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Question
From the following data, find the regression equation of Y on X and estimate Y when X = 10.
| X | 1 | 2 | 3 | 4 | 5 | 6 |
| Y | 2 | 4 | 7 | 6 | 5 | 6 |
Solution
| X = xi | Y = yi | `"x"_"i"^2` | xi yi |
| 1 | 2 | 1 | 2 |
| 2 | 4 | 4 | 8 |
| 3 | 7 | 9 | 21 |
| 4 | 6 | 16 | 24 |
| 5 | 5 | 25 | 25 |
| 6 | 6 | 36 | 36 |
| 21 | 30 | 91 | 116 |
From the table, we have
n = 6, ∑ xi = 21, ∑ yi = 30, `sum "x"_"i"^2 = 91`, ∑ xi yi = 116
`bar x = (sum x_i)/"n" = 21/6 = 3.5`
`bar y = (sum y_i)/"n" = 30/6 = 5`
Now, `"b"_"YX" = (sum"x"_"i" "y"_"i" - "n" bar "x" bar "y")/(sum "x"_"i"^2 - "n" bar"x"^2)`
`= (116 - 6xx3.5xx5)/(91 - 6(3.5)^2) = (116 - 105)/(91 - 73.5) = 11/17.5 = 0.63`
Also, `"a" = bar y - "b"_"YX" bar x`
= 5 - 0.63 × 3.5
= 5 - 2.205 = 2.8
The regression equation of Y on X is,
Y = a + bYX X
∴ Y = 2.8 + 0.63 X
For X = 10,
Y = 2.8 + 0.63 × 10
= 2.8 + 6.3 = 9.1
∴ The value of Y when X =10 is 9.1
Notes
The answer in the textbook is incorrect.
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