Use the method of least squares to fit a trend line to the data in Problem 6 below. Also, obtain the trend value for the year 1975 - Mathematics and Statistics

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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  
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Solution

In the given problem, n = 15 (odd), middle t – values is 1969, h = 1

u = `("t" - "middle value")/"h"`

= `("t" - 1969)/1`

= t – 1969

We obtain the following table:

Year 
t

Production
yt
u = t − 1969 u2 uyt Trend Value
1962 0 −  49 0 − 0.6
1963 0 − 6 36 0 0.2
1964 1  − 5 25 − 5 1
1965 1 − 4 16 − 4 1.8
1966 2 − 3 9 − 6 2.6
1967 3 − 2 4 − 6 3.4
1968 4 − 1 1 − 4 4.2
1969 5 0 0 0 5
1970 6 1 1 6 5.8
1971 8 2 4 16 6.6
1972 9 3 9 27 7.4
1973 9 4 16 36 8.
1974 8 5 25 40 9
1975 9 6 36 54 9.8
1976 10 7 49 70 10.6
Total 75 0 280 224  

From the table, n = 15, ∑yt = 75, ∑u = 0, ∑u2 = 280, ∑uyt = 224

The two normal equations are:

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

∴ 75 = 15a' + b'(0)   ......(i)

and

224 = a′(0) + b′(280)  .....(ii)

From (i), a′ = `75/15` = 5

From (ii), b′= `224/280` = 0.8

∴ The equation of the trend line is yt = a′ + b′u

i.e., yt = 5 + 0.8 u, where u = t – 1969

Now, for t = 1975, u = 1975 – 1969 = 6

∴  yt = 5 + 0.8 × 6 = 9.8

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

RELATED QUESTIONS

Obtain the trend values for the above data using 3-yearly moving averages.


Choose the correct alternative :

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Fill in the blank :

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


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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

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Year 1977 1978 1979 1980 1981 1982 1983 1984
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Year 1980 1985 1990 1995 2000 2005 2010
IMR 10 7 5 4 3 1 0

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Least squares method of finding trend is very simple and does not involve any calculations


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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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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  
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  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`


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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

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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

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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  

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Year Production Year Production
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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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Fit a trend line to the following data by the method of least square :

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

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