Question

The following table shows the number of salesmen working for a certain concern:

Year	1992	1993	1994	1995	1996
No. of salesman	46	48	42	56	52

Use the method of least squares to fit a straight line and estimate the number of salesmen in 1997

Answer 1

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

Year (x)	No. of Salesman (y)	X = (x – 1994)	x2	XY	Trends Value (Yt)
1992	46	– 2	4	– 92	44.8
1993	48	– 1	1	– 48	46.8
1994	42	0	0	0	48.8
1995	56	1	1	56	50.8
1996	52	2	4	104	52.8
N = 5	`sum"Y"` = 244	`sum"X"` = 0	`sum"X"^2` = 10	`sum"XY"` = 20	`sum"Yt"` = 244

a = `(sum"Y")/"n" = 244/5` = 48.8

b = `(sum"XY")/(sum"X"^2) = 20/10` = 2

Therefore, the required equation of the straight line trend is given by

Y = a + bX

Y = 48.8 + 2X

i.e Y = 48.8 + 2(x – 1994)

When x = 1992

Yt = 48.8 + 2(1992 – 1994)

Y = 48.8 + 2(– 2)

= 48.8 – 4

= 44.8

When x = 1993

Yt = 48.8 + 2(1993 – 1994)

Y = 48.8 + 2(– 1)

= 48.8 – 2

= 46.8

When x = 1994

Yt = 48.8 + 2(1994 – 1994)

Y = 48.8 + 2(0)

= 48.8

When x = 1995

Yt = 48.8 + 2(1995 – 1994)

Y = 48.8 + 2(1)

= 50.8

When x = 1996

Yt = 48.8 + 2(1996 – 1994)

Y = 48.8 + 2(2)

= 48.8 + 4

= 52.8

When x = 1997

Yt = 48.8 + 2(1997 – 1994)

Y = 48.8 + 2(3)

= 48.8 + 6

= 54.8