Question

Following table shows the amount of sugar production (in lac tons) for the years 1971 to 1982

Year	1971	1972	1973	1974	1975	1976
Production	1	0	1	2	3	2
Year	1977	1978	1979	1980	1981	1982
Production	4	6	5	1	4	10

Fit a trend line by the method of least squares

Answer 1

In the given problem, n = 12 (even), two middle t − values are 1976 and 1977, h = 1

u = `("t" - "mean of two middle values")/("h"/2)`

= `("t" - 1976.5)/(1/2)`

= 2(t − 1976.5)

We obtain the following table:

Year  t	Poduction yt	u = 2(t − 1976.5	u2	uyt	Trend Value
1971	1	– 11	121	– 11	0.1535
1972	0	– 9	81	– 0	0.7165
1973	1	– 7	49	– 7	1.2795
1974	2	– 5	25	– 10	1.8425
1975	3	– 3	9	– 9	2.4055
1976	2	– 1	1	– 2	2.9685
1977	4	1	1	4	3.5315
1978	6	3	9	18	4.0945
1979	5	5	25	25	4.6575
1980	1	7	49	7	5.2205
1981	4	9	81	36	5.7835
1982	10	11	121	110	6.3465
Total	39	0	572	161

From the table, n = 12, ∑y_t = 39, ∑u = 0, ∑u² = 572, ∑uy_t = 161

The two normal equations are: 

∑y_t = na' + b'∑u and ∑uy_t = a'∑u + b'∑u²

∴ 39 = 12a' + b'(0)   ......(i)

and 161 = a'(0) + b'(572)    ......(ii)

From (i), a′ = `39/12` = 3.25

From (ii), b′ = `161/572` = 0.2815

∴ The equation of the trend line is y_t = a′ + b′u

i.e., y_t = 3.25+ 0.2815 u,

where u = 2(t − 1976.5)