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Question
Obtain the trend line for the above data using 5 yearly moving averages.
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Solution
Construct the following table to obtain 5 yearly moving averages in data of problem 1.
| Year T | Production yt (in' 000 tonnes) |
5 – yearly moving total | 5 – yearly moving averages trend value |
| 1962 | 0 | – | – |
| 1963 | 0 | – | – |
| 1964 | 1 | 6 | 1.2 |
| 1965 | 1 | 8 | 1.6 |
| 1966 | 4 | 12 | 2.4 |
| 1967 | 2 | 20 | 4 |
| 1968 | 4 | 26 | 5.2 |
| 1969 | 9 | 32 | 6.4 |
| 1970 | 7 | 38 | 7.6 |
| 1971 | 10 | – | – |
| 1972 | 8 | – | – |
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Solution: Let us fit equation of trend line for above data.
Let the equation of trend line be y = a + bx .....(i)
Here n = 7(odd), middle year is `square` and h = 5
| Year | IMR (y) | x | x2 | x.y |
| 1980 | 10 | – 3 | 9 | – 30 |
| 1985 | 7 | – 2 | 4 | – 14 |
| 1990 | 5 | – 1 | 1 | – 5 |
| 1995 | 4 | 0 | 0 | 0 |
| 2000 | 3 | 1 | 1 | 3 |
| 2005 | 1 | 2 | 4 | 2 |
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| Total | 30 | 0 | 28 | – 44 |
The normal equations are
Σy = na + bΣx
As, Σx = 0, a = `square`
Also, Σxy = aΣx + bΣx2
As, Σx = 0, b =`square`
∴ The equation of trend line is y = `square`
Obtain the trend values for the following data using 5 yearly moving averages:
| Year | 2000 | 2001 | 2002 | 2003 | 2004 |
| Production xi |
10 | 15 | 20 | 25 | 30 |
| Year | 2005 | 2006 | 2007 | 2008 | 2009 |
| Production xi |
35 | 40 | 45 | 50 | 55 |
