Question

The coefficient of correlation between the values denoted by X and Y is 0.5. The mean of X is 3 and that of Y is 5. Their standard deviations are 5 and 4 respectively.
Find:
(i) the two lines of regression.
(ii) the expected value of Y, when X is given 14.
(iii) the expected value of X, when Y is given 9.

Answer 1

`barx = 3, bary = 5, σ_x = 5, σ_y = 4, r = 0.5`

b_yx = `r. (σ_y)/(σ_x) = 0.5 xx (4)/(5) = 0.4`,

b_xy = `r. (σ_x)/(σ_y) = 0.5 xx (5)/(4) = 0.625`

(i) Regression equation,

`y - bary = b_(yx)  ( x - barx)`
y - 5 = (0.4) (x - 3)
y = 0.4x + 3.8

`x - barx = b_(xy)  ( y - bary)`
x - 3 = (0.625) (y - 5)
x = 0.625y - 0.125

Thus two lines of regression are y = 0.4x + 3.8 and x = 0.625y - 0.125.

(ii) 
x = 14
y = 0.4 x 14 + 3.8
= 5.6 + 3.8 = 9.4

Thus, the expected value of Y, when X = 14 is 9.4

(iii) 
y = 9
x = 0.625 x 9 - 0.125 = 5.625 - 0.125 = 5.5

Thus, the expected value of X, when Y = 9 is 5.5