Chapter 14 – Time Series Analysis and Forecasting
Subjective Short Answer
60. What are the forecasts for April through July based on a three-month weighted moving average applied to the
following past demand data and using the weights: 5, 3,and 2 (largest weight is for most recent data)?
Month
Demand
Forecast
January
40
February
45
March
57
April
60
May
75
June
87
July
Demand
January
40
February
45
March
57
April
60
May
75
June
87
July
1
61. Actual sales for January through April are shown below.
Observation
Month
Actual Sales (A)
Forecast Sales (F)
1
January
18
2
February
23
3
March
20
4
April
16
5
May
Use exponential smoothing with α = 0.2 to calculate smoothed averages and forecast sales for May from the above data.
Assume the forecast for the initial period (January) is 18. Show all of your computations.
Month
Actual Sales (A)
Forecast Sales (F)
1
January
18
18.00
2
February
23
18.00
3
March
20
19.00
4
April
16
19.20
5
May
18.56
1
62. The actual demand for a product and the forecast for the product are shown below. Calculate MAD and MSE. Show
all of your computations.
Observation
Actual Demand (A)
Forecast (F)
1
35
—
2
30
35
3
26
30
4
34
26
5
28
34
6
38
28
Chapter 14 – Time Series Analysis and Forecasting
Actual Sales (A)
Forecast (F)
Error (A–F)
1
35
2
30
35
3
26
30
4
34
26
5
28
34
6
38
28
1
63. The quarterly sales (in thousands of copies) for a specific educational software over the past three years are given in
the following table. The trend for these data is T = 174 + 4t (t represents time, where t=1 for Quarter 1 of 2012 and t=12
for Quarter 4 of 2014).
2012
2013
2014
Quarter 1:
170
180
190
Quarter 2:
111
96
120
Quarter 3:
270
280
290
Quarter 4:
250
220
223
a.
Using the trend equation given above, compute the multiplicative seasonal index for Quarter 3).
b.
Using the trend equation given and your seasonal index from part a), forecast sales for the third
quarter of 2015.
2012
2013
2014
Averaged Q3 Index
1
64. The sales records of a company over a period of seven years are shown below.
Year (t)
Sales (Millions of Dollars)
1
12
2
16
3
17
4
19
5
18
6
21
7
22
a.
Develop a linear trend expression for the above time series.
b.
Forecast sales for period 10.
b.
1
65. Student enrollment at a university over the past six years is given below.
Year (t)
Enrollment (In 1,000s)
1
6.30
2
7.70
3
8.00
4
8.20
5
8.80
6
8.00
a.
Develop a linear trend expression for the above time series.
b.
Forecast enrollment for year 10.
b.
10,063
1
66. The following time series shows the sales of a clothing store over a 10–week period.
Week
Sales ($1,000s)
1
15
2
16
3
19
4
18
5
19
6
20
7
19
8
22
9
15
10
21
a.
Compute a 4-week moving average for the above time series.
b.
Compute the mean square error (MSE) for the 4–week moving average forecast.
c.
Use α = 0.3 to compute the exponential smoothing values for the time series.
d.
Forecast sales for week 11.
a.
17, 18, 19, 19, 20, 19
b.
7.67
c.
15.00, 15.00, 15.30, 16.40, 16.89, 17.52, 18.26, 19.38, 18.07, 18.95
d.
$19,560
1
67. The following time series shows the number of units of a particular product sold over the past six months.
Month
Units Sold (1000s)
1
8
2
3
3
4
4
5
5
12
6
10
a.
Compute a 3-month moving average (centered) for the above time series.
b.
Compute the mean square error (MSE) for the 3-month moving average.
Chapter 14 – Time Series Analysis and Forecasting
c.
Use α = 0.2 to compute the exponential smoothing values for the time series.
d.
Forecast the sales volume for month 7.
a.
5, 4, 7
b.
c.
8, 8, 7, 6.4, 6.12, 7.296
1
68. The sales volumes of CMM, Inc., a computer firm, for the past 8 years is given below.
Year (t)
Sales ($Millions)
1
2
2
3
3
5
4
4
5
6
6
8
7
9
8
9
a.
Develop a linear trend expression for the above time series.
b.
Forecast sales for period 9.
b.
$10,568,000
1
69. The sales records of a major auto manufacturer over the past ten years are shown below.
Year (t)
Cars Sold (1000s of Units)
1
195
2
200
3
250
4
270
5
320
6
380
7
440
8
460
9
500
10
500
Develop a linear trend expression and project the sales (the number of cars sold) for time period t = 11
1
70. The following data show the quarterly sales of Amazing Graphics, Inc. for the years 6 through 8. Clearly, there is
seasonality in the sales figures. The approximated trend equation is T = 2.00 +0t, meaning is no trend effect present.
Year
Quarter
Sales
6
1
2.38
2
1.47
3
2.41
4
1.65
7
1
2.38
2
1.39
3
2.34
4
1.68
8
1
2.41
2
1.55
3
2.54
4
1.72
a.
Compute the seasonal indexes for the four quarters.
b.
Using the seasonal indexes developed in Part a), compute the sales forecasts for the four quarters of year 9.
a.
Seasonal Factors: 1.20; 0.74; 1.22; 0.84
b.
Sales Forecasts (Year 9): 2.40; 1.48; 2.44; 1.68
1
71. John has collected the following information on the amount of tips he has collected from parking cars the last seven
nights.
Day
Tips
1
18
2
22
3
17
4
18
5
28
6
20
7
12
a.
Compute the 3-day moving averages for the time series.
b.
Compute the mean square error for the forecasts.
c.
Compute the mean absolute deviation for the forecasts.
d.
Forecast John’s tips for day 7.
a.
19, 19, 21, 22, 20
b.
45.75
c.
5.25
d.
22
1
72. The following information has been collected on the sales of greeting cards for the past 6 weeks.
Week
Sales
1
105
2
90
3
95
4
110
5
105
6
100
a.
Produce exponential smoothing forecasts for the series using a smoothing constant of .2.
b.
Compute the mean square error for the forecasts produced with a smoothing constant of .2.
c.
What is the forecast of sales for week 7?
d.
Is a smoothing constant of .2 or .3 better for the sales data? Explain.
a.
105, 105, 102, 100.6, 102.48, 102.984
73. Consider the following annual series on the number of people assisted by a county human resources department.
Year
People (in 100s)
1
22
2
24
3
28
4
24
5
22
6
24
7
20
8
26
9
24
10
28
11
26
a.
Prepare 3-year moving average values to be used as forecasts for periods 4 through 11.
Calculate the mean squared error (MSE) measure of forecast accuracy for periods 4 through 11.
b.
Use a smoothing constant of .4 to compute exponential smoothing values to be used as
forecasts for periods 2 through 11. Calculate the MSE.
c.
Compare the results in Parts a and b.
a.
24.667, 25.333, 24.667, 23.333, 22, 23.333, 23.333, 26, MSE = 7.667
b.
22, 22.8, 24.88, 24.528, 23.5168, 23.71, 22.226, 23.7356, 23.8414, 25.505, MSE = 8.405
c.
The forecasts produced in Part a are better than those produced in Part b.
1
74. The temperature in Chicago has been recorded for the past seven days. You are given the information below.
Day
Temperature
1
82
2
80
3
84
4
83
5
80
6
79
7
82
a.
Produce exponential smoothing forecasts for the series using a smoothing constant of .2.
b.
Compute the mean square error for the forecasts produced with a smoothing constant of .2.
c.
What is the forecasted temperature for day 8?
d.
Is a smoothing constant of .2 or .3 better for the temperature data? Explain.
a.
82, 81.6, 82.08, 82.264, 81.8112, 81.249
b.
4.033
c.
81.399
d.
A smoothing constant of 0.2 is better because the MSE is lower when 0.2 is used.
1
75. The yearly series below exhibits a long-term trend. Use regression analysis to produce forecasts for years 11 and 12.
b.
75.523
c.
102.39
d.
0.2 is better since the MSE is smaller
1
Chapter 14 – Time Series Analysis and Forecasting
Year
Time Series Value
1
120
2
132
3
148
4
152
5
160
6
175
7
182
8
190
9
195
10
205
T = 115.2 + 9.218182t
1
76. The following time series gives the number of units sold during 5 years at a boat dealership.
Year
Quarter
Number of Units
1
1
300
2
240
3
240
4
290
2
1
350
2
300
3
280
4
320
3
1
410
2
400
3
390
4
410
4
1
490
2
450
3
440
4
510
5
1
540
2
530
3
520
4
540
a.
Find the four-quarter centered moving averages.
b.
Plot the series and the moving averages on a graph.
c.
Compute the seasonal-irregular component.
d.
Compute the seasonal factors for all four quarters.
e.
Compute the deseasonalized time series for sales.
f.
Calculate the linear trend from the deseasonalized sales.
g.
Forecast the number of units sold in each quarter of year 6.
515, 528.75
Chapter 14 – Time Series Analysis and Forecasting
77. Below you are given information on John’s income for the past 7 years.
Year
Income (In 1000s)
1
15.0
2
16.2
3
17.1
4
18.1
5
18.8
6
19.2
7
20.5
a.
Use regression analysis to obtain an expression for the linear trend component.
b.
Forecast John’s income for the next 5 years.
a.
T = 14.3857 + 0.86429t
b.
21.3, 22.2, 23.0, 23.9, 24.8
1
78. You are given the following information on the quarterly profits for Ajax Corporation.
Year
Quarter
Quarterly Profits
1
1
150
2
120
3
160
4
150
2
1
150
2
130
3
180
4
160
3
1
170
2
140
d.
1.1132, 0.9954, 0.9056, 0.9858
T = 216.2993 + 17.35763t
g.
646.56, 595.42, 557.42, 623.90
1
Chapter 14 – Time Series Analysis and Forecasting
3
200
4
180
4
1
200
2
150
3
230
4
200
a.
Find the four-quarter centered moving averages.
b.
Compute the seasonal-irregular component.
c.
Compute the seasonal factors for all four quarters.
d.
Represent the deseasonalized series.
a.
145, 146.25, 150, 153.75, 157.5, 161.25, 165, 170, 176.25, 181.25, 186.25, 192.5
b.
1.103, 1.026, 1, 0.846, 1.143, 0.992, 1.03, 0.824, 1.135, 0.993, 1.074, 0.779
c.
1.04, 0.82, 1.132, 1.008
1
79. Below you are given information on crime statistics for Middletown.
Year
Quarter
Number of Crimes Committed
1
1
10
2
20
3
25
4
5
2
1
10
2
30
3
35
4
25
3
1
20
2
40
3
35
4
15
4
1
20
2
50
3
45
4
35
The seasonal factors for these data are
Quarter
Seasonal Factor St
1
.589
2
1.351
3
1.335
4
.726
a.
Deseasonalize the series.
b.
Obtain an estimation of the linear trend for this series.
c.
Use the seasonal and trend components to forecast the number of crimes for each quarter of
Year 5.
33.71, 48.21
b.
T = 13.5155 + 1.603765t
c.
24.02, 57.26, 58.72, 33.1
POINTS:
1
80. Below you are given the seasonal factors and the estimated trend equation for a time series. These values were
computed on the basis of 5 years of quarterly data.
Quarter
Seasonal Factor St
1
1.2
2
.9
3
.8
4
1.1
T = 126.23 – 1.6t
Produce forecasts for all four quarters of year 6 by using the seasonal and trend components.
ANSWER:
111.156, 81.927, 71.544, 96.613
POINTS:
1
81. The following data show the quarterly sales of a major auto manufacturer (introduced in exercise 4) for the years 8
through 10.
Year
Quarter
Sales
8
1
160
2
180
3
190
4
170
9
1
200
2
210
3
260
4
230
10
1
210
2
240
3
290
4
260
a.
Compute the four-quarter moving average values for the above time series.
b.
Compute the seasonal factors for the four quarters.
c.
Use the seasonal factors developed in Part b to adjust the forecast for the effect of season for
year 9.
a.
180.00, 188.75, 201.25, 217.50, 226.25, 231.25, 238.75, 245.25
b.
0.935, 0.975, 1.1, 0.945
c.
213.90, 215.38, 236.36, 243.39
POINTS:
1
82.
Connie Harris, in charge of office supplies at First Capital Mortgage Corp., would like to predict the quantity of paper
used in the office photocopying machines per month. She believes that the number of loans originated in a month
influence the volume of photocopying performed. She has compiled the following recent monthly data:
Number of Loans
Sheets of Photocopy
Originated in Month
Paper Used (000’s)
25
16
25
13
35
18
40
25
Chapter 14 – Time Series Analysis and Forecasting
40
21
45
22
50
24
60
25
a. Develop the least-squares estimated regression equation that relates sheets of photocopy paper used to loans originated.
b). Use the regression equation developed in part (a) to forecast the amount of paper used in a month when 65 loan
originations are expected.
b. The forecast is y = 7.5 + .325x = 7.5 + .325(65) = 28,625 sheets.
1
83.
The number of haircuts performed each day at KwikKuts in the last four weeks are listed below.
Week
Monday
Tuesday
Wednesday
Thursday
Friday
1
122
122
103
133
98
2
127
130
106
137
97
3
126
131
111
151
104
4
135
135
110
146
107
a. Plot the sales data. Do you see both trend and seasonality components in the data?
b. Forecast the number of haircuts to be performed in each workday of week 6.
1
84.
Four months ago, the Bank Drug Company introduced Jeffrey William brand designer bandages. Advertised using the
slogan, “What the best dressed cuts are wearing”, weekly sales for this period (in 1000’s) have been as follows:
Week
Sales
Week
Sales
Week
Sales
1
12.8
7
20.6
12
23.8
2
14.6
8
18.5
13
25.1
3
15.2
9
19.9
14
24.7
4
16.1
10
23.6
15
26.5
5
15.8
11
24.2
16
28.9
6
17.2
a) Plot a graph of sales vs. weeks. Does linear trend appear reasonable?
b) Assuming linear trend, forecast sales for weeks 17, 18, 19, and 20.
b. Week 17 — 29.0; Week 18 — 30.0; Week 19 — 31.0; Week 20 — 32.0
1
85.
Weekly sales of the Weber Dicamatic food processor for the past ten weeks have been:
Week
Sales
Week
Sales
1
980
6
990
2
1040
7
1030
3
1120
8
1260
Chapter 14 – Time Series Analysis and Forecasting
4
1050
9
1240
5
960
10
1100
a. Determine, on the basis of minimizing the mean square error, whether a three period or four period simple moving
average model gives a better forecast for this problem.
b. For each model, forecast sales for week 11.
1
86. Delta Corp’s plant in Austin has been experiencing imbalances in its inventory of components used in the production
of a line of computer printers. Both stock shortages and overstock conditions are occurring.
The production analysis group is studying the pattern of demand for component PS2400, a power supply used in many of
Delta’s products. The group believes that the most recent 12 weeks of demand for the PS2400 is representative of the
future weekly demand:
Week
Demand
(Units)
Week
Demand
(Units)
Week
Demand
(Units)
Week
Demand
(Units)
1
159
4
161
7
203
10
168
2
217
5
173
8
195
11
198
3
186
6
157
9
188
12
159
a. Use a four-week moving average to develop a forecast of the demand for the PS2400 component in week 13.
b. Use a four-week weighted moving average with weights of .4 (for the most recent datum), .3, .2, and .1 to forecast the
demand for the PS2400 component in week 13.
1