Question

For 5 observations of pairs of (X, Y) of variables X and Y the following results are obtained. ∑X = 15, ∑Y = 25, ∑X² = 55, ∑Y² = 135, ∑XY = 83. Find the equation of the lines of regression and estimate the values of X and Y if Y = 8; X = 12.

Answer 1

N = 5, ΣX = 15, ΣY = 25, ΣX² = 55, ΣY² = 135, ΣXY = 83, `bar"X" = 15/5` = 3, `bar"Y" = 25/5` = 5.

b_yx = `("N"sum"XY" - (sum"X")(sum"Y"))/("N"(sum"X"^2) - (sum"X")^2)`

= `(5(83) - (15)(25))/(5(55) - (15)^2)`

= `(415 - 375)/(275 - 225)`

= `40/50`

= 0.8

Regression line of Y on X:

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

Y – 5 = 0.8(X – 3)

Y = 0.8X – 2.4 + 5

Y = 0.8X + 2.6

When X = 12, Y = 0.8X + 2.6

Y = (0.8)12 + 2.6

= 9.6 + 2.6

= 12.2

b_xy = `("N"sum"XY" - (sum"X")(sum"Y"))/("N"(sum"Y"^2) - (sum"Y")^2)`

= `(5(83) - (15)(25))/(5(135) - (25)^2)`

= `(415 - 375)/(675 - 625)`

= `40/50`

= 0.8

Regression line of X on Y:

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

X – 3 = 0.8(Y – 5)

X = 0.8Y – 4 + 3

X = 0.8Y – 1

When Y = 8, X = 0.8Y – 1

X = (0.8)8 – 1

= 6.4 – 1

= 5.4