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 - Mathematics and Statistics

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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

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Solution

In the given problem, n = 11 (odd), middle t- values is 1981, h = 1

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

= `("t" - 1981)/1`

= t – 1981

We obtain the following table:

Year 

t

Production

yt

u = t − 1981 u2 uyt Trend Value
1976 0 − 5 25 0 1.6819
1977 4 − 4 16 − 16 2.4728
1978 4 − 3 9 − 12 3.2637
1979 2 − 2 4 − 4 4.0546
1980 6 − 1 1 − 6 4.8455
1981 8 0 0 0 5.6364
1982 5 1 1 5 6.4273
1983 9 2 4 18 7.2182
1984 4 3 9 12 8.0091
1985 10 4 16 40 8.8
1986 10 5 25 50 9.5909
Total 62 0 110 87 87

From the table, n = 11, ∑yt = 62, ∑u = 0, ∑u2 = 110, ∑uyt = 87

The two normal equations are:

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

∴ 62 = 11a' + b'(0)   .....(i)

and

87 = a'(0) + b'(110)  .....(ii)

From (i), a′ = `62/11` = 5.6364

From (ii), b′ = `87/110` = 0.7909

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

i.e., yt = 5.6364+ 0.7909 u,

where u = t – 1981

Now, for t = 1990,

u = 1990 – 1981

= 9

∴ yt = 5.6364 + 0.7909 × 9

= 12.7545

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

RELATED QUESTIONS

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Year 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976
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Year 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982
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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
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Production 4 6 5 1 4 10

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Year 1976 1977 1978 1979 1980 1981
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Year 1962 1963 1964 1965 1966 1967 1968 1969
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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

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

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The normal equations are

Σy = na + bΣx

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Year Production (y) x x2 xy
2006 19 – 9 81 – 171
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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
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Year 2000 2001 2002 2003 2004
Production
xi
10 15 20 25 30
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xi
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