Question

Complete the following activity to fit a trend line to the following data by the method of least squares.

Year	1975	1976	1977	1978	1979	1980	1981	1982	1983
Number of deaths	0	6	3	8	2	9	4	5	10

Solution:

Here n = 9. We transform year t to u by taking u = t - 1979. We construct the following table for calculation :

Year t	Number of deaths xt	u = t - 1979	u2	uxt
1975	0	- 4	16	0
1976	6	- 3	9	- 18
1977	3	- 2	4	- 6
1978	8	- 1	1	- 8
1979	2	0	0	0
1980	9	1	1	9
1981	4	2	4	8
1982	5	3	9	15
1983	10	4	16	40
	`sumx_t` =47	`sumu`=0	`sumu^2=60`	`square`

The equation of trend line 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 = 47, sumu= 0, sumu^2 = 60`

By putting these values in normal equations, we get

47 = 9a' + b' (0)       ...(3)

40 = a'(0) + b'(60)      ...(4)

From equation (3), we get a' = `square`

From equation (4), we get b' = `square`

∴ the equation of trend line is x_t = `square`

Answer 1

Here n = 9. We transform year t to u by taking u = t - 1979. We construct the following table for calculation :

Year t	Number of deaths xt	u = t - 1979	u2	uxt
1975	0	- 4	16	0
1976	6	- 3	9	- 18
1977	3	- 2	4	- 6
1978	8	- 1	1	- 8
1979	2	0	0	0
1980	9	1	1	9
1981	4	2	4	8
1982	5	3	9	15
1983	10	4	16	40
	`sumx_t` =47	`sumu`=0	`sumu^2=60`	`bb(sumux_t=40)`

The equation of trend line 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 = 47, sumu= 0, sumu^2 = 60`

By putting these values in normal equations, we get

47 = 9a' + b' (0)       ...(3)

40 = a'(0) + b'(60)      ...(4)

From equation (3), we get a' = 5.2222

From equation (4), we get b' = 0.6667

Putting a' = 5.2222 and b' = 0.6667 in x_t = a' + b'u, we get

∴ the equation of trend line is x_t = 5.2222 + 0.6667u