Question

1. A University has recorded the following freshmen enrollment of students in each Academic Year shown below.

Year                Enrollment

2015                     662

2016                     596

2017                     570

2018                     541

2019                     496

      a.   What is the forecast for 2020 using a three period moving average?

b. What is the forecast for 2020 using a weighted moving average, in which the weights are .6, .3, .1?

c. What is the forecast for 2020 using a linear trend?

Extra credit (10 points) – This part is not required. Use the mean absolute deviation (MAD) to determine which method is most accurate.

Answer 1

Let the endowment for year n be An and Forecast be Fn

(a) For 3 period moving average, Fn = (An-1 + An-2 + An-3)/3

Forecast for 2020 = F2020 = (A2019 + A2018 + A2017)/3 = (496 + 541 + 570)/3 = 535.67

(b)

For 3 period weighted moving average, Fn = 0.60An-1 + 0.30An-2 + 0.10An-3

Forecast for 2020 = F2020 = 0.60A2019 + 0.30A2018 + 0.10A2017 = 0.60*496 + 0.30*541 + 0.10*570 = 516.9

(c)

Let the year be denoted by x and enrollment by y

Let the Regression line be y = bo + b1x

where,

bo = ( Σy Σx2 - Σx Σxy ) / ( nΣx2 - (Σx)2 )

b1 = ( nΣxy - ΣxΣy ) / ( nΣx2 - (Σx)2 )

year	x	y	x2	xy	y2
2015	1	662	4060225	1333930	438244
2016	2	596	4064256	1201536	355216
2017	3	570	4068289	1149690	324900
2018	4	541	4072324	1091738	292681
2019	5	496	4076361	1001424	246016
Total	15	2865	20341455	5778318	1657057

bo = ( Σy Σx2 - Σx Σxy ) / ( nΣx2 - (Σx)2 ) = ( 2865*20341455 - 15*5778318 ) / ( 5*20341455 - 152 ) = 572.15

b1 = ( nΣxy - ΣxΣy ) / ( nΣx2 - (Σx)2 ) = ( 5*5778318 - 15*2865 ) / ( 5*20341455 - 152 ) = 0.28

=> y = 572.15 + 0.28x

Hence forecast for 2020 (x = 6) = 572.15 + 0.28*6 = 573.83

(d) MAD is the average of absolute error

Error = An - Fn

for 3 period moving average -

year	Enrollment	Forecast	Error	Absolute Error
2015	662
2016	596
2017	570
2018	541	609.33	-68.33	68.33
2019	496	569.00	-73.00	73.00
			MAD	70.67

For weighted moving average

year	Enrollment	Forecast	Error	Absolute Error
2015	662
2016	596
2017	570
2018	541	587.00	-46.00	46.00
2019	496	555.20	-59.20	59.20
			MAD	52.60

For Linear trend -

	year	Enrollment	Forecast	Error	Absolute Error
1	2015	662	572.43	89.57	89.57
2	2016	596	572.71	23.29	23.29
3	2017	570	572.99	-2.99	2.99
4	2018	541	573.27	-32.27	32.27
5	2019	496	573.55	-77.55	77.55
				MAD	45.13

Linear trend menthod is the most accurate since it has the lowest MAD