Question

The correlation coefficient between x and y is 0.6. If the variance of x is 225, the variance of y is 400, mean of x is 10 and mean of y is 20, find

1. the equations of two regression lines.
2. the expected value of y when x = 2.

Answer 1

Given: r = 0.6, `barx = 10, bary = 20`, var. (x) = (σ_x²) = 225 and var. (y) = (σ_y²) = 400.

i. `σ_x = sqrt(225)`

= 15 

And `σ_y = sqrt(400)`

= 20

`b_(yx) = r σ_y/σ_x`

= `0.6 xx 20/15`

= `4/5`

And `b_(xy) = r σ_x/σ_y`

= `0.6 xx 15/20`

= `9/20`

So, line of regression of y on x is

`y - bary = b_(yx) (x - barx)`

⇒ `y - 20 = 4/5 (x - 10)`

⇒ 5y – 100 = 4x – 40

⇒ 4x – 5y + 60 = 0   ...(i)

Line of regression of x on y is

`x - barx = b_(xy) (y - bary)`

⇒ `x - 10 = 9/20 (y - 20)`

⇒ 20x – 200 = 9y – 180

⇒ 20x – 9y – 20 = 0

ii. At x = 2

4 × 2 – 5y + 60 = 0   ...[Putting x = 2 in equation (i)]

⇒ 5y = 68

∴ `y = 68/5`