Obtain trend values for data in Problem 1 using 4-yearly centred moving averages - Mathematics and Statistics

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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
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Solution

Construct the following table for obtaining 4-yearly centred moving average for the data.

Year 

t

Production

yt

4-yearly moving
total

 

4-yearly moving average 2 unit moving total 4-yearly centred
moving averages trend value
1971 1        
           
1972 0        
    4 1    
1973 1     2.5 1.25
    6 1.5    
1974 2     3..5 1.75
    8 2    
1975 3     4.75 2.375
    11 2.75    
1796 2     6.5 3.25
    15 3.75    
1977 4     8 4
    17 4.25    
1978 6     8.25 4.125
    16 4    
1979 5     8 4
    16 4    
1980 1     9 4.5
    20 5    
1981 4        
           
1982 10        
Concept: Measurement of Secular Trend
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Chapter 2.4: Time Series - Q.4

RELATED QUESTIONS

Obtain the trend values for the data in using 4-yearly centered moving averages.

Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985
Index 0 2 3 3 2 4 5 6 7 10

Fit a trend line to the data in Problem 7 by the method of least squares. Also, obtain the trend value for the year 1990.


Choose the correct alternative :

We can use regression line for past data to forecast future data. We then use the line which_______.


Fill in the blank :

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


State whether the following is True or False :

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


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 :

Fit a trend line to data in Problem 4 by the method of least squares.


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.


Solve the following problem :

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010.

Year 1980 1985 1990 1995 2000 2005 2010
IMR 10 7 5 4 3 1 0

Fit a trend line to the above data by graphical method.


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


The simplest method of measuring trend of time series is ______


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


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 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  

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.

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010

Year 1980 1985 1990 1995
IMR 10 7 5 4
Year 2000 2005 2010  
IMR 3 1 0  

Fit a trend line by the method of least squares

Solution: Let us fit equation of trend line for above data.

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

Here n = 7(odd), middle year is `square` and h = 5

Year IMR (y) x x2 x.y
1980 10 – 3 9 – 30
1985 7 – 2 4 – 14
1990 5 – 1 1 – 5
1995 4 0 0 0
2000 3 1 1 3
2005 1 2 4 2
2010 0 3 9 0
Total 30 0 28 – 44

The normal equations are

Σy = na + bΣx

As, Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx2

As, Σx = 0, b =`square`

∴ The equation of trend line is y = `square`


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 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`


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

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:


The following table shows gross capital information (in Crore ₹) for years 1966 to 1975:

Years 1966 1967 1968 1969 1970
Gross Capital information 20 25 25 30 35
Years 1971 1972 1973 1974 1975
Gross Capital information 30 45 40 55 65

Obtain trend values using 5-yearly moving values.


The publisher of a magazine wants to determine the rate of increase in the number of subscribers. The following table shows the subscription information for eight consecutive years:

Years 1976 1977 1978 1979
No. of subscribers
(in millions)
12 11 19 17
Years 1980 1981 1982 1983
No. of subscribers
(in millions)
19 18 20 23

Fit a trend line by graphical method.


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