Question

Following table gives the number of road accidents (in thousands) due to overspeeding in Maharashtra for 9 years. Complete the following activity to find the trend by the method of least squares.

Year	2008	2009	2010	2011	2012	2013	2014	2015	2016
Number of accidents	39	18	21	28	27	27	23	25	22

Solution:

We take origin to 18, we get, the number of accidents as follows:

Year	Number of accidents xt	t	u = t - 5	u2	u.xt
2008	21	1	-4	16	-84
2009	0	2	-3	9	0
2010	3	3	-2	4	-6
2011	10	4	-1	1	-10
2012	9	5	0	0	0
2013	9	6	1	1	9
2014	5	7	2	4	10
2015	7	8	3	9	21
2016	4	9	4	16	16
	`sumx_t=68`	-	`sumu=0`	`sumu^2=60`	`square`

The equation of trend is x_t =a'+ b'u.

The normal equations are,

`sumx_t=na^'+b^'sumu             ...(1)`

`sumux_t=a^'sumu+b^'sumu^2      ...(2)`

Here, n = 9, `sumx_t=68,sumu=0,sumu^2=60,sumux_t=-44`

Putting these values in normal equations, we get

68 = 9a' + b'(0)     ...(3)

∴ a' = `square`

-44 = a'(0) + b'(60)          ...(4)

∴ b' = `square`

The equation of trend line is given by

x_t = `square`

Answer 1

We take origin to 18, we get, the number of accidents as follows:

Year	Number of accidents xt	t	u = t - 5	u2	u.xt
2008	21	1	-4	16	-84
2009	0	2	-3	9	0
2010	3	3	-2	4	-6
2011	10	4	-1	1	-10
2012	9	5	0	0	0
2013	9	6	1	1	9
2014	5	7	2	4	10
2015	7	8	3	9	21
2016	4	9	4	16	16
	`sumx_t=68`	-	`sumu=0`	`sumu^2=60`	`bb(sumux_t=-44)`

The equation of trend is x_t =a'+ b'u.

The normal equations are,

`sumx_t=na^'+b^'sumu             ...(1)`

`sumux_t=a^'sumu+b^'sumu^2      ...(2)`

Here, n = 9, `sumx_t=68,sumu=0,sumu^2=60,sumux_t=-44`

Putting these values in normal equations, we get

68 = 9a' + b'(0)     ...(3)

∴ 68 = 9a'

∴ a' = 7.56

-44 = a'(0) + b'(60)          ...(4)

∴ -44 = 60b'

∴ b' = -0.73

Putting a' = 7.56 and b' = -0.73 in the equation x_t = a' + b'u, we get the equation of trend as

x_t = 7.56 - 0.73u