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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.
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Solution
| 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
bxy = `("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
byx = `("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
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