Question

1. The dean of a school of business is forecasting total student enrollment for this year (2019)'s summer session classes based on the following historical data:

YEAR	TOTAL ENROLLMENT
	y
2015	2,000
2016	2,200
2017	2,800
2018	3,000

 

a) What is this year's forecast using a three-year simple moving average?

b) What is this year's forecast using a three-year weighted moving average with weights of 0.5, 0.3, and 0.22

c) What is this year's forecast using exponential smoothing with alpha=0.4, if year 2017's smoothed forecast was 2,600?

d) What is the slope (b) of the least squares trend line for these data?

e) What is the Y-intercept (a) of the least squares trend line for these data?

f) What is this year's forecast using the least squares trend line for these data?

Answer 1

(a) For Period t, let the actual enrollment be A_t and Forecast be F_t

For 3 year moving average, F_t = (A_t-1 + A_t-2 + A_t-3)/3

=> F₂₀₁₉ = (A₂₀₁₈ + A₂₀₁₇ + A_t2016)/3 = (3000 + 2800 + 2200)/3 = 2666.67

(b)

For 3 year weighted moving average, F_t = 0.5A_t-1 + 0.3A_t-2 + 0.2A_t-3

=> F₂₀₁₉ = 0.5A₂₀₁₈ + 0.3A₂₀₁₇ + 0.2A_t2016 = 0.5*3000 + 0.3*2800 + 0.2*2200 = 2780

(c)

Exponential forecasting coefficient ? = 0.40,

For Period t, Actual Demand is = A_t

Forecast = F_t

Exponential Forecast F_t = F_t-1 + ? (A_t-1 - F_t-1)

given, F₂₀₁₇ = 2600

A₂₀₁₇ = 2800

=> F₂₀₁₈ = F₂₀₁₇ + ? (A₂₀₁₇ - F₂₀₁₇) = 2600 + 0.4(2800 - 2600) = 2680

F₂₀₁₉ = F₂₀₁₈ + ? (A₂₀₁₈ - F₂₀₁₈) = 2680 + 0.4(3000 - 2680) = 2808

(d)

Let the year be denoted by x and enrollment by y

Let the Regression line be y = b_o + b₁x

where,

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

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

Year	x	Enrollment (y)	x2	xy
2015	1	2000	1	2000
2016	2	2200	4	4400
2017	3	2800	9	8400
2018	4	3000	16	12000
Total	10	10000	30	26800

=> b_o = ( 10000*30 - 10*26800 ) / ( 4*30 - 10² ) = 1600

b₁ = ( 4*26800 - 10*10000 ) / ( 4*30 - 10² ) = 360

=> y = 1600 + 360x

Hence, slope = b₁ = 360

(e) Y - intercept = b_o = 1600

(f) For Year 2019, x = 5

=> y = 1600 + 360*5 = 3400