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Question
The following table shows the number of salesmen working for a certain concern:
| Year | 1992 | 1993 | 1994 | 1995 | 1996 |
| No. of salesman |
46 | 48 | 42 | 56 | 52 |
Use the method of least squares to fit a straight line and estimate the number of salesmen in 1997
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Solution
Computation of trend values by the method of least squares (ODD Years)
| Year (x) |
No. of Salesman (y) |
X = (x – 1994) | x2 | XY | Trends Value (Yt) |
| 1992 | 46 | – 2 | 4 | – 92 | 44.8 |
| 1993 | 48 | – 1 | 1 | – 48 | 46.8 |
| 1994 | 42 | 0 | 0 | 0 | 48.8 |
| 1995 | 56 | 1 | 1 | 56 | 50.8 |
| 1996 | 52 | 2 | 4 | 104 | 52.8 |
| N = 5 | `sum"Y"` = 244 | `sum"X"` = 0 | `sum"X"^2` = 10 | `sum"XY"` = 20 | `sum"Yt"` = 244 |
a = `(sum"Y")/"n" = 244/5` = 48.8
b = `(sum"XY")/(sum"X"^2) = 20/10` = 2
Therefore, the required equation of the straight line trend is given by
Y = a + bX
Y = 48.8 + 2X
i.e Y = 48.8 + 2(x – 1994)
When x = 1992
Yt = 48.8 + 2(1992 – 1994)
Y = 48.8 + 2(– 2)
= 48.8 – 4
= 44.8
When x = 1993
Yt = 48.8 + 2(1993 – 1994)
Y = 48.8 + 2(– 1)
= 48.8 – 2
= 46.8
When x = 1994
Yt = 48.8 + 2(1994 – 1994)
Y = 48.8 + 2(0)
= 48.8
When x = 1995
Yt = 48.8 + 2(1995 – 1994)
Y = 48.8 + 2(1)
= 50.8
When x = 1996
Yt = 48.8 + 2(1996 – 1994)
Y = 48.8 + 2(2)
= 48.8 + 4
= 52.8
When x = 1997
Yt = 48.8 + 2(1997 – 1994)
Y = 48.8 + 2(3)
= 48.8 + 6
= 54.8
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