Question

You are given the following data:

Details	X	Y
Arithmetic Mean	36	85
Standard Deviation	11	8

If the Correlation coefficient between X and Y is 0.66, then find

1. the two regression coefficients,
2. the most likely value of Y when X = 10.

Answer 1

`bar"X"` = 36, `bar"Y"` = 85, σ_x = 11, σ_y = 8, r = 0.66

(i) The two regression coefficients are,

b_yx = `"r"(sigma_"y")/(sigma_"x") = 0.66 xx 8/11` = 0.48

b_xy = `"r"(sigma_"x")/(sigma_"y") = 0.66 xx 11/8` = 0.9075 = 0.91

(ii) Regression equation of X on Y:

`"X" - bar"X" = "b"_"xy"("Y" - bar"Y")`

X – 36 = 0.91(Y – 85)

X – 36 = 0.91Y – 77.35

X = 0.91Y – 77.35 + 36

X = 0.91Y – 41.35

Regression line of Y on X:

`"Y" - bar"Y" = "b"_"yx"("X" - bar"X")`

Y – 85 = 0.48(X – 36)

Y = 0.48X – 17.28 + 85

Y = 0.48X + 67.72

The most likely value of Y when X = 10

Y = 0.48(10) + 67.72 = 72.52