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Following table shows the amount of sugar production (in lac tons) for the years 1971 to 1982
Year | 1971 | 1972 | 1973 | 1974 | 1975 | 1976 |
Production | 1 | 0 | 1 | 2 | 3 | 2 |
Year | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 |
Production | 4 | 6 | 5 | 1 | 4 | 10 |
Fit a trend line by the method of least squares
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Solution
In the given problem, n = 12 (even), two middle t − values are 1976 and 1977, h = 1
u = `("t" - "mean of two middle values")/("h"/2)`
= `("t" - 1976.5)/(1/2)`
= 2(t − 1976.5)
We obtain the following table:
Year t |
Poduction y_{t} |
u = 2(t − 1976.5 | u^{2} | uy_{t} | Trend Value |
1971 | 1 | – 11 | 121 | – 11 | 0.1535 |
1972 | 0 | – 9 | 81 | – 0 | 0.7165 |
1973 | 1 | – 7 | 49 | – 7 | 1.2795 |
1974 | 2 | – 5 | 25 | – 10 | 1.8425 |
1975 | 3 | – 3 | 9 | – 9 | 2.4055 |
1976 | 2 | – 1 | 1 | – 2 | 2.9685 |
1977 | 4 | 1 | 1 | 4 | 3.5315 |
1978 | 6 | 3 | 9 | 18 | 4.0945 |
1979 | 5 | 5 | 25 | 25 | 4.6575 |
1980 | 1 | 7 | 49 | 7 | 5.2205 |
1981 | 4 | 9 | 81 | 36 | 5.7835 |
1982 | 10 | 11 | 121 | 110 | 6.3465 |
Total | 39 | 0 | 572 | 161 |
From the table, n = 12, ∑y_{t} = 39, ∑u = 0, ∑u^{2} = 572, ∑uy_{t} = 161
The two normal equations are:
∑y_{t} = na' + b'∑u and ∑uy_{t} = a'∑u + b'∑u^{2}
∴ 39 = 12a' + b'(0) ......(i)
and 161 = a'(0) + b'(572) ......(ii)
From (i), a′ = `39/12` = 3.25
From (ii), b′ = `161/572` = 0.2815
∴ The equation of the trend line is y_{t} = a′ + b′u
i.e., y_{t} = 3.25+ 0.2815 u,
where u = 2(t − 1976.5)
RELATED QUESTIONS
Fit a trend line to the data in Problem 4 above by the method of least squares. Also, obtain the trend value for the index of industrial production for the year 1987.
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.
Choose the correct alternative :
We can use regression line for past data to forecast future data. We then use the line which_______.
Choose the correct alternative :
What is a disadvantage of the graphical method of determining a trend line?
Fill in the blank :
The complicated but efficient method of measuring trend of time series is _______.
State whether the following is True or False :
Moving average method of finding trend is very complicated and involves several calculations.
Fit a trend line to the following data by the method of least squares.
Year | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 |
Production | 0 | 4 | 9 | 9 | 8 | 5 | 4 | 8 | 10 |
Solve the following problem :
Following table shows the amount of sugar production (in lac tonnes) for the years 1971 to 1982.
Year | 1971 | 1972 | 1973 | 1974 | 1975 | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 |
Production | 1 | 0 | 1 | 2 | 3 | 2 | 3 | 6 | 5 | 1 | 4 | 10 |
Fit a trend line to the above data by graphical method.
Obtain trend values for the following data using 4-yearly centered moving averages.
Year | 1971 | 1972 | 1973 | 1974 | 1975 | 1976 |
Production | 1 | 0 | 1 | 2 | 3 | 2 |
Year | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 |
Production | 3 | 6 | 5 | 1 | 4 | 10 |
Solve the following problem :
Following data shows the number of boxes of cereal sold in years 1977 to 1984.
Year | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 |
No. of boxes in ten thousand | 1 | 0 | 3 | 8 | 10 | 4 | 5 | 8 |
Fit a trend line to the above data by graphical method.
Solve the following problem :
Fit a trend line to data by the method of least squares.
Year | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 |
Number of boxes (in ten thousands) | 1 | 0 | 3 | 8 | 10 | 4 | 5 | 8 |
Solve the following problem :
Fit a trend line to data in Problem 13 by the method of least squares.
Choose the correct alternative:
Moving averages are useful in identifying ______.
The method of measuring trend of time series using only averages is ______
State whether the following statement is True or False:
The secular trend component of time series represents irregular variations
State whether the following statement is True or False:
Moving average method of finding trend is very complicated and involves several calculations
State whether the following statement is True or False:
Least squares method of finding trend is very simple and does not involve any calculations
Obtain trend values for data, using 4-yearly centred moving averages
Year | 1971 | 1972 | 1973 | 1974 | 1975 | 1976 |
Production | 1 | 0 | 1 | 2 | 3 | 2 |
Year | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 |
Production | 4 | 6 | 5 | 1 | 4 | 10 |
The following table gives the production of steel (in millions of tons) 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 |
Obtain the trend value for the year 1990
Obtain the trend values for the data, using 3-yearly moving averages
Year | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 |
Production | 0 | 4 | 4 | 2 | 6 | 8 |
Year | 1982 | 1983 | 1984 | 1985 | 1986 | |
Production | 5 | 9 | 4 | 10 | 10 |
Use the method of least squares to fit a trend line to the data given below. Also, obtain the trend value for the year 1975.
Year | 1962 | 1963 | 1964 | 1965 | 1966 | 1967 | 1968 | 1969 |
Production (million barrels) |
0 | 0 | 1 | 1 | 2 | 3 | 4 | 5 |
Year | 1970 | 1971 | 1972 | 1973 | 1974 | 1975 | 1976 | |
Production (million barrels) |
6 | 8 | 9 | 9 | 8 | 7 | 10 |
Obtain trend values for data, using 3-yearly moving averages
Solution:
Year | IMR | 3 yearly moving total |
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 | – | – |
Fit equation of trend line for the data given below.
Year | Production (y) | x | x^{2} | xy |
2006 | 19 | – 9 | 81 | – 171 |
2007 | 20 | – 7 | 49 | – 140 |
2008 | 14 | – 5 | 25 | – 70 |
2009 | 16 | – 3 | 9 | – 48 |
2010 | 17 | – 1 | 1 | – 17 |
2011 | 16 | 1 | 1 | 16 |
2012 | 18 | 3 | 9 | 54 |
2013 | 17 | 5 | 25 | 85 |
2014 | 21 | 7 | 49 | 147 |
2015 | 19 | 9 | 81 | 171 |
Total | 177 | 0 | 330 | 27 |
Let the equation of trend line be y = a + bx .....(i)
Here n = `square` (even), two middle years are `square` and 2011, and h = `square`
The normal equations are Σy = na + bΣx
As Σx = 0, a = `square`
Also, Σxy = aΣx + bΣx^{2}
As Σx = 0, b = `square`
Substitute values of a and b in equation (i) the equation of trend line is `square`
To find trend value for the year 2016, put x = `square` in the above equation.
y = `square`
Complete the table using 4 yearly moving average method.
Year | Production | 4 yearly moving total |
4 yearly centered total |
4 yearly centered moving average (trend values) |
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 | – | – |
Obtain the trend values for the following data using 5 yearly moving averages:
Year | 2000 | 2001 | 2002 | 2003 | 2004 |
Production x_{i} |
10 | 15 | 20 | 25 | 30 |
Year | 2005 | 2006 | 2007 | 2008 | 2009 |
Production x_{i} |
35 | 40 | 45 | 50 | 55 |
Following table shows the amount of sugar production (in lakh tonnes) for the years 1931 to 1941:
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: