# Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010 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 leas - Mathematics and Statistics

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Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010

 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

Solution: Let us fit equation of trend line for above data.

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

Here n = 7(odd), middle year is square and h = 5

 Year IMR (y) x x2 x.y 1980 10 – 3 9 – 30 1985 7 – 2 4 – 14 1990 5 – 1 1 – 5 1995 4 0 0 0 2000 3 1 1 3 2005 1 2 4 2 2010 0 3 9 0 Total 30 0 28 – 44

The normal equations are

Σy = na + bΣx

As, Σx = 0, a = square

Also, Σxy = aΣx + bΣx2

As, Σx = 0, b =square

∴ The equation of trend line is y = square

#### Solution

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

Here n = 7(odd), middle year is 1995 and h = 5

x = ("t" - "middle year")/"h"

= ("t" - 1995)/5

We obtain the following table:

 Year IMR (y) x x2 x.y 1980 10 – 3 9 – 30 1985 7 – 2 4 – 14 1990 5 – 1 1 – 5 1995 4 0 0 0 2000 3 1 1 3 2005 1 2 4 2 2010 0 3 9 0 Total 30 0 28 – 44

From the table, n = 7, Σyt = 30, Σx = 0, Σx2 = 28, Σxyt = – 44

The normal equations are

Σy = na + bΣx

∴ 30 = 7a + bΣx

As, Σx = 0, a = 30/7 = 4.2857

Also, Σxy = aΣx + bΣx

∴ – 44 = aΣx + b′(28)

As, Σx = 0, b =(-44)/28 =  – 1.5714

∴ The equation of trend line is y = a + bx

∴ The equation of trend line is y = 4.2857 – 1.5714x

Concept: Measurement of Secular Trend
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Chapter 2.4: Time Series - Q.5
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