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Question
The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.
| Year | 1962 | 1963 | 1964 | 1965 | 1966 | 1967 | 1968 | 1969 | 1970 | 1971 | 1972 | 1973 | 1974 | 1975 | 1976 |
| Production (Million Barrels) |
0 | 0 | 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 8 | 9 | 10 |
i. Obtain trend values for the above data using 5-yearly moving averages.
ii. Plot the original time series and trend values obtained above on the same graph.
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Solution
i.
| Year t |
Production (millions of barrels) yt | 5–yearly moving total |
5–yearly moving averages trend value |
| 1962 | 0 | – | – |
| 1963 | 0 | – | – |
| 1964 | 1 | 4 | 0.8 |
| 1965 | 1 | 7 | 1.4 |
| 1966 | 2 | 11 | 2.2 |
| 1967 | 3 | 15 | 3 |
| 1968 | 4 | 20 | 4 |
| 1969 | 5 | 25 | 5 |
| 1970 | 6 | 30 | 6 |
| 1971 | 7 | 35 | 7 |
| 1972 | 8 | 38 | 7.6 |
| 1973 | 9 | 41 | 8.2 |
| 1974 | 8 | 44 | 8.8 |
| 1975 | 9 | – | – |
| 1976 | 10 | – | – |
ii.
Taking year on X-axis and production trend on Y-axis, we plot the points for production corresponding to years to get the graph of time series and plot the points for trend values corresponding to years to get the graph of trend as shown in the adjoining figure. Production: 5 yearly moving Average: ---------------
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Σy = na + bΣx
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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`
