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Question
From the following data estimate y when x = 125.
| X | 120 | 115 | 120 | 125 | 126 | 123 |
| Y | 13 | 15 | 14 | 13 | 12 | 14 |
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Solution
In order to estimate y, we have to find the regression equation of Y on X.
| X = xi | Y = yi | `"x"_"i"^2` | xi yi |
| 120 | 13 | 14400 | 1560 |
| 115 | 15 | 13225 | 1725 |
| 120 | 14 | 14400 | 1680 |
| 125 | 13 | 15625 | 1625 |
| 126 | 12 | 15876 | 1512 |
| 123 | 14 | 15129 | 1722 |
| 729 | 81 | 88655 | 9824 |
From table, we get
n = 6, ∑ xi = 729, ∑ yi = 81, ∑ `"x"_"i"^2` = 88655, ∑ xi yi = 9824
∴ `bar"x" = (sum "x"_"i")/"n" = 729/6 = 121.5`
∴ `bar"y" = (sum "y"_"i")/"n" = 81/6 = 13.5`
Now,
`"b"_"YX" = (sum"x"_"i" "y"_"i" - "n" bar "x" bar "y")/(sum "x"_"i"^2 - "n" bar"x"^2)`
`= (9824 - 6 xx 121.5 xx 13.5)/(88655 - 6(121.5)^2)`
`= (9824 - 9841.5)/(88655 - 88573.5)`
= `(- 17.5)/81.5`
= - 0.22
Also, `"a" = bar"y" - "b"_"YX" bar"x"`
= 13.5 - (- 0.22) × 121.5
= 13.5 + 26.73
= 40.23
∴ The regression equation of Y on X is
Y = a + bYX X
∴ Y = 40.23 - 0.22 X
For X = 125,
Y = 40.23 - 0.22 × 125 = 40.23 - 27.5 = 12.73
∴ The estimated value of Y is 12.73 for X = 125.
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