Question

Solve the following problem :

Fit a trend line to data in Problem 13 by the method of least squares.

Answer 1

In the given problem, n = 9 (odd), middle t – value is 1979, h – 1

u = `"t - middle value"/"h" = ("t" - 1979)/(1)` = t – 1979

We obtain the following table.

Year t	No. of deaths  yt	u = t –  1979	u2	uyt	Trend Value
1975	0	–4	16	0	2.5554
1976	6	–3	9	–18	3.2221
1977	3	–2	4	–6	3.8888
1978	8	–1	1	–8	4.5555
1979	2	0	0	0	5.2222
1980	9	1	1	9	5.8887
1981	4	2	4	8	6.5556
1982	5	3	9	15	7.2223
1983	10	4	16	40	7.8890
Total	47	0	60	40

From the table, n = 9, `sumy_"t" = 47, sumu = 0, sumu^2 = 60,sumuy_"t" = 40`

The two normal equations are: `sumy_"t" = "na"' + "b"' sumu  "and" sumuy_"t", = a'sumu + b'sumu^2`

∴ 47 = 9a' + b'(0)            ...(i)   and
40 = a'(0) + b'(60)           ...(ii)

From (i), a' = `(47)/(9)` = 5.2222

From (ii), b' = `(40)/(60)` = 0.6667
∴  The equation of the trend line is y_t = a' + b'u
i.e., y_t = 5.2222 + 0.6667 u, where u = t – 1979.