Fit equation of trend line for the data given below. Year Production (y) x x2 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 16 201 - Mathematics and Statistics

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Fit equation of trend line for the data given below.

Year Production (y) x x2 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 + bΣx2

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`

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Solution

Let the equation of trend line be y = a + bx   .....(i)

Here n = 10 (even), two middle years are 2010 and 2011, and h = 2 

The normal equations are Σy = na + bΣx

As Σx = 0,

∴ 177 = 10a + b(0)

∴ a = `177/10`

a = 17.7 

Also, Σxy = aΣx + bΣx2

As Σx = 0, 

27 = a(0) + b(330)

∴ b = `27/330`

= 0.08

b = 0.1

Substitute values of a and b in equation (i) the equation of trend line is y = 17.7 + 0.1x

To find trend value for the year 2016, put x = 11 in the above equation.

y = 17.7 + 1.1

y = 18.8

Concept: Measurement of Secular Trend
  Is there an error in this question or solution?
Chapter 2.4: Time Series - Q.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.


Choose the correct alternative :

What is a disadvantage of the graphical method of determining a trend line?


Fill in the blank :

The method of measuring trend of time series using only averages is _______


State whether the following is True or False :

Moving average method of finding trend is very complicated and involves several calculations.


State whether the following is True or False :

All the three methods of measuring trend will always give the same results.


Solve the following problem :

Obtain trend values for the following data using 5-yearly moving averages.

Year 1974 1975 1976 1977 1978 1979 1980 1981 1982
Production 0 4 9 9 8 5 4 8 10

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 in Problem 16 by the method of least squares.


Solve the following problem :

Obtain trend values for data in Problem 19 using 3-yearly moving averages.


Solve the following problem :

Following tables shows the wheat yield (‘000 tonnes) in India for years 1959 to 1968.

Year 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968
Yield 0 1 2 3 1 0 4 1 2 10

Fit a trend line to the above data by the method of least squares.


The complicated but efficient method of measuring trend of time series is ______


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


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


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  

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
Production
(million barrels)
0 0 1 1 2 3 4 5
Year 1970 1971 1972 1973 1974 1975 1976  
Production
(million barrels)
6 7 8 9 8 9 10  
  1. Obtain trend values for the above data using 5-yearly moving averages.
  2. Plot the original time series and trend values obtained above on the same graph.

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

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
xi
10 15 20 25 30
Year 2005 2006 2007 2008 2009
Production
xi
35 40 45 50 55

The complicated but efficient method of measuring trend of time series is ______.


Following table gives the number of road accidents (in thousands) due to overspeeding in Maharashtra for 9 years. Complete the following activity to find the trend by the method of least squares.

Year 2008 2009 2010 2011 2012 2013 2014 2015 2016
Number of accidents 39 18 21 28 27 27 23 25 22

Solution:

We take origin to 18, we get, the number of accidents as follows:

Year Number of accidents xt t u = t - 5 u2 u.xt
2008 21 1 -4 16 -84
2009 0 2 -3 9 0
2010 3 3 -2 4 -6
2011 10 4 -1 1 -10
2012 9 5 0 0 0
2013 9 6 1 1 9
2014 5 7 2 4 10
2015 7 8 3 9 21
2016 4 9 4 16 16
  `sumx_t=68` - `sumu=0` `sumu^2=60` `square`

The equation of trend 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=68,sumu=0,sumu^2=60,sumux_t=-44`

Putting these values in normal equations, we get

68 = 9a' + b'(0)     ...(3)

∴ a' = `square`

-44 = a'(0) + b'(60)          ...(4)

∴ b' = `square`

The equation of trend line is given by

xt = `square`


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