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Question
The following table gives the aptitude test scores and productivity indices of 10 workers selected at random.
| Aptitude score (X) | 60 | 62 | 65 | 70 | 72 | 48 | 53 | 73 | 65 | 82 |
| Productivity Index (Y) | 68 | 60 | 62 | 80 | 85 | 40 | 52 | 62 | 60 | 81 |
Obtain the two regression equations and estimate the test score when the productivity index is 75.
Solution
Here, X = Aptitude score, Y = Productivity index
| X = xi | Y =yi | `"x"_"i" - bar"x"` | `bar"y"_"i" - bar"y"` | `("x"_"i" - bar"x")^2` | `("y"_"i" - bar"y")^2` | `("x"_"i" - bar"x")("y"_"i" - bar"y")` |
| 60 | 68 | -5 | 3 | 25 | 9 | -15 |
| 62 | 60 | -3 | -5 | 9 | 25 | 15 |
| 65 | 62 | 0 | -3 | 0 | 9 | 0 |
| 70 | 80 | 5 | 15 | 25 | 225 | 75 |
| 72 | 85 | 7 | 20 | 49 | 400 | 140 |
| 48 | 40 | -17 | -25 | 289 | 625 | 425 |
| 53 | 52 | -12 | -13 | 144 | 169 | 156 |
| 73 | 62 | 8 | -3 | 64 | 9 | -24 |
| 65 | 60 | 0 | -5 | 0 | 25 | 0 |
| 82 | 81 | 17 | 16 | 289 | 256 | 272 |
| 650 | 650 | - | - | 894 | 1752 | 1044 |
From the table, we have
n = 10, ∑ xi = 650, ∑ yi = 650
∴ `bar"x" = (sum "x"_"i")/"n" = 650/10 = 65`
`bar"y" = (sum "y"_"i")/"n" = 650/10 = 65`
Since the mean of X and Y are whole numbers, we will use the formula
`"b"_"YX" = (sum ("x"_"i" - bar"x")("y"_"i" - bar"y"))/(sum("x"_"i" - bar"x")^2) and "b"_"XY" = (sum ("x"_"i" - bar"x")("y"_"i" - bar"y"))/(sum("y"_"i" - bar"y")^2)`
From the table, we have
`sum ("x"_"i" - bar"x")("y"_"i" - bar"y") = 1044, sum ("x"_"i" - bar"x")^2 = 894, sum ("y"_"i" - bar"y") = 1752`
`"b"_"XY" = (sum ("x"_"i" - bar"x")("y"_"i" - bar"y"))/(sum("x"_"i" - bar"x")^2) = 1044/1752 = 0.59`
Now, `"a"' = bar"x" - "b"_"XY" bar"y"`
= 65 - 0.59 × 65 = 65 - 38.35 = 26.65
∴ The regression equation of Aptitude score (X) on productivity index (Y) is
X = a' + bXY Y
∴ X = 26.65 + 0.59 Y
For Y = 75,
X = 26.65 + 0.59 × 75 = 26.65 + 44.25 = 70.9
∴ The test score is 70.9 when productivity index is 75
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