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Question
Fit a trend line to the data in Problem 7 by the method of least squares. Also, obtain the trend value for the year 1990.
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Solution
In the given problem, n = 11 (odd), middle t- values is 1981, h = 1
u = `"t - middle value"/"h" = ("t" - 1981)/(1)` = t – 1981
We obtain the following table.
| Year t |
Production yt |
u = t–1981 | u2 | uyt | Trend Value |
| 1976 | 0 | –5 | 25 | 0 | 1.6819 |
| 1977 | 4 | –4 | 16 | –16 | 2.4728 |
| 1978 | 4 | –3 | 9 | –12 | 3.2637 |
| 1979 | 2 | –2 | 4 | –4 | 4.0546 |
| 1980 | 6 | –1 | 1 | –6 | 4.8455 |
| 1981 | 8 | 0 | 0 | 0 | 5.6364 |
| 1982 | 5 | 1 | 1 | 5 | 6.4273 |
| 1983 | 9 | 2 | 4 | 18 | 7.2182 |
| 1984 | 4 | 3 | 9 | 12 | 8.0091 |
| 1985 | 10 | 4 | 16 | 40 | 8.8 |
| 1986 | 10 | 5 | 25 | 50 | 9.5909 |
| Total | 62 | 0 | 110 | 87 |
From the table, n = 11, `sumy_"t" = 62, sumu = 0, sumu^2 = 110, sumuy_"t" = 87`
The two normal equations are : `sumy_"t" = "na"' + "b"' sumu "and" sumuy_"t" = "a"' sumu + "b"'sumu^2`
∴ 62 = 11a' + b'(0) ...(i) and
87 = a'(0) + b'(110) ...(ii)
From (i), a' = `(62)/(11)` = 5.6364
From (ii), b' = `(87)/(110)` = 0.7909
∴ The equation of the trend line is yt = a' + b'u
i.e., yt = 5.6364 + 0.7909 u, where u = t – 1981
∴ Now, For t = 1990, u = 1990 – 1981= 9
∴ yt = 5.6364 + 0.7909 x 9 = 12.7545.
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Here n = 9. We transform year t to u by taking u = t - 1979. We construct the following table for calculation :
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| `sumx_t` =47 | `sumu`=0 | `sumu^2=60` | `square` |
The equation of trend line is xt= a' + b'u.
The normal equations are,
`sumx_t = na^' + b^' sumu` ...(1)
`sumux_t = a^'sumu + b^'sumu^2` ...(2)
Here, n = 9, `sumx_t = 47, sumu= 0, sumu^2 = 60`
By putting these values in normal equations, we get
47 = 9a' + b' (0) ...(3)
40 = a'(0) + b'(60) ...(4)
From equation (3), we get a' = `square`
From equation (4), we get b' = `square`
∴ the equation of trend line is xt = `square`
