Question

Following table shows the amount of sugar production (in lakh tonnes) for the years 1931 to 1941:

Year	Production	Year	Production
1931	1	1937	8
1932	0	1938	6
1933	1	1939	5
1934	2	1940	1
1935	3	1941	4
1936	2

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

Answer 1

Let y_t be the trend line represented by the equation

y_t = a + bt

Let u = `(t - "Midvalue")/h`

Midvalue = 1936, h = 1

∴ u = `(t - 1936)/1` = t – 1936

Year (t)	yt	u	u2	u.yt
1931	1	– 5	25	– 5
1932	0	– 4	16	0
1933	1	– 3	09	– 3
1934	2	– 2	04	– 4
1935	3	– 1	01	– 3
1936	2	0	00	0
1937	8	1	01	8
1938	6	2	04	12
1939	5	3	09	15
1940	1	4	16	04
1941	4	5	25	20
	`sumy_t` = 33	`sumu` = 0	`sumu^2` = 110	44

The equation of trend line becomes,

y_t = a' + b'u     .......(1)

The normal equations are

`sumy_t = na^' + b^'sumu`  .......(2)

`sumu.y_t = u^'sumu + b^'sumu^2`  ......(3)

From equation (2), we get

∴ Normal equations are

33 = 11a' + 0.b'

⇒ 11a' = 33

⇒ a' = 3

From equation (3), we get

44 = a'.0 + 110.b'

⇒ b' = `44/110` = 0.4

∴ b' = 0.4

∴ The equation of the trend line is given by y_t = 3 + (0.4)u.