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Question
A departmental store gives trafnfng to the salesmen in service followed by a test. It is experienced that the performance regarding sales of any salesman is linearly related to the scores secured by him. The following data gives the test scores and sales made by nine (9) salesmen during a fixed period.
| Test scores (X) | 16 | 22 | 28 | 24 | 29 | 25 | 16 | 23 | 24 |
| Sales (Y) (₹ in hundreds) | 35 | 42 | 57 | 40 | 54 | 51 | 34 | 47 | 45 |
(a) Obtain the line of regression of Y on X.
(b) Estimate Y when X = 17.
Solution
| X = xi | Y = yi | `x_i - barx` | `y_i - bary` | `(x - barx)^2` | `(x_i - barx)(y_i - bary)` |
| 16 | 35 | -7 | -10 | 49 | 70 |
| 22 | 42 | -1 | -3 | 1 | 3 |
| 28 | 57 | 5 | 12 | 25 | 60 |
| 24 | 40 | 1 | -5 | 1 | -5 |
| 29 | 54 | 6 | 9 | 36 | 54 |
| 25 | 51 | 2 | 6 | 4 | 12 |
| 16 | 34 | -7 | -11 | 49 | 77 |
| 23 | 47 | 0 | 2 | 0 | 0 |
| 24 | 45 | 1 | 0 | 1 | 0 |
| 207 | 405 | 0 | 0 | 166 | 271 |
`therefore n = 9 , Σx_i = 207 , Σy_i = 405`
`barx = (Σx_i)/n = 207/9 = 23 , bary = (Σy_i)/n = 405/9 = 45`
(a) Line of regression Y on X is
`y - bary = b_(yx)(x - barx)` .....(i)
Where , `b_(yx) = (Σ(x_i - barx)(y_i - bary))/(Σ(x_i - barx)^2 )`
= `271/166`
= 1.6325
From (i) equation of regression line Y on X is
(y - 45) = 1.6325 (x - 23)
y - 45 = -1.6325(23) + 1.6325x
y = 74525 + 1.6325x
(b) Estimate of Y when X = 17 is
y = 7.4525 + ( 1.6325) (17)
= 7.4525 + 27.7525
= 35.205
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