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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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RELATED QUESTIONS
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| Year | 1962 | 1963 | 1964 | 1965 | 1966 | 1967 | 1968 | 1969 |
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| 1980 | 10 | – 3 | 9 | – 30 |
| 1985 | 7 | – 2 | 4 | – 14 |
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| 2000 | 3 | 1 | 1 | 3 |
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The normal equations are
Σy = na + bΣx
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3-yearly moving average (trend value) |
| 1980 | 10 | – | – |
| 1985 | 7 | `square` | 7.33 |
| 1990 | 5 | 16 | `square` |
| 1995 | 4 | 12 | 4 |
| 2000 | 3 | 8 | `square` |
| 2005 | 1 | `square` | 1.33 |
| 2010 | 0 | – | – |
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4 yearly centered total |
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| 2006 | 19 | – | – | |
| `square` | ||||
| 2007 | 20 | – | `square` | |
| 72 | ||||
| 2008 | 17 | 142 | 17.75 | |
| 70 | ||||
| 2009 | 16 | `square` | 17 | |
| `square` | ||||
| 2010 | 17 | 133 | `square` | |
| 67 | ||||
| 2011 | 16 | `square` | `square` | |
| `square` | ||||
| 2012 | 18 | 140 | 17.5 | |
| 72 | ||||
| 2013 | 17 | 147 | 18.375 | |
| 75 | ||||
| 2014 | 21 | – | – | |
| – | ||||
| 2015 | 19 | – | – |
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| Year | Production | Year | Production |
| 1931 | 1 | 1937 | 8 |
| 1932 | 0 | 1938 | 6 |
| 1933 | 1 | 1939 | 5 |
| 1934 | 2 | 1940 | 1 |
| 1935 | 3 | 1941 | 4 |
| 1936 | 2 |
Complete the following activity to fit a trend line by method of least squares:
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