Following table shows the amount of sugar production (in lac tons) for the years 1971 to 1982 Year 1971 1972 1973 197 1975 1976 Production 1 0 1 2 3 2 Year 1977 1978 1979 1980 1981 1982 Production 4 - Mathematics and Statistics

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Sum

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

yt

u = 2(t − 1976.5  u2 uyt 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, ∑yt = 39, ∑u = 0, ∑u2 = 572, ∑uyt = 161

The two normal equations are: 

∑yt = na' + b'∑u and ∑uyt = a'∑u + b'∑u2

∴ 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 yt = a′ + b′u

i.e., yt = 3.25+ 0.2815 u,

where u = 2(t − 1976.5)

Concept: Measurement of Secular Trend
  Is there an error in this question or solution?
Chapter 2.4: Time Series - Q.4

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 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`


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

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:


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