Question

The data about Cost Inflation Index (CII) declared by the central government are as follows. The year 1981 – 82 is the base for this index. Find the estimate of this index for the year 2015-16 by fitting the linear equation to these data.

Year	2007-08	2008-09	2009-10	2010-11	2011-12	2012-13	2013-14	2014-15
CII	551	582	632	711	785	852	939	1024

Answer 1

Here, n = 8. So t = 1, 2,…, 8 and y = Cost Inflation Index (CII)

We fit the equation of linear trend ŷ = a + bt

The table for calculating the values of a and b is prepared as follows:

\(t̄ = \frac{Σt}{n}\)

Putting n = 8 and Σt = 36, we get

\(t̄ = \frac{36}{8}=4.5\)

\(ȳ = \frac{Σy}{n}\)

Putting n = 8 and Σy = 6076, we get

\(ȳ = \frac{6076}{8}=759.5\)

\(Now, b = \frac{nΣty−(Σt)(Σy)}{nΣt^2−(Σt)^2}\)

Putting n = 8, Σty = 30257, Σt = 36, Σy = 6076 and Σt² = 204 in the formula,

a = ȳ – bt̄
Putting ȳ = 759.5, b = 69.4 and t̄ = 4.5, we get
a = 759.5 – 69.4 (4.5)
= 759.5 – 312.3
= 447.2

Equation of linear trend:
Putting a = 447.2 and b = 69.4 in ŷ = a + bt,
we get
ŷ = 447.2 + 69.4t

Estimate of CII for the year 2015-16:
We take t = 9, corresponding to the year 2015-16
Putting t = 9 in ŷ = 447.2 + 69.41, we get
ŷ = 447.2 + 69.4 (9)
= 447.2 + 4 624.6
= 1071.8

Hence, the estimate of CII for the year 2015-16 obtained is 1071.8.