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प्रश्न
Explain the method of fitting a straight line
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उत्तर
The method of fitting a straight line is as follows
Procedure:
(i) The straight-line trend is represented by the equation Y = a + bX ........(1)
Where Y is the actual value, X is time, a, b are constants.
(ii) The constants ‘a and ‘b’ are estimated by solving the following two normal Equations
`sum"Y"` = na + b `sum"X"` ........(2)
`sum"XY" = "a"sum"X" + "b"sum"X"^2` .........(3)
Where ‘n’ = number of years given in the data.
(iii) By taking the mid-point of the time as the origin, we get `sum"X"` = 0
(iv) When `sum"X"` = 0, the two normal equations reduces to
`sum"Y"` = na + b(0), a `(sum"Y")/"n" = bar"Y"`
`sum"XY"` = a(0) + `"b"sum"X"^2`, b = `(sum"XY")/(sum"X"^2)`
The constant ‘a’ gives the mean of Y and ‘b gives the rate of change (slope),
(v) By substituting the values of ‘a and ‘b’ in the trend equation (1), we get the Line of Best Fit.
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संबंधित प्रश्न
Define Time series
State the uses of time series
Find the trend of production by the method of a five-yearly period of moving average for the following data:
| Year | Production ('000) |
| 1979 | 126 |
| 1980 | 123 |
| 1981 | 117 |
| 1982 | 128 |
| 1983 | 125 |
| 1984 | 124 |
| 1985 | 130 |
| 1986 | 114 |
| 1987 | 122 |
| 1988 | 129 |
| 1989 | 118 |
| 1990 | 123 |
The following table gives the number of small-scale units registered with the Directorate of Industries between 1985 and 1991. Show the growth on a trend line by the free hand method.
| Year | No. of units (in '000) |
| 195 | 10 |
| 986 | 22 |
| 1987 | 36 |
| 198 | 62 |
| 1989 | 55 |
| 1990 | 0 |
| 1991 | 34 |
| 1992 | 50 |
The annual production of a commodity is given as follows:
| Year | production (in tones) |
| 1995 | 155 |
| 1996 | 162 |
| 1997 | 171 |
| 19988 | 182 |
| 1999 | 158 |
| 2000 | 880 |
| 2001 | 178 |
Fit a straight line trend by the method of least squares
Use the method of monthly averages to find the monthly indices for the following data of production of a commodity for the years 2002, 2003 and 2004
| 2002 | 2003 | 2004 |
| 15 | 20 | 18 |
| 18 | 18 | 25 |
| 17 | 16 | 21 |
| 19 | 13 | 11 |
| 16 | 12 | 14 |
| 20 | 15 | 16 |
| 21 | 22 | 19 |
| 18 | 16 | 20 |
| 17 | 18 | 1 |
| 15 | 20 | 16 |
| 14 | 17 | 18 |
| 18 | 15 | 20 |
Fit a straight line trend by the method of least squares to the following data
| Year | 1980 | 1981 | 1982 | 1983 | 1984 | 1985 | 1986 | 1987 |
| Sales | 50.3 | 52.7 | 49.3 | 57.3 | 56.8 | 60.7 | 62.1 | 58.7 |
A bullet of mass m and velocity a is fired into a large block of wood of mass M The final velocity of the system is
The nth term of the series 2 + 4 + 7 + 11 + ..... is
Sum of the first n terms of the series `1/2 + 3/4 + 7/8 + 15/16 +`......... is equal to:
