4. In a time-series decomposition of sales (in millions of units), the following trend has been estimated:

CMAT = 4.7 * 0.37(T)

The seasonal indices have been found to be:

For the coming year the time index and cycle factors are:

a. From this information prepare a forecast for each quarter of the coming year.

b. Actual sales for the year you forecast in part (a) were 17.2, 13.2, 10.8, and 14.2 for quarters 1, 2, 3, and 4, respectively. Use these actual sales figures along with your forecasts to calculate the root-mean-squared error for the forecast period.

5. A tanning parlor located in a major shopping center near a large New England city has the following history of customers over the last four years (data are in hundreds of customers):

a. Construct a table in which you show the actual data (given in the table), the centered moving average, the centered moving-average trend, the seasonal factors, and the cycle factors for every quarter for which they can be calculated in years 1 through 4.

b. Determine the seasonal index for each quarter.

c. Do the best you can to project the cycle factor through 2008.

d. Make a forecast for each quarter of 2008.

e. The actual numbers of customers served per quarter in 2008 were 6.8, 5.1, 4.7, and 6.5 for quarters 1 through 4, respectively (numbers are in hundreds). Calculate the

RMSE for 2008.

f. Prepare a time-series plot of the actual data, the centered moving averages, the long-term trend, and the values predicted by your model for 2004 through 2008 (where data are available).

8. Kim Brite and Larry Short have developed a series of exclusive mobile-home parks in which each unit occupies a site at least 100  150 feet. Each site is well landscaped to provide privacy and a pleasant living environment. Kim and Larry are considering opening more such facilities, but to help manage their cash flow they need better forecasts of mobile-home shipments (MHS), since MHS appears to influence their vacancy rates and the rate at which they can fill newly opened parks. They have 16 years of data on mobile-home shipments, beginning with 1988Q1 and ending with 2003Q4, as shown:

Assuming that Kim Brite and Larry Short have hired you as a forecasting consultant:

a. Provide a time-series plot of the actual MHS data along with the deseasonalized data. Write a brief memo in which you report the nature and extent of the seasonality in the data. Include seasonal indices in your report.

b. Develop a long-term linear trend for the data, based on the centered moving averages. Let time equal 1 for 1988Q1 in your trend equation. On the basis of this trend, does the future look promising for Brite and Short?

c. One of the things Ms. Brite and Mr. Short are concerned about is the degree to which MHS is subject to cyclical fluctuations. Calculate cycle factors and plot them in a time-series graph, including projections of the cycle factor through 2004. In evaluating the cycle factor, see whether interest rates appear to have any effect on the cyclical pattern. The rate for 1988Q1 through 2003Q4 is provided in the following table, should you wish to use this measure of interest rates.

d. Demonstrate for Ms. Brite and Mr. Short how well your time-series decomposition model follows the historical pattern in the data by plotting the actual values of MHS and those estimated by the model in a single time-series plot.

e. Prepare a forecast for 2004 and calculate the root-mean-squared error (RMSE), given the actual values of MHS for 2004 shown:

13. a. Use the following data on millions of dollars of jewelry sales (JS) to prepare a time-series decomposition forecast of JS for the four quarters of 2005:

The actual data for 2005 are:

b. Evaluate your model in terms of fit and accuracy using RMSE.

c. Plot your forecast values of JS along with the actual values.

d. Look at the seasonal indices, and explain why you think they do or do not make sense.

e. Compare the results from your time-series decomposition model with those obtained using a Winters’ exponential smoothing model in terms of both fit and accuracy.