Question

The heights (in cm.) of a group of fathers and sons are given below:

Heights of fathers:	158	166	163	165	167	170	167	172	177	181
Heights of Sons:	163	158	167	170	160	180	170	175	172	175

Find the lines of regression and estimate the height of the son when the height of the father is 164 cm.

Answer 1

Heights of fathers (X)	Heights of Sons (Y)	dx = X − 168	dy = Y − 169	dx2	dy2	dxdy
158	163	− 10	− 6	100	36	− 60
166	158	− 2	− 11	4	121	22
163	167	− 5	− 2	25	4	10
165	170	− 3	1	9	1	− 3
167	160	− 1	− 9	1	81	9
170	180	2	11	4	121	22
167	170	− 1	1	1	1	−1
172	175	4	6	16	36	24
177	172	9	3	81	9	27
181	175	13	6	169	36	78
1686	1690	6	0	410	446	248

N = 10, ∑X = 1686, ∑Y = 1690, ∑dx² = 410, ∑y² = 446, ∑dxdy = 248, `bar"X" = 1686/10` = 168.6, `bar"Y" = 1690/10` = 169

b_xy = `("N"sum"dxdy" - (sum"dx")(sum"dy"))/("N"sum"dy"^2 - (sum"dy")^2)`

= `(10(248) - 6(0))/((10)(446) - 0^2)`

= `2480/4460`

= 0.556

Regression equation of X on Y

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

X – 168.6 = 0.556 (Y – 169)

X – 168.6 = 0.556Y – 93.964

 X = 0.556Y – 93.964 + 168.6

X = 0.556Y + 76.636

X = 0.556Y + 74.64

b_yx = `("N"sum"dxdy" - (sum"dx")(sum"dy"))/("N"sum"dx"^2 - (sum"dx")^2)`

= `(10(248) - 0)/(10(410) - 6^2)`

= `2480/(4100 - 36)`

= `2480/4064`

= 0.610

Regression equation of Y on X

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

Y − 169 = 0.610 (X − 168.6)

Y – 169 = 0.610X – 102.846

Y = 0.610X – 102.846 + 169

Y = 0.610X + 66.154 ………(1)

To get son’s height (Y) when the father height is X = 164 cm.

Put X = 164 cm in equation (1) we get

Son’s height = 0.610 × 164 + 66.154

= 100.04 + 66.154 

= 166.19 cm