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प्रश्न
Use the method of least squares to fit a trend line to the data in Problem 1 above. Also, obtain the trend value for the year 1975.
उत्तर
In the given problem, n = 11 (odd), middle t- values is 1967, h = 1
u = `"t - middle value"/"h" = ("t" - 1967)/1` = t – 1967
We obtain the following table.
| Year (t) | Production(yt) (in '000 tonnes) |
u = t 1967 | u2 | uyt | Trend value |
| 1962 | 0 | –5 | 25 | 0 | –1.0002 |
| 1963 | 0 | –4 | 16 | 0 | 0.0362 |
| 1964 | 1 | –3 | 9 | –3 | 1.0726 |
| 1965 | 1 | –2 | 4 | –2 | 2.1090 |
| 1966 | 4 | –1 | 1 | –4 | 3.1454 |
| 1967 | 2 | 0 | 0 | 0 | 4.1818 |
| 1968 | 4 | 1 | 1 | 4 | 5.2182 |
| 1969 | 9 | 2 | 4 | 18 | 6.2546 |
| 1970 | 7 | 3 | 9 | 21 | 7.2910 |
| 1971 | 10 | 4 | 16 | 40 | 8.3274 |
| 1972 | 8 | 5 | 25 | 40 | 9.3638 |
| Total | 46 | 0 | 110 | 114 |
From the table, n = 11, `sumy_"t" = 46, sumu = 0, sumu^2 = 110,sumuy_"t" = 114`
The two normal equations are: `sumy_"t" = "na"' + "b"' sumu "and" sumuy_"t", = a'sumu + b'sumu^2`
∴ 46 = 11a' + b'(0) ...(i) and
114 = a' = `(46)/(11)` = 4.1818
From (ii), b' = `(114)/(110)` = 1.0364
∴ The equation of the trend line is yt = a' + b'u
i.e., yt = 4.1818 + 1.0364 u, where u = t – 1967
Now, for t = 1975, u = 1975 – 1967 = 8
∴ yt = 4.1818 + 1.0364 x 8 = 12.473.
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संबंधित प्रश्न
The following table gives the production of steel (in millions of tonnes) for years 1976 to 1986.
| Year | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 | 1985 | 1986 |
| Production | 0 | 4 | 4 | 2 | 6 | 8 | 5 | 9 | 4 | 10 | 10 |
Fit a trend line to the above data by the graphical method.
Choose the correct alternative :
Which of the following can’t be a component of a time series?
An overall upward or downward pattern in an annual time series would be contained in which component of the times series?
Choose the correct alternative :
The following trend line equation was developed for annual sales from 1984 to 1990 with 1984 as base or zero year. Y = 500 + 60X (in 1000 Rs). The estimated sales for 1984 (in 1000 Rs) is:
Fill in the blank :
_______ component of time series is indicated by a smooth line.
_______ component of time series is indicated by periodic variation year after year.
_______ component of time series is indicated by a long wave spanning two or more years.
Fill in the blank :
_______ component of time series is indicated by up and down movements without any pattern.
State whether the following is True or False :
The secular trend component of time series represents irregular variations.
State whether the following is True or False :
Seasonal variation can be observed over several years.
Cyclical variation can occur several times in a year.
Choose the correct alternative:
Which of the following can’t be a component of a time series?
Which component of time series refers to erratic time series movements that follow no recognizable or regular pattern?
Choose the correct alternative:
The following trend line equation was developed for annual sales from 1984 to 1990 with 1984 as base or zero year.
Y = 500 + 60X (in 1000 ₹). The estimated sales for 1984 (in 1000 ₹) is
Choose the correct alternative:
An overall upward or downward pattern in an annual time series would be contained in which component of the times series?
______ components of time series is indicated by a smooth line
Cyclical variation can occur several times in a year.
