 2y - 5 = 0 ------(i)

Let y = - x/2 + 5/2 be the regression line of y on x [ from equation 1]

2x + 3y - 8

x = -(3/2)y + 8/2 be the regressiopn line of x on y

⇒ b_yx = -1/2 and b_x_y = -3/2

b_xy = Regression line of y on x

b_yx = Regression line of x on y

We know that regression coefficient = r = √(b_yx × b_xy)

⇒ r = √(-1/2 × -3/2)

Since both bxy and byx are negative r is also negative

∴ r = -√3/2 < 1

σ_x = \(\sqrt{12}\) = 2√3

We know that b_yx = r (σ_y/σ_x)

⇒ -1/2 = -√3/2 (σ_y/2√3)

⇒ σ_y = 2

∴ var(y) = variance of y =(2)² = 4

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