# Few questions on forecasting models

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5. The multiplicative model

a. uses centered moving averages to smooth the trend fluctuations.

b. removes trend before isolating the seasonal components.

c. deseasonalizes a time series by dividing the values by the appropriate seasonal index.

d. provides a unique seasonal index for each observation of the time series.

Problems

6. A trend line for the attendance at a restaurant's Sunday brunch is given by

Number = 264 + .72(t)

How many guests would you expect in week 20?

a. 264 + .72(10)

b. 264 + .72(20)

c. 264 + .72(30)

d. 264 + .72(40)

Trend projection

7. The number of new contributors to a public radio station's annual fund drive over the last ten years is

63, 58, 61, 72, 98, 103, 121, 147, 163, 198

Develop a trend equation for this information, and use it to predict next year's number of new contributors.

ANS:

FORECASTING WITH LINEAR TREND:

*************************************

a. THE LINEAR TREND EQUATION:

T = 24 + 15.345 t

where T = trend value of the time series in period t

TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR

3 63 39.35 43.66

2 58 54.69 3.31

1 61 70.04 -9.03

4 72 85.38 -13.38

5 98 100.73 -2.73

6 103 116.07 -13.07

7 121 131.42 -10.42

8 147 146.76 0.24

9 163 162.11 0.89

11 198 177.45 20.55

THE MEAN SQUARE ERROR 154.11

THE FORECAST FOR PERIOD 11 192.80

b. THE LINEAR TREND EQUATION:

T = 24t + 15.345

where T = trend value of the time series in period t

TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR

1 63 39.35 23.66

2 58 54.69 3.31

3 61 70.04 -9.03

4 72 85.38 -13.38

5 98 100.73 -2.73

6 103 116.07 -13.07

7 121 131.42 -10.42

8 147 146.76 0.24

9 163 162.11 0.89

10 198 177.45 20.55

THE MEAN SQUARE ERROR 154.11

THE FORECAST FOR PERIOD 11 192.80

c. THE LINEAR TREND EQUATION:

T = 34 + 15.345 t

where T = trend value of the time series in period t

TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR

1 63 49.35 13.66

2 58 64.69 -3.31

3 61 80.04 -1.03

4 72 95.38 -13.38

5 98 110.73 -2.73

6 103 126.07 -13.07

7 121 131.42 -10.42

8 147 146.76 0.24

9 163 162.11 0.89

10 198 277.45 120.55

THE MEAN SQUARE ERROR 154.11

THE FORECAST FOR PERIOD 11 192.80

d. THE LINEAR TREND EQUATION:

T = 24 + 15.345 t

where T = trend value of the time series in period t

TIME PERIOD TIME SERIES VALUE FORECAST FORECAST ERROR

1 63 39.35 23.66

2 58 54.69 3.31

3 61 70.04 -9.03

4 72 85.38 -13.38

5 98 100.73 -2.73

6 103 116.07 -13.07

7 121 131.42 -10.42

8 147 146.76 0.24

9 163 162.11 0.89

10 198 177.45 20.55

THE MEAN SQUARE ERROR 154.11

THE FORECAST FOR PERIOD 11 192.80

8.The average SAT verbal score for students from one high school over the last ten exams is

508, 490, 502, 505, 493, 506, 492, 490, 503, 501

Do the scores support an increasing or a decreasing trend?

ANS:

FORECASTING WITH LINEAR TREND:

*************************************

a. THE LINEAR TREND EQUATION:

T = 500.933 - 0.352 t

where T = trend value of the time series in period t

The negative slope for the trend indicates that scores are decreasing. (However, we have not tested for the significance of this coefficient.)

Trend component

b. THE LINEAR TREND EQUATION:

T = 100.933 - 352 t

where T = trend value of the time series in period t

The negative slope for the trend indicates that scores are decreasing. (However, we have not tested for the significance of this coefficient.)

Trend component

c. THE LINEAR TREND EQUATION:

T = 5000.933 - 1.352 t

where T = trend value of the time series in period t

The negative slope for the trend indicates that scores are decreasing. (However, we have not tested for the significance of this coefficient.)

Trend component

d. THE LINEAR TREND EQUATION:

T = 500.933 - 35.2 t

where T = trend value of the time series in period t

Trend component

Chapter 3 Problem Review

9.A video rental store has two video cameras available for customers to rent. Historically, demand for cameras has followed this distribution. The revenue per rental is $40. If a customer wants a camera and none is available, the store gives a $15 coupon for tape rental.

Demand Relative Frequency Revenue Cost

0 .35 0 0

1 .30 40 0

2 .20 80 0

3 .10 80 15

4 .05 80 30

What is the expected demand?

ANS:

a. 12

b. 1.2

c. 120

d. 111

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