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Business Statistics

David F. Groebner, Patrick W. Shannon, Phillip C. Fry, Kent D. Smith

Chapter 16

Analyzing and Forecasting Time-Series Data - all with Video Answers

Educators

Chapter Questions

00:37

Problem 1

What is meant by time-series data? Give an example.

Hossam Mohamed
Hossam Mohamed
Numerade Educator
00:37

Problem 2

Explain the difference between time-series data and cross-sectional data. Are these two types of data sets mutually exclusive? What do they have in common? How do they differ?

Hossam Mohamed
Hossam Mohamed
Numerade Educator

Problem 3

What are the differences between quantitative and qualitative forecasting techniques? Under what conditions is it appropriate to use a quantitative technique?

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Problem 4

Provide an example of a business decision that requires (1) a short-term forecast, (2) a medium-term forecast, and (3) a long-term forecast.

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Problem 5

What is meant by the trend component of a time series? How is a linear trend different from a nonlinear trend?

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Problem 6

Must a seasonal component be associated with the seasons (fall, spring, summer, winter) of the year? Provide an example of a seasonal effect that is not associated with the seasons of the year.

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Problem 7

A Greek entrepreneur followed the olive harvests. He noted that olives ripen in September. Each March he would try to determine if the upcoming olive harvest would be especially bountiful. If his analysis indicated it would, he would enter into agreements with the owners of all the olive oil presses in the region. In exchange for a small deposit months ahead of the harvest, he would obtain the right to lease the presses at market prices during the harvest. If he was correct about the harvest and demand for olive oil presses boomed, he could make a great deal of money. Identify the following quantities in the context of this scenario:
a. forecasting horizon
b. category that applies to the forecasting horizon identified in part a
c. forecasting period
d. forecasting interval

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03:26

Problem 8

Consider the following median selling prices (\$thousands) for homes in a community:
$$
\begin{array}{rc}
\hline \text { Year } & \text { Price } \\
\hline 1 & 320 \\
2 & 334 \\
3 & 329 \\
4 & 344 \\
5 & 358 \\
6 & 347 \\
7 & 383 \\
8 & 404 \\
9 & 397 \\
10 & 411 \\
\hline
\end{array}
$$
a. Use year 1 as a base year and construct a simple index number to show how the median selling price has increased.
b. Determine the actual percentage growth in the median selling price between the base year and year 10 .
c. Determine the actual percentage growth in the median selling price between the base year and year 5 .
d. Determine the actual percentage growth in the median selling price between year 5 and year 10 .

Bryan Luo
Bryan Luo
Numerade Educator
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Problem 9

The following values represent advertising rates paid by a regional catalog retailer that advertises either on radio or in newspapers:
$$
\begin{array}{ccc}
\hline \text { Year } & \text { Radio Rates (\$) } & \text { Newspaper Rates (\$) } \\
\hline 1 & 300 & 400 \\
2 & 310 & 420 \\
3 & 330 & 460 \\
4 & 346 & 520 \\
5 & 362 & 580 \\
6 & 380 & 640 \\
7 & 496 & 660 \\
\hline
\end{array}
$$

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
01:21

Problem 10

Using 1999 as the base year, construct a separate index for each component in the construction of a house.

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 11

Plot both series of data and comment on the trend you see in both plots.

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01:13

Problem 12

Construct a Paasche index for 2004 using the data. Use 1999 as the base year and assume that in $200460 \%$ of the cost of a townhouse was in materials.

Carson Merrill
Carson Merrill
Numerade Educator
02:25

Problem 13

Construct a Laspeyres index using the data, assuming that in $1999,40 \%$ of the cost of a townhouse was labor.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:09

Problem 14

Retail Forward, Inc., is a global management consulting and market research firm specializing in retail intelligence and strategies. One of its press releases (June Consumer Outlook: Spending Plans Show Resilience, June 1, 2006) divulged the result of the Retail Forward ShopperScape ${ }^{\text {TM }}$ survey conducted each month from a sample of 4,000 U.S. primary household shoppers. A measure of consumer spending is represented by the figure at the top of the page:
a. Describe the type of index used by Retail Forward to explore consumer spending.
b. Determine the actual percentage change in the Future Spending Index between December 2005 and June 2006.
c. Determine the actual percentage change in the Future Spending Index between June 2005 and June 2006.

Ashley Jordon
Ashley Jordon
Numerade Educator
04:59

Problem 15

The Energy Information Administration (EIA), created by Congress in 1977, is a statistical agency of the U.S. Department of Energy. It provides data, forecasts, and analyses to promote sound policymaking and public understanding regarding energy and its interaction with the economy and the environment. The price of the sources of energy is becoming more and more important as our natural resources are consumed. The file entitled Prices contains data for the period 1993-2008 concerning the price of gasoline ( $\$ / \mathrm{gal}$.$) ,$ natural gas (\$/cu. ft.), and electricity (cents/ kilowatt hr.).
a. Using 1993 as the base, calculate an aggregate energy price index for these three energy costs.
b. Determine the actual percentage change in the aggregate energy prices between 1993 and 2008.
c. Determine the actual percentage change in the aggregate energy prices between 1998 and 2008.

Tim Schmuhl
Tim Schmuhl
Numerade Educator
02:41

Problem 16

The federal funds rate is the interest rate charged by banks when banks borrow "overnight" from each other. The funds rate fluctuates according to supply and demand and is not under the direct control of the Federal Reserve Board, but is strongly influenced by the Fed's actions. The file entitled The Fed contains the federal funds rates for the period 1955-2008.
a. Construct a time-series plot for the federal funds rate for the period 1955-2008.
b. Describe the time-series components that are present in the data set.
c. Indicate the recurrence periods for any seasonal or cyclical components.

Denae Mcneily
Denae Mcneily
Numerade Educator
02:11

Problem 17

The Census Bureau of the Department of Commerce released the U.S. retail e-commerce sales for the period Fourth Quarter 1999-Fourth Quarter 2008. The file entitled E-Commerce contains that data.
a. Using the fourth quarter of 1999 as the base, calculate a Laspeyres Index for the retail sales for the period of Fourth Quarter 1999-Fourth Quarter 2008.
b. Determine the actual percentage change in the retail sales for the period Fourth Quarter 1999-First Quarter 2004.
c. Determine the actual percentage change in the retail sales for the period First Quarter 2004-First Quarter 2006.

Harshita Goel
Harshita Goel
Numerade Educator

Problem 18

Based on the Durbin-Watson statistic, is there evidence of autocorrelation in these data? Use a linear trend model.

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Problem 19

Using the multiplicative model, estimate the $T \times C$ portion by computing a 12 -month moving average and then the centered 12 -month moving average.

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Problem 20

Estimate the $S \times I$ portion of the multiplicative model by finding the ratio-to-moving-averages for the timeseries data. Determine whether these ratio-to-movingaverages are stable from year to year.

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Problem 21

Extract the irregular component by taking the normalized average of the ratio-to-moving-averages. Make a table that shows the normalized seasonal indexes. Interpret what the index for January means relative to the index for July.

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Problem 22

Based on your work in the previous three problems,
a. Determine a seasonally adjusted linear trend forecasting model. Compare this model with an unadjusted linear trend model. Use both models to forecast Tran's sales for period 49.
b. Which of the two models developed has the lower $M A D$ and lower MSE?

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Problem 23

Consider the following set of sales data, given in millions of dollars:
$$
\begin{array}{ll}
\hline \text { 1st quarter } 152 & \text { 1st quarter 217 } \\
\text { 2nd quarter 162 } & \text { 2nd quarter 209 } \\
\text { 3rd quarter 157 } & \text { 3rd quarter 202 } \\
\text { 4th quarter 167 } & \text { 4th quarter 221 } \\
\hline
\end{array}
$$
$$
\begin{array}{ll}
\hline \text { 1st quarter 182 } & \text { 1st quarter 236 } \\
\text { 2nd quarter 192 } & \text { 2nd quarter 242 } \\
\text { 3rd quarter 191 } & \text { 3rd quarter 231 } \\
\text { 4th quarter 197 } & \text { 4th quarter 224 } \\
\hline
\end{array}
$$
a. Plot these data. Based on your visual observations, what time-series components are present in the data?
b. Determine the seasonal index for each quarter.
c. Fit a linear trend model to the data and determine the MAD and MSE values. Comment on the adequacy of the linear trend model based on these measures of forecast error.
d. Provide a seasonally unadjusted forecast using the linear trend model for each quarter of the year 2010 .
e. Use the seasonal index values computed in part $b$ to provide seasonal adjusted forecasts for each quarter of 2010 .

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Problem 24

Examine the following time series:
$$
\begin{array}{ccccccccccc}
\hline \boldsymbol{t} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\boldsymbol{y}_{\boldsymbol{t}} & 52 & 72 & 58 & 66 & 68 & 60 & 46 & 43 & 17 & 3 \\
\hline
\end{array}
$$
a. Produce a scatter plot of this time series. Indicate the appropriate forecasting model for this time series.
b. Construct the equation for the forecasting model identified in part a.
c. Produce forecasts for time periods $11,12,13$, and 14 .
d. Obtain the forecast bias for the forecasts produced in part c if the actual time series values are $-35,-41$, -79 , and -100 for periods $11-14$, respectively.

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Problem 25

Examine the following quarterly data:
$$
\begin{array}{ccccccccccccc}
\hline \boldsymbol{t} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\boldsymbol{y}_{\boldsymbol{t}} & 2 & 12 & 23 & 20 & 18 & 32 & 48 & 41 & 35 & 52 & 79 & 63 \\
\hline
\end{array}
$$
a. Compute the four-period moving averages for this set of data.
b. Compute the centered moving averages from the moving averages of part a.
c. Compute the ratio-to-moving-averages values.
d. Calculate the seasonal indexes. Normalize them if necessary.
e. Deseasonalize the time series.
f. Produce the trend line using the deseasonalized data.
g. Produce seasonally adjusted forecasts for each of the time periods $13,14,15$, and 16 .

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05:35

Problem 26

"The average college senior graduated this year with more than $$\$ 19,000$$ in debt" was the beginning sentence of a recent article in USA Today. The majority of students have loans that are not due until the student leaves school. This can result in the student ignoring the size of debt that piles up. Federal loans obtained to finance college education are steadily mounting. The data given here show the amount of loans (\$million) for the last 13 academic years, with year 20 being the most recent.
$$
\begin{array}{crcccc}
\hline \text { Year } & \text { Amount } & \text { Year } & \text { Amount } & \text { Year } & \text { Amount } \\
\hline 1 & 9,914 & 8 & 16,221 & 15 & 37,228 \\
2 & 10,182 & 9 & 22,557 & 16 & 39,101 \\
3 & 12,493 & 10 & 26,011 & 17 & 42,761 \\
4 & 13,195 & 11 & 28,737 & 18 & 49,360 \\
5 & 13,414 & 12 & 31,906 & 19 & 57,463 \\
6 & 13,890 & 13 & 33,930 & 20 & 62,614 \\
7 & 15,232 & 14 & 34,376 & & \\
\hline
\end{array}
$$
a. Produce a time-series plot of these data. Indicate the time-series components that exist in the data.
b. Conduct a test of hypothesis to determine if there exists a linear trend in these data. Use a significance level of 0.10 and the $p$-value approach.
c. Provide a $90 \%$ prediction interval for the amount of federal loans for the 26th academic year.

Georgiann Andersen
Georgiann Andersen
Numerade Educator
02:02

Problem 27

The average monthly price of regular gasoline in Southern California is monitored by the Automobile Club of Southern California's monthly Fuel Gauge Report. The prices of the time period July 2004 to June 2006 are given here.
$$
\begin{array}{ccrccc}
\hline \text { Month } & \text { Price (\$) } & \text { Month } & \text { Price (\$) } & \text { Month } & \text { Price (\$) } \\
\hline 7 / 04 & 2.247 & 5 / 05 & 2.532 & 3 / 06 & 2.626 \\
8 / 04 & 2.108 & 6 / 05 & 2.375 & 4 / 06 & 2.903 \\
9 / 04 & 2.111 & 7 / 05 & 2.592 & 5 / 06 & 3.417 \\
10 / 04 & 2.352 & 8 / 05 & 2.774 & 6 / 06 & 3.301 \\
11 / 04 & 2.374 & 9 / 05 & 3.031 & & \\
12 / 04 & 2.192 & 10 / 05 & 2.943 & & \\
1 / 05 & 1.989 & 11 / 05 & 2.637 & & \\
2 / 05 & 2.130 & 12 / 05 & 2.289 & & \\
3 / 05 & 2.344 & 1 / 06 & 2.357 & & \\
4 / 05 & 2.642 & 2 / 06 & 2.628 & & \\
\hline
\end{array}
$$
a. Produce a time-series plot of the average price of regular gas in Southern California. Identify any time-series components that exist in the data.
b. Identify the recurrence period of the time series. Determine the seasonal index for each month within the recurrence period.
c. Fit a linear trend model to the deseasonalized data.
d. Provide a seasonally adjusted forecast using the linear trend model for July 2006 and July 2010.

Erika Bustos
Erika Bustos
Numerade Educator

Problem 28

Manuel Gutierrez correctly predicted the increasing need for home health care services due to the country's aging population. Five years ago, he started a company offering meal delivery, physical therapy, and minor housekeeping services in the Galveston area. Since that time he has opened offices in seven additional Gulf State cities. Manuel is currently analyzing the revenue data from his first location for the first five years of operation.
(TABLE CAN'T COPY)
a. Plot these data. Based on your visual observations, what time-series components are present in the data?
b. Determine the seasonal index for each month.
c. (1) Fit a linear trend model to the deseasonalized data for the years 2005-2009 and determine the MAD and MSE for forecasts for each of the months in 2010. (2) Conduct a test of hypothesis to determine if the linear trend model fits the existing data. (3) Comment on the adequacy of the linear trend model based on the measures of forecast error and the hypothesis test you conducted.
d. Manuel had hoped to reach $$\$ 2,000,000$$ in revenue by the time he had been in business for 10 years. From the results in part c, is this a feasible goal based on the historical data provided? Consider and comment on the size of the standard error for this prediction. What makes this value so large? How does it affect your conclusion?
e. Use the seasonal index values computed in part b to provide seasonal adjusted forecasts for each month of the year 2010.

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Problem 29

A major brokerage company has an office in Miami, Florida. The manager of the office is evaluated based on the number of new clients generated each quarter. The following data reflect the number of new customers added during each quarter between 2006 and 2009.
(TABLE CAN'T COPY)
a. Plot the time series and discuss the components that are present in the data.
b. Referring to part a, fit the linear trend model to the data for the years 2006-2008. Then use the resulting model to forecast the number of new brokerage customers for each quarter in the year 2009. Compute the MAD and MSE for these forecasts and discuss the results.
c. Using the data for the years 2006-2008, determine the seasonal indexes for each quarter.
d. Develop a seasonally unadjusted forecast for the four quarters of year 2009.
e. Using the seasonal indexes computed in part d, determine the seasonally adjusted forecast for each quarter for the year 2009. Compute the MAD and MSE for these forecasts.
f. Examine the values for the MAD and MSE in parts $b$ and e. Which of the two forecasting techniques would you recommend the manager use to forecast the number of new clients generated each quarter? Support your choice by giving your rationale.

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Problem 30

Logan Pickens is a plan/build construction company specializing in resort area construction projects. Plan/build companies typically have a cash flow problem since they tend to be paid in lump sums when projects are completed or hit milestones. However, their expenses, such as payroll, must be paid regularly. Consequently, such companies need bank lines of credit to finance their initial costs, but in 2009 lines of credit were difficult to negotiate. The data file LoganPickens contains month-end cash balances for the past 16 months.
a. Plot the data as a time-series graph. Discuss what the graph implies concerning the relationship between cash balance and the time variable, month.
b. Fit a linear trend model to the data. Compute the coefficient of determination for this model and show the trend line on the time-series graph. Discuss the appropriateness of the linear trend model. What are the strengths and weaknesses of the model?
c. Referring to part b, compute the MAD and MSE for the 16 data points.
d. Use the $t^2$ transformation approach and recompute the linear model using the transformed time variable. Plot the new trend line against the transformed data. Discuss whether this model appears to provide a better fit than did the model without the transformation. Compare the coefficients of determination for the two models. Which model seems to be superior, using the coefficient of determination as the criterion?
e. Refer to part d. Compute the MAD and MSE for the 16 data values. Discuss how these compare to those that were computed in part c, prior to transformation. Do the measures of fit ( $R^2, M S E$, or $M A D$ ) agree on the best model to use for forecasting purposes?

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Problem 31

Refer to Problem 16-30.
a. Use the linear trend model (without transformation) for the first 15 months and provide a cash balance forecast for month 16 . Then make the $t^2$ transformation and develop a new linear trend forecasting model based on months $1-15$. Forecast the cash balance for month 16 . Now compare the accuracy of the forecasts with and without the transformation. Which of the two forecast models would you prefer? Explain your answer.
b. Provide a $95 \%$ prediction interval for the cash balance forecast for month 16 using the linear trend model both with and without the transformation. Which interval has the widest width? On this basis, which procedure would you choose?

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02:41

Problem 32

The federal funds rate is the interest rate charged by banks when banks borrow "overnight" from each other. The funds rate fluctuates according to supply and demand and is not under the direct control of the Federal Reserve Board, but is strongly influenced by the Fed's actions. The file entitled The Fed contains the federal funds rates for the period 1955-2008.
a. Produce a scatter plot of the federal funds rate for the period 1955-2008. Identify any time-series components that exist in the data.
b. Identify the recurrence period of the time series. Determine the seasonal index for each month within the recurrence period.
c. Fit a nonlinear trend model that uses coded years and coded years squared as predictors for the deseasonalized data.
d. Provide a seasonally adjusted forecast using the nonlinear trend model for 2010 and 2012.
e. Diagnose the model.

Denae Mcneily
Denae Mcneily
Numerade Educator

Problem 33

The Census Bureau of the Department of Commerce released the U.S. retail e-commerce sales ("Quarterly Retail E-Commerce Sales 1st Quarter 2006," May 18, 2006) for the period of Fourth Quarter 1999-Fourth Quarter 2008. The file entitled E-Commerce contains those data.
a. Produce a time-series plot of this data. Indicate the time-series components that exist in the data.
b. Conduct a test of hypothesis to determine if there exists a linear trend in these data. Use a significance level of 0.10 and the $p$-value approach.
c. Provide forecasts for the e-commerce retail sales for the next four quarters.
d. Presume the next four quarters exhibit e-commerce retail sales of $35,916,36,432,35,096$, and 36,807 , respectively. Produce the forecast bias. Interpret this number in the context of this exercise.

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Problem 34

The following table represents two years of data:
$$
\begin{array}{llll}
\hline \text { 1st quarter } & 242 & \text { 1st quarter } & 272 \\
\text { 2nd quarter } & 252 & \text { 2nd quarter } & 267 \\
\text { 3rd quarter } & 257 & \text { 3rd quarter } & 276 \\
\text { 4th quarter } & 267 & \text { 4th quarter } & 281 \\
\hline
\end{array}
$$
a. Prepare a single exponential smoothing forecast for the first quarter of year 3 using an alpha value of 0.10 . Let the initial forecast value for quarter 1 of year 1 be 250 .
b. Prepare a single exponential smoothing forecast for the first quarter of year 3 using an alpha value of 0.25 . Let the initial forecast value for quarter 1 of year 1 be 250 .
c. Calculate the $M A D$ value for the forecasts you generated in parts $a$ and $b$. Which alpha value provides the smaller $M A D$ value at the end of the 4 th quarter in year 2 ?

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Problem 35

The following data represent enrollment in a major at your university for the past six semesters (Note: semester 1 is the oldest data; semester 6 is the most recent data):
$$
\begin{array}{cc}
\hline \text { Semester } & \text { Enrollment } \\
\hline 1 & 87 \\
2 & 110 \\
3 & 123 \\
4 & 127 \\
5 & 145 \\
6 & 160 \\
\hline
\end{array}
$$
a. Prepare a graph of enrollment for the six semesters.
b. Based on the graph you prepared in part a, does it appear that a trend is present in the enrollment figures?
c. Prepare a single exponential smoothing forecast for semester 7 using an alpha value of 0.35 . Assume that the initial forecast for semester 1 is 90 .
d. Prepare a double exponential smoothing forecast for semester 7 using an alpha value of 0.20 and a beta value of 0.25 . Assume that the initial smoothed constant value for semester 1 is 80 and the initial smoothed trend value for semester 1 is 10 .
e. Calculate the $M A D$ values for the simple exponential smoothing model and the double exponential smoothing model at the end of semester 6 . Which model appears to be doing the better job of forecasting course enrollment? Don't include period 1 in the calculation.

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Problem 36

The following data represent the average number of employees in outlets of a large consumer electronics retailer:
$$
\begin{array}{lllllllllll}
\hline \text { Year } & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\
\text { Number } & 20.6 & 17.3 & 18.6 & 21.5 & 23.2 & 19.9 & 18.7 & 15.6 & 19.7 & 20.4 \\
\hline
\end{array}
$$
a. Construct a time-series plot of this time series. Does it appear that a linear trend exists in the time series?
b. Calculate forecasts for each of the years in the time series. Use a smoothing constant of 0.25 and single exponential smoothing.
c. Calculate the MAD value for the forecasts you generated in part $b$.
d. Construct a single exponential smoothing forecast for 2011 . Use a smoothing constant of 0.25 .

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Problem 37

A brokerage company is interested in forecasting the number of new accounts the office will obtain next month. It has collected the following data for the past 12 months:
$$
\begin{array}{cc}
\hline \text { Month } & \text { Accounts } \\
\hline 1 & 19 \\
2 & 20 \\
3 & 21 \\
4 & 25 \\
5 & 26 \\
6 & 24 \\
7 & 24 \\
8 & 21 \\
9 & 27 \\
10 & 30 \\
11 & 24 \\
12 & 30 \\
\hline
\end{array}
$$
a. Produce a time-series plot for these data. Specify the exponential forecasting model that should be used to obtain next month's forecast.
b. Assuming a double exponential smoothing model, fit the least squares trend to the historical data, to determine the smoothed constant-process value and the smoothed trend value for period 0 .
c. Produce the forecasts for periods 1 through 12 using $\alpha=0.15, \beta=0.25$. Indicate the number of new accounts the company may expect to receive next month based on the forecast model.
d. Calculate the MAD for this model.

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Problem 38

With tax revenues declining in many states, school districts have been searching for methods of cutting costs without affecting classroom academics. One district has been looking at the cost of extracurricular activities ranging from band trips to athletics. The district business manager has gathered the past six months' costs for these activities as shown here.
$$
\begin{array}{lc}
\hline \text { Month } & \text { Expenditures (\$) } \\
\hline \text { September } & 23,586.41 \\
\text { October } & 23,539.22 \\
\text { November } & 23,442.06 \\
\text { December } & 23,988.71 \\
\text { January } & 23,727.13 \\
\text { February } & 23,799.69 \\
\hline
\end{array}
$$
Using this past history, prepare a single exponential smoothing forecast for March using an $\alpha$ value of 0.25 .

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01:20

Problem 39

"The average college senior graduated this year with more than $$\$ 19,000$$ in debt" was the beginning sentence of a recent article in USA Today. The majority of students have loans that are not due until the student leaves school. This can result in the student ignoring the size of debt that piles up. Federal loans obtained to finance college education are steadily mounting. The data given here show the amount of loans (\$million) for the last 20 academic years, with year 20 being the most recent.
$$
\begin{array}{cccccc}
\hline \text { Year } & \text { Amount } & \text { Year } & \text { Amount } & \text { Year } & \text { Amount } \\
\hline 1 & 9,914 & 8 & 16,221 & 15 & 37,228 \\
2 & 10,182 & 9 & 22,557 & 16 & 39,101 \\
3 & 12,493 & 10 & 26,011 & 17 & 42,761 \\
4 & 13,195 & 11 & 28,737 & 18 & 49,360 \\
5 & 13,414 & 12 & 31,906 & 19 & 57,463 \\
6 & 13,890 & 13 & 33,930 & 20 & 62,614 \\
7 & 15,232 & 14 & 34,376 & & \\
\hline
\end{array}
$$
a. Produce a time-series plot for these data. Specify the exponential forecasting model that should be used to obtain next year's forecast.
b. Assuming a double exponential smoothing model, fit the least squares trend to the historical data to determine the smoothed constant-process value and the smoothed trend value for period 0 .
c. Using data for periods 1 through 20 and using $\alpha=0.20$ and $\beta=0.30$, forecast the total student loan volume for the year 21 .
d. Calculate the $M A D$ for this model.

Tim Schmuhl
Tim Schmuhl
Numerade Educator

Problem 40

The human resources manager for a medium-sized business is interested in predicting the dollar value of medical expenditures filed by employees of her company for the year 2011. From her company's database she has collected the following information showing the dollar value of medical expenditures made by employees for the previous seven years:
$$
\begin{array}{cc}
\hline \text { Year } & \text { Medical Claims } \\
\hline 2004 & \$ 405,642.43 \\
2005 & \$ 407,180.60 \\
2006 & \$ 408,203.30 \\
2007 & \$ 410,088.03 \\
2008 & \$ 411,085.64 \\
2009 & \$ 412,200.39 \\
2010 & \$ 414,043.90 \\
\hline
\end{array}
$$
a. Prepare a graph of medical expenditures for the years 2004-2010. Which forecasting technique do you think is most appropriate for this time series, single exponential smoothing or double exponential smoothing? Why?
b. Use an $\alpha$ value of 0.25 and a $\beta$ value of 0.15 to produce a double exponential forecast for the medical claims data. Use linear trend analysis to obtain the starting values for $C_0$ and $T_0$.
c. Compute the MAD value for your model for the years 2004 to 2010. Also produce a graph of your forecast values.

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Problem 41

. Retail Forward, Inc., is a global management consulting and market research firm specializing in retail intelligence and strategies. One of its press releases (June Consumer Outlook: Spending Plans Show Resilience, June 1, 2006) divulged the result of the Retail Forward ShopperScape ${ }^{\mathrm{TM}}$ survey conducted each month from a sample of 4,000 U.S. primary household shoppers. A measure of consumer spending is represented by the following figure:
a. Construct a time-series plot of these data. Does it appear that a linear trend exists in the time series?
b. Calculate forecasts for each of the months in the time series. Use a smoothing constant of 0.25 .
c. Calculate the $M A D$ value for the forecasts you generated in part $b$.
d. Construct a single exponential smoothing forecast for July 2006. Use a smoothing constant of 0.25 .

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Problem 42

The National Association of Theatre Owners is the largest exhibition trade organization in the world, representing more than 26,000 movie screens in all 50 states and in more than 20 countries worldwide. Its membership includes the largest cinema chains and hundreds of independent theater owners. It publishes statistics concerning the movie sector of the economy. The file entitled Flicks contains data on average U.S. ticket prices ($$\$ $$). One concern is the rapidly increasing price of tickets.
a. Produce a time-series plot for these data. Specify the exponential forecasting model that should be used to obtain next year's forecast.
b. Assuming a double exponential smoothing model, fit the least squares trend to the historical data to determine the smoothed constant-process value and the smoothed trend value for period 0 .
c. Use $\alpha=0.20$ and $\beta=0.30$ to forecast the average yearly ticket price for the year 2010 .
d. Calculate the $M A D$ for this model.

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01:42

Problem 43

Inflation is a fall in the market value or purchasing power of money. Measurements of inflation are prepared and published by the Bureau of Labor Statistics of the Department of Labor, which measures average changes in prices of goods and services. The file entitled CPI contains the monthly CPI and inflation rate for the period January 2000-December 2005.
a. Construct a plot of this time series. Does it appear that a linear trend exists in the time series? Specify the exponential forecasting model that should be used to obtain next month's forecast.
b. Assuming a single exponential smoothing model, calculate forecasts for each of the months in the time series. Use a smoothing constant of 0.15 .
c. Calculate the $M A D$ value for the forecasts you generated in part $b$.
d. Construct a single exponential smoothing forecast for January 2006. Use a smoothing constant of 0.15 .

MA
Melissa A
Numerade Educator

Problem 44

The sales manager at Grossmieller Importers in New York City needs to determine a monthly forecast for the number of men's golf sweaters that will be sold so that he can order an appropriate amount of packing boxes. Grossmieller ships sweaters to retail stores throughout the United States and Canada. Shirts are packed six to a box. Data for the past 12 months are contained in the data file called Grossmieller.
a. Plot the sales data using a time-series plot. Based on the graph, what time series components are present? Discuss.
b. (1) Use a single exponential smoothing model with $\alpha=0.30$ to forecast sales for month 17. Assume that the initial forecast for period 1 is 36,000 .
(2) Compute the $M A D$ for this model. (3) Graph the smoothing-model-fitted values on the time-series plot.
c. (1) Referring to part b, try different alpha levels to determine which smoothing constant value you would recommend. (2) Indicate why you have selected this value and then develop the forecast for month 17. (3) Compare this to the forecast you got using $\alpha=0.30$ in part b .

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Problem 45

Referring to Problem 16-44, in which the sales manager for Grossmieller Imports of New York City needs to forecast monthly sales,
a. Discuss why a double exponential smoothing model might be preferred over a single exponential smoothing model.
b. (1) Develop a double exponential smoothing model using $\alpha=0.20$ and $\beta=0.30$ as smoothing constants. To obtain the starting values, use the regression trend line approach discussed in this section. (2) Determine the forecast for month 17. (3) Also compute the MAD for this model. (4) Graph the fitted values on the time-series graph.
c. Compare the results for this double exponential smoothing model with the "best" single exponential smoothing model developed in part c of Exercise 16-44. Discuss which model is preferred.
d. Referring to part $b$, try different alpha and beta values in an attempt to determine an improved forecast model for monthly sales. For each model, show the forecast for period 17 and the MAD. Write a short report that compares the different models.
e. Referring to part d and to part c for Exercise 16-44, write a report to the Grossmieller sales manager that indicates your choice for the forecasting model, complete with your justification for the selection.

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