Question

Student enrollment at a college over the past five years is given below.

Year(t)	Enrollment(y)
1	18.9
2	23.1
3	24
4	24.6
5	26.4

Required:

a. Develop a linear trend equation for this time series.

b. Forecast enrollment for year 12.

c. Forecast enrollment for year 15.

Answer 1

Year (ti)	Enrollment (Yi)	Deviation Xi= ti - 3	Xi * Yi
1 2 3 4 5	18.9 23.1 24 24.6 26.4	-2 -1 0 1 2	-37.8 -23.1 0 24.6 52.8	4 1 0 1 4
				= 10

Mean of X =

Mean of Y =

Using the method of Least square,

Linear equation is given by -

Yi = a + b*Xi

where, a and b are unknown constants.

Normal equations are given by -

Solving these equation we get,

and,  

Putting values in this formula, we get,

a = 23.4 - 1.65 * 0 = 23.4

So, equation for the linear trend is -

Y = 23.4 + 1.65 * X

Or, Y = 23.4 + 1.65 * (t - 3)

= 23.4 + 1.65t - 4.95

= 28.35 + 1.65t

= So, the linear trend equation for the time series is given by -

Yi = 28.35 + 1.65 * ti

b) Enrollment for year 12, i.e, t = 12 is -

Y = 28.35 + 1.65 * 12 = 48.15

c) Enrollment for Year 15, i.e, t = 15 is -

Y = 28.35 + 1.65 * 15 = 53.1