Question

The information about death rate of a state in different years is given in the following table. Fit a linear equation to find trend and hence estimate the death rate for the year 2017.

Year	2009	2010	2011	2012	2013	2014	2015
Death rate	7.6	6.9	7.1	7.3	7.2	6.9	6.9

Answer 1

Here, n = 7; So t = 1, 2, …………, 7 and y = Death rate
We fit a linear equation ŷ = a + bt to find trend.

The table to calculate the values of a and b is prepared as follows:

\(\bar t=\frac{\sum t}{n}\)

Putting n = 7 and Σt = 28, we get

\(t̄ =\frac{28}{7}=4\)

\(\bar y=\frac{\sum y}{n}\)

Putting n = 7 and Σy = 49.9, we get

\(ȳ = \frac{49.9}{7}=7.13\)

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

Putting n = 7, Σty = 197.6, Σt = 28, Σy = 49.9 and Σt² = 140 in the formula,

a = ȳ – bt̄
Putting ȳ = 7.13, b = – 0.07 and t̄ = 4, we get
a = 7.13 – (-0.07) (4)
= 7.13 + 0.28
= 7.41

Equation of Linear trend:
Putting a = 7.41 and b = – 0.07 in ŷ = a + bt,
we get
ŷ = 7.41 – 0.07t

Estimate of death rate for the year 2017:
For the year 2017, corresponding t = 9
Putting t = 9 in ŷ = 7.41 – 0.07, we get
ŷ = 7.41 – 0.07 (9)
= 7.41 – 0.63
= 6.78

Hence, the estimate of death rate for the year 2017 obtained is 6.78.