Chapter: Time Series· Mathematics & Statistics (Commerce) Part 2

Following table shows the all India Infant Mortality Rates (per '000) for years 1980 to 1986:Years1980198119821983198419851986IMR10653310Fit a trend line to the above data by the method of least squares.

Q.5 (B) (ii) Following table shows the all India Infant Mortality Rates (per '000) for years 1980 to 1986:

Years	1980	1981	1982	1983	1984	1985	1986
IMR	10	6	5	3	3	1	0

Fit a trend line to the above data by the method of least squares.

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Answer: Trend line \(y=4-1.5u\), where \(u=t-1983\)

Here \(n=7\) (odd), so take \(u=t-1983\) (the middle year).

Year \(t\)	IMR \(y\)	\(u=t-1983\)	\(u^2\)	\(uy\)	Trend value \(4-1.5u\)
1980	10	\(-3\)	9	\(-30\)	8.5
1981	6	\(-2\)	4	\(-12\)	7
1982	5	\(-1\)	1	\(-5\)	5.5
1983	3	0	0	0	4
1984	3	1	1	3	2.5
1985	1	2	4	2	1
1986	0	3	9	0	\(-0.5\)
Total	28	0	28	\(-42\)

Normal equations: \(\sum y=na+b\sum u\) and \(\sum uy=a\sum u+b\sum u^2\).
Since \(\sum u=0\): \(a=\frac{\sum y}{n}=\frac{28}{7}=4\), \(b=\frac{\sum uy}{\sum u^2}=\frac{-42}{28}=-1.5\).

The equation of the trend line is \(y=4-1.5u\), where \(u=t-1983\). The trend values are shown in the last column; the IMR shows a falling trend.

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Following table shows the all India Infant Mortality Rates (per '000) for years 1980 to 1986:Years1980198119821983198419851986IMR10653310Fit a trend line to the above data by the method of least squares.

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