# 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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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

#### 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 TrendValue 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
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Chapter 2.4: Time Series - Q.4
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