Question

The annual production of a commodity is given as follows:

Year	production (in tones)
1995	155
1996	162
1997	171
19988	182
1999	158
2000	880
2001	178

Fit a straight line trend by the method of least squares

Answer 1

Computation of trend values by the method of least squares. (ODD years)

Year (x)	Production (in tones) (Y)	X = x – 1998	X2	XY	Trend values (Yt)
1995	155	– 3	9	– 465	159.57
1996	162	– 2	4	– 324	162.86
1997	171	– 1	1	– 171	166.14
1998	182	0	0	0	169.43
1999	158	1	1	158	172.72
2000	180	2	4	360	176.00
2001	178	3	9	534	179.29
N = 7	`sum"Y"` = 1186	`sum"X"` = 0	`sum"X"^2` = 28	`sum"XY"` = 92	`sum"Y"_"t"` = 1186.01

a =`(sum"Y")/"N" = 1186/7` = 169.429

b = `(sum"XY")/(sum"X"^2) = 92/28` = 3.286

Therefore, the required equation of the straight-line trend is given by Y = a + bX

i. Y = 169.429 + 3.286 X or Y = 169.429 + 3.286 (x – 1998)

The trends values are obtained by

When x = 1995, Y_t = 169.429 + 3.286(1995 – 1998) = 159.57

When x = 1996, Y_t = 169.429 + 3.286(1996 – 1998) = 162.86

When x = 1997, Y_t = 169.429 + 3.286(1997 – 1998) = 166.14

When x = 1998, Y_t = -169.429 + 3.286(1998 – 1998) = 169.43

When x = 1999, Y_t = 169.429 + 3.286(1999 – 1998) = 172.72

When x = 2000, Y_t = 169.429 + 3.286(2000 – 1998) = 176.00

When x = 2001, Y_t = 169.429 + 3.286(2001 – 1998) = 179.29