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Question
Calculate the regression equations of X on Y and Y on X from the following data:
| X | 10 | 12 | 13 | 17 | 18 |
| Y | 5 | 6 | 7 | 9 | 13 |
Solution
| X = xi | Y = yi | `"x"_"i"^2` | `"y"_"i"^2` | xi yi |
| 10 | 5 | 100 | 25 | 50 |
| 12 | 6 | 144 | 36 | 72 |
| 13 | 7 | 169 | 49 | 91 |
| 17 | 9 | 289 | 81 | 153 |
| 18 | 13 | 324 | 169 | 234 |
| 70 | 40 | 1026 | 360 | 600 |
From the table, we have,
n = 5, ∑ xi = 70, ∑ yi = 40, ∑ xi yi = 600, `sum"x"_"i"^2 = 1026`, `sum"y"_"i"^2 = 360
`bar"x" = sum"x"_"i"/"n" = 70/5 = 14`,
`bar"y" = sum"y"_"i"/"n" = 40/5 = 8`
Now, for regression equation of X on Y
`"b"_"XY" = (sum"x"_"i" "y"_"i" - "n" bar "x" bar "y")/(sum "y"_"i"^2 - "n" bar"y"^2)`
`= (600 - 5 xx 14 xx 8)/(360 - 5(8)^2) = (600 - 560)/(360 - 320) = 40/40 = 1`
Also, `"a"' = bar"x" - "b"_"XY" bar"y" = 14 - 1(8) = 14 - 8 = 6`
∴ The regression equation of X on Y is
X = a' + bXYY
∴ X = 6 + Y
Now, for regression equation of Y on X
`"b"_"YX" = (sum"x"_"i" "y"_"i" - "n" bar "x" bar "y")/(sum "x"_"i"^2 - "n" bar"x"^2)`
`= (600 - 5(14)(8))/(1026 - 5(14)^2) = (600- 560)/(1026 - 980) = 40/46 = 0.87`
Also, a = `bar"y" - "b"_"YX" bar"x"`
`= 8 - 0.87 xx 14 = 8 - 12.18 = - 4.18`
∴ The regression equation of Y on X is
Y = a + bYX X
∴ Y = - 4.18 + 0.87X
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