Problems/Solutions

Problems/Solutions

Problems/Solutions
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Problems

1. The data below consist of the closing price of the common stock of the American Telephone and Telegraph Corporation on 10 recent trading days.

Time(t)	Price	Time(t)	Price
1	$24.10	6	$22.73
2	23.80	7	22.60
3	23.39	8	21.76
4	22.90	9	22.14
5	22.10	10	21.69

Using a five-period moving average, forecast the price of the stock for period 10.
What is the error of the forecast in #1-a?
Using a five-period moving average, forecast the price of the stock for period 11.

2. A product is manufactured in distinct batches of various sizes. The cost accountant wished to obtain an equation to use for estimating the cost of a batch. He obtained data on a number of batches, consisting of the size of the batch, measured in number of pieces, and the total cost of the batch, consisting of the setup cost and the variable costs of labor, material, etc. The total cost is stated in thousands of dollars.

Which is the dependent variable? The independent variable?
Draw the scatterplot of this data. Does a straight line look like a reasonable fit?
Obtain the Image11 , xy, and Image12 columns.
What is the value of the slope?
What is the interpretation of the slope?

	*Size of*	*Cost of*
	*Batch*	*Batch*
	20	$1.4
	30	3.4
	40	4.1
	50	3.8
	70	6.7
	80	6.6
	100	7.8
	120	10.4
	150	11.7
	650	$55.9

What is the value of the y intercept?
What is the interpretation of the y intercept?
Estimate the cost of a batch of 125 pieces.
What is the value of the coefficient of correlation? Does it appear to indicate a high degree of association between the size of the batch and the cost?
What is the value of Image13 ?

3. The president of the Rich and Greene College of Business Administration wishes to forecast the enrollment for next fall. The enrollment is measured in Full Time Equivalents (FTE), which represent the number of full-time students, which is equivalent to the existing mixture of full-time and part-time students. Data representing the fall enrollment for the past ten years is given below:

	*time(t)*	*Enrollment*
	1	907
	2	981
	3	1014
	4	1015
	5	1050
	6	1071
	7	1123
	8	1118
	9	1175
	10	1216

Draw a scatterplot. Does the data appear to contain a linear trend?
Obtain the t, y, and ty columns.
What is the value of the slope?
What is the interpretation of the slope?
What is the value of the y intercept?
What is the interpretation of the y intercept?
Forecast the enrollment for next fall.

4. The table below contains data on the monthly amount, in millions of dollars, which was spent by "leading national advertisers" for advertising apparel and accessories in magazines, in four recent years.

		Year
	Month	1	2	3	4
	January	$6.7	$7.9	$ 8.8	$ 7.4
	February	6.2	8.4	10..3	17.4
	March	12.1	15.1	20. 4	26.1
	April	14.4	15.9	17.3	26.6
	May	11.1	11.8	15.7	17.0
	June	7.4	5.5	9.0	10.4
	July	6.4	7.6	8.9	7.9
	August	12.9	13.0	20.0	24.7
	September	21.1	23.2	32.6	35.6
	October	15.4	17.2	24.2	24.8
	November	16.5	16.7	22.0	22.2
	December	11.6	11.9	16.9	19.8

Source: Survey of Current Business, U.S. Department of Commerce, Washington, D.C., various dates.

Use this data to obtain a set of 12 monthly seasonal indexes.

Obtain the set of 37 12-month moving averages. The first value will fall between June and July of Year 1, and the last value will fall between June and July of Year 4.
Obtain the set of 36 centered 12 -month moving averages. The first value will fall in July of Year 1, and the last value in June of Year 4.
Obtain the 36 ratios of the observed values to the centered moving averages. Group them by months.
Find the mean of the ratios for each month.
Adjust the means to add to 12. These are the monthly seasonal indexes.
Describe the seasonal pattern in magazine advertising for apparel and accessories which is revealed by the monthly seasonal indexes.

5. This problem utilizes the data in Problem #4 to make forecasts using the naive methods in your textbook.

Use the actual value in the previous period to forecast the amount which will be spent on advertising apparel and accessories for year 5.
Use the actual value in the same month of the previous year to forecast the amount which will be spent on advertising apparel and accessories for January of year 5.

6. Use the data for year 1 in Problem #4 for this problem. Calculate the single-factor exponential smoothing forecasts for advertising for the last 11 months of Year 1. Use the first actual value as the starting forecast, and Image14 .

Construct the worksheet for single-factor smoothing and obtain the forecasts for the months of February through December.
Draw the graph of the actual values and the forecasted values.

7. Use your work in Problem #6 in this problem, which involves the mean squared error (MSE), the mean absolute deviation (MAD), and the tracking signal. Use the tracking signal to estimate when a change occurs in the rate of spending on advertising apparel and accessories.

Obtain the actual column, the forecasted column, and the error column for single-factor smoothing of the data for year 1 in Problem #4. (The actual column will have 12 entries, and the forecasted column will have 11 entries.)
Obtain squared error column.
Obtain the MSE for year 1.
Obtain the absolute error column.
Obtain the MAD for year 1.
Beginning with the absolute error of 0.50 for February of year 1, obtain updated estimates of MAD for the months of February through December. Use Image15 .
Obtain the tracking signal column.
Plot the tracking signals on a control chart with limits of ±4.
What conclusion can be drawn from the tracking signal?

8. Solve problem #7-a, on an EXCEL spreadsheet.

Obtain the cumulative (A - F) column on the same spreadsheet.
Obtain the |A-F| column.
Obtain the cumulative |A - F| column.
Obtain the MAD column.
Obtain the tracking signal column.
Did you get the same answers?

9. Use the data for Year I in Problem 4 for this problem. Calculate the trend-adjusted exponential smoothing forecasts for advertising for the last 8 months of Year 1. Use Image16 and use the first four actual values to obtain the starting values.

What is the starting estimate of the trend, Image17 ?
What is the initial forecast,Image18 ?
Construct the worksheet for trend-adjusted smoothing and obtain the forecasts for the months of May through December.
Draw the graph of the actual values and the forecasted values.

Solutions

1.

A forecast for t = 10 will require the previous five prices.

₅

Error = Actual - Forecast = 21.69 ? 22.27 = -$0.58
MA₅ =( 22.73 + 22.60 + 21.76 + 22.14 + 21.69)/5 = $22.18

2.

The size will be the independent variable (X) and the cost will be the dependent variable (Y).

Image33 ; Image34 ; Image35 ; Image36 ; Image37
Image38
It is estimated that each additional piece in a batch costs Image39 .
Image40
It is estimated that the cost of setting up to produce a batch is 0.5937(10³) = $593.70.
Image41
Image42Image43

This high value confirms the impression given by the scatterplot and indicates a strong linear relationship.

Image44

3.

Image45 ; Image46 ; Image47 ; Image48
Image49
It is estimated that the enrollment increases by 30.33 FTE per year, on the average.
Image50
It is estimated that the enrollment at t = 0 was 900.2 FTE.
Image51 FTE.

4.

The first 12-month total is: 6.7 + 6.2 + . . .+ 11.6 = 141.8 and the average is 11.82. The second total is 143.0 and the average is 11.92. The last 12-month total is 239.9 and the average is 19.99.
The first centered 12-month moving average is (11.82 +11.92)/2 = 11.87. Here is the complete set:

		Years
		1	2	3	4
	January		12.43	14.32	19.09
	February		12.48	14.66	19.24
	March		12.57	15.34	19.56
	April		12.74	16.03	19.71
	May		12.82	16.54	19.74
	June		12.84	16.97	19.87
	July	11.87	12.89	17.12
	August	12.00	13.01	17.36
	September	12.23	13.31	17.89
	October	12.42	13.59	18.52
	November	12.51	13.81	18.96
	December	12.46	14.12	19.07

The first ratio is 6.4/11.87 = .5392. The last ratio is 10.4/19.87 = .5234. Each month has three ratios.

Here are the totals of each month's ratios and the seasonal indexes:

		Total		Mean		Index
	January	1.6377		.5459		.5459
	February	2.2801		.7600		.7600
	March	3.8656		1.2885		1.2886
	April	3.6768		1.2256		1.2257
	May	2.7308		.9103		.9103
	June	1.4820		.4940		.4940
	July	1.6487		.5496		.5496
	August	3.2254		1.0751		1.0752
	September	5.2906		1.7635		1.7636
	October	3.8122		1.2707		1.2708
	November	3.6886		1.2295		1.2296
	December	2.6600		.8867		.8867
				11.9994		12.0000

The adjustment factor = 12.0000/11.9994 = 1.00005; multiply each mean by this factor.

Advertising for apparel and accessories is highest in the spring and fall months and lowest in the winter and summer months. It is, perhaps, surprising that advertising is low in December.

5.

Image52
Image53

6.

Image54
Image55
Image56

Here is the complete table:

Month

t

A_t

F_t

(A_t-F_t)

January

February

March

April

May

June

July

August

September

October

November

December

1

2

3

4

5

6

7

8

9

10

11

12

6.7

6.2

12.1

14.4

11.1

7.4

6.4

12.9

21.1

15.4

16.5

11.6

-

6.7

6.6

7.7

9.04

9.45

9.04

8.51

9.39

11.73

12.46

13.27

-

-5.5

5.5

6.7

2.06

-2.05

-2.64

4.39

11.71

3.67

4.04

1.67

7. a,b,d. Here is a portion of the table:

t

A

F

A - F

(A-F)²

abs(A-F)

1

2

3

.

.

12

6.7

6.2

12.1

.

.

11.6

6.70

6.60

.

.

13.27

-0.50

5.50

.

.

-1.67

0.25

30.25

.

.

2.79

279.77

0.50

5.50

.

.

1.67

44.93

c.

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d.

<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif::Image58::/sites/dl/free/0072443901/24520/Image58.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif">Image58 (1.0K)</a>Image58

f,g. Here are the calculations for the first few rows.

Image59
Image60
Image61
Image62
Image63
Image64

Here is the beginning and ending of the table.

t

A - F

Σ (A - F)

Image65

MAD

TS

2

3

4

.

.

12

-0.50

5.50

6.70

.

.

-1.67

-.50

5.00

11.70

.

.

31.21

0.50

5.50

6.70

.

.

1.67

0.50

1.50

2.54

.

.

3.81

-1.00

3.33

4.61

.

.

8.19

h.

i. Since the tracking signal is seldom within the limits, the value of a = .20 is a poor choice. In fact, the use of single-factor smoothing may be inappropriate with this time series. Trend -adjusted smoothing may be preferable, since a trend appears to be present.

8.

Image66
In cell C-5: =B4
In cell D-5:=B5-C5
In cell C-6: =B5+0.20*D5
In cell D-6: =B6-C6
In cell C-7: =C6+0.20*D6
etc.

In cell E-5: =D5
In cell E-6: = SUM(D5:D6)
In cell E-7: = SUM(D5:D7)
etc.
In cell F5: =ABS(D5)
In cell F6: =ABS(D6)
In cell F7: =ABS(D7)
etc.
In cell G5: =F5
In cell G6: =SUM(F5:F6)
In cell G7: =SUM(F5:F7)
etc.
In cell H5: =F5
In cell H6: =H5+0.20*(F6-H5)
In cell H7: = H6+0.20*(F7-H6)
etc.
In cell I-5: =E5/H5
In cell I-6: =E6/H6
In cell I-7: =E7/H7
etc.
Did you get the same answers?

9.

Image67
Image68
Here are the calculations for the first two rows of the table.
Image69
Image70
Image71
. Image72
Image73
Image74

Here is the complete table.

	t	A	TAF	(A -TAF)
	5	11.1	16.97	-5.87
	6	7.4	18.37	-10.97
	7	6.4	18.52	-12.12
	8	12.9	18.00	-5.10
	9	21.1	18.40	2.70
	10	15.4	20.16	-4.76
	11	16.5	20.97	-4.47
	12	11.6	21.65	-10.05

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