Business Forecasting

1. The Carol music company has been in business for 5 years. During that time, sales of electric organs have increased from 12 units in the first year to 76 units in the most recent year. Carol, Pinto the firm’s owner, wants to develop a forecast of organ sales for the coming year. The historical data follow.

Year 1 2 3 4 5Sales

12 28 34 50 76

a. Show a graph of this time series. Does a linear trend appear to be present?b. Develop the equation for a linear trend component for the time series. What is the average increase in

sales that the firm has been realizing per year?The quarterly sales data follow:

YearQuarter

1Quarter2 Quarter3

Quarter4

Total yearly sales

1 4 2 1 5 122 6 4 4 14 283 10 3 5 16 344 12 9 7 22 505 18 10 13 35 76

c. Compute the SI for the four quarters.d. When does Carol Music experience the largest seasonal effect? Does this appear reasonable? Explain.e. Deseasonalize the data and use the deseasonalized time series to identify the trend.f. Use the results of part e. to develop a quarterly forecast for next year based on trend.g. Use the seasonal indexes developed in part c. to adjust the forecasts developed in part f. to account for

the effect of season.

2. Hind Marine has been an authorized dealer for C&D marine radios for the past 7 years. The following table reports the number of radios sold each year.

Year 1 2 3 4 5 6 7Number Sold 3

550 7

590 105 11

0130

a. Show a graph of this time series. Does a linear trend appear to be present?b. Develop the equation for the linear trend component of the time series.c. Use the linear trend developed in part b. to develop a forecast for annual sales in year 8.Suppose the quarterly sales values for the 7 years of historical data are as follow.

Year

Quarter1 Quarter2 Quarter3Quarter

4Total Yearly Sales

1 6 15 10 4 352 10 18 15 7 503 14 26 23 12 754 19 28 25 18 905 22 34 28 21 1056 24 36 30 20 1107 28 40 35 27 130

d. Show the 4-qtr MA values for this time series. Plot both the original time series and the MA series on the same graph.

e. Compute the seasonal indexes for the 4-qtrs.f. When does Hind marine experience the largest seasonal effect? Does this seem reasonable? Explain.g. Deseasonalize the data and use the deseasonalized time series to identify the trend.h. Use the results of part g. to develop a quarterly forecast for next year based on trend.

i. Use the seasonal indexes developed in part e. to adjust the forecasts developed in part h. to account for the effect of season.

3. Use exploratory data analysis to determine whether there is a trend / or seasonality in mobile home shipments (MHS). The data by quarter are shown in the table (c2p9). On the basis of your analysis, do you think there is a significant trend in MHS? Is there seasonality? What forecasting methods might be appropriate for MHS? According to the guidelines in table 2.1.

4. Home sales are often considered an important determinant of the future health of the economy. Thus, there is widespread interest in being able to forecast total houses sold (THS). The quarterly data for THS are shown in table (c2p10) in thousands of units.a. Prepare a time series plot of THS. Describe what you see in this plot in terms of trend and seasonality?b. Calculate and plot the first eight autocorrelation coefficients for THS. What does this autocorrelation

structure suggest abut the trend?c. De-trend the data by calculating the first differences.

DTHSt = THSt – THSt-1

Calculate and plot the first eight autocorrelation coefficients for DTHS. Is there a trend in DTHS?

11. As the world’s economy becomes increasingly interdependent, various exchange rates between currencies have become important in making business decisions. For many US businesses, in Japanese exchange rates (In Yen/ US dollar) is an important decision variable. This exchange rate (EXRJ) is shown in the c1p9 table by month for a two year period.a. prepare a time series plot of this series, use the Naïve forecasting model to forecast EXRJ for each

month from year1 M2 (February) through year3 M1(January). Calculate the RMSE for the period from year 1 M2 through year2 M12.

b. On the basis of a time series plot of these data and the autocorrelation structure of EXRJ, would you say the data are stationary? Explain your answer. (c2p11)

12. The data in the table c3p12 represent warehouse club and the superstore sales on a monthly basis.a. Prepare a time series plot of the data, and visually inspect that plot to determine the characteristics you

see in this series.b. Use a smoothing model to develop a forecast of sales for the next 12 months, and explain why you

selected that model. Plot the actual and forecast values. Determine the RMSE for your model during the historical period.

13. The data in the table c3p13 are for retail sales in the book stores by quarter.a. Plot these data and examine the plot. Does this view of the data suggest a particular smoothing model?

Do the data appear to be seasonal? Explain.b. use a smoothing method to forecast the four quarters of 2005. Plot the actual and forecast values.

14. Forecasting Food and Beverage Sales.The vintage restaurant in Khandala, a resort community near Lonavala, Maval. The restaurant, which is owned and operated by Rocky Johnson, has just completed its third year of operation. During the time Rockey has sought to establish the reputation for the restaurant as a high-quality dining establishment that specializes in fresh sea food. Efforts by Rockey and his staff have proven successful, and his restaurant has become one of the best and the fastest growing restaurant in Khandala.Rockry has concluded that to plan for the growth of the restaurant in the future, he needs to develop a system that will enable him to forecast food and beverage sales by month for up to one year in advance. Rockey has the following data ($ 1000) on total food and beverage dales for the three years of operation.

Month Year1 Year2

Year3

Jan 242 263 282

Feb 235 238 255Mar 232 247 265Apr 178 193 205May 184 193 210Jun 140 149 160Jul 145 157 166Aug 152 161 174Sept 110 122 126Oct 130 130 148Nov 152 167 173Dec 206 230 235

Perform an analysis of the ales data for the vintage restaurant. Prepare a report for Rockey that summarize your findings, forecasts, and recommendations. Include: a. A graph of time series.b. An analysis of the seasonality of the data. Indicate the seasonal indexes for each month, and comment

on the high and low seasonal sales months. Do the seasonal indexes make intuitive sense? Discuss.c. A forecast of sales for January to December of the fourth yeard. Recommendations as to when the system that you have developed should be updated to account for the

new sales data.e. Any detailed calculations of your analysis in the appendix of your report.

Assume that January sales for the fourth year turn out to be $295,000. What was your forecast error? If this is a large error, Rockey may be puzzled about the difference between your forecast and the actual sales value. What can you do to resolve his uncertainty in the forecasting procedure?

1. The time series component which reflects a regular, multi-year pattern of being above and below the trend line isa. a trend b. seasonal c. cyclical d. irregularAnswer: c

2. The time series component that reflects variability during a single year is calleda. a trend b. seasonal c. cyclical d. irregularAnswer: b

3. The time series component that reflects variability due to natural disasters is calleda. a trend b. seasonal c. cyclical d. irregular Answer: d

4. The time series component that reflects gradual variability over a long time period is calleda. a trend b. seasonal c. cyclical d. irregularAnswer: a

5. The trend component is easy to identify by usinga. MA b. exponential smoothingc. regression analysis d. the Delphi approachAnswer: c

6. The forecasting method that is appropriate when the time series has no significant trend, cyclical, or seasonal effect isa. MA b. MSEc. mean average deviation d. qualitative forecasting methodsAnswer: a

7. If data for a time series analysis is collected on an annual basis only, which component may be ignored?a. trend b. seasonal c. cyclical d. irregularAnswer: b

8. For the following time series, you are given the moving average forecast.Time Period Time Series Value Moving Average Forecast

1 23 ----2 17 ----3 17 ----4 26 195 11 206 23 187 17 20

The MSE equals: a. 0 b. 6 c. 41 d. 164Answer: c

9. If the estimate of the trend component is 158.2, the estimate of the seasonal component is 94%, the estimate of the cyclical component is 105%, and the estimate of the irregular component is 98%, then the multiplicative model will produce a forecast ofa. 1.53 b. 1.53% c. 153.02 d. 153,020,532Answer: c

10. Below you are given the first four values of a time series.Time Period Time Series Value

1 182 203 254 17

Using a 4-period moving average, the forecasted value for period 5 isa. 2.5 b. 17 c. 20 d. 10Answer: c

11. Below you are given the first two values of a time series. You are also given the first two values of the exponential smoothing forecast.

Exponential SmoothingTime Period (t) Time Series Value ( Yt ) Forecast ( Ft)

1 18 182 22 18

If the smoothing constant equals 0.3, then the exponential smoothing forecast for time period three isa. 18 b. 19.2 c. 20 d. 40Answer: b

12. The following linear trend expression was estimated using a time series with 17 time periods.Tt = 129.2 + 3.8t The trend projection for time period 18 is: a. 68.4 b. 193.8 c. 197.6 d. 6.84Answer: c

Exhibit 1: Below you are given the first five values of a quarterly time series. The multiplicative model is appropriate and a 4-quarter MA will be used.

Year Quarter Time Series Value Yt

1 1 36 2 24 3 16 4 202 1 4413. Refer to Exhibit 1. An estimate of the trend component times the cyclical component (T2Ct) for Quarter

3 of Year 1, when a four-quarter moving average is used, isa. 24 b. 25 c. 26 d. 28Answer: b

14. Refer to Exhibit 1. An estimate of the seasonal-irregular component for Quarter 3 of Year 1 is

a. 0.64 b. 1.5625 c. 5.333 d. 30Answer: a

15. You are given the following information on the seasonal-irregular component values for a quarterly time series:

Seasonal-IrregularQuarter Component Values (StIt)

1 1.23, 1.15, 1.162 0.86,0.89, 0.83 3 0.77, 0.72, 0.794 1.20, 1.13, 1.17

The SI for Quarter-1 is: a. 0.997 b. 1.18 c. 4 d. 3Answer: b

16. Below you are given some values of a time series consisting of 26 time periods.Time Period Time Series Value

1 372 483 504 63 . . .

23 10524 10725 11226 114

The estimated regression equation for these data is: Yt = 16.23 + 0.52Yt-1 + 0.37Yt-2

The forecasted value for time period 27 is: a. 53.23 b. 109.5 c. 116.65 d. 116.95Answer: d

17. A group of observations measured at successive time intervals is known asa. a trend component b. a time seriesc. a forecast d. an additive time series modelAnswer: b

18. A component of the time series model that results in the multi-period above-trend and below-trend behavior of a time series isa. a trend component b. a cyclical componentc. a seasonal component d. an irregular componentAnswer: b

19. The model that assumes that the actual time series value is the product of its components is thea. forecast time series model b. multiplicative time series modelc. additive time series model d. None of these alternatives is correct.Answer: b

20. A method of smoothing a time series that can be used to identify the combined trend/cyclical component isa. MA b. the percent of trendc. exponential smoothing d. the trend/cyclical indexAnswer: a

21. A method that uses a weighted average of past values for arriving at smoothed time series values is known asa. a smoothing average b. MAc. an exponential average d. an exponential smoothingAnswer: d

22. In the linear trend equation, T = b0 + b1t, b1 represents the

a. trend value in period t b. intercept of the trend linec. slope of the trend line d. point in timeAnswer: c

23. In the linear trend equation, T = b0 + b1t, b0 represents thea. time b. slope of the trend linec. trend value in period-1 d. the Y-interceptAnswer: d

24. A parameter of the exponential smoothing model which provides the weight given to the most recent time series value in the calculation of the forecast value is known as thea. MSE b. MADc. smoothing constant d. None of these alternatives is correct.Answer: c

25. One measure of the accuracy of a forecasting model isa. the smoothing constant b. a deseasonalized time seriesc. the MSE d. None of these alternatives is correct.Answer: c

26. A qualitative forecasting method that obtains forecasts through "group consensus" is known as thea. Autoregressive model b. Delphi approachc. MAD d. None of these alternatives is correct.Answer: b

Exhibit 2: Consider the following time series.t 1 2 3 4Yi 4 7 9 10

27. Refer to Exhibit 2. The slope of linear trend equation, b1, isa. 2.5 b. 2.0 c. 1.0 d. 1.25Answer: b

28. Refer to Exhibit 2. The intercept, b0, isa. 2.5 b. 2.0 c. 1.0 d. 1.25Answer: a

29. Refer to Exhibit 2. The forecast for period 5 isa. 10 b. 2.5 c. 12.5 d. 4.5Answer: c

30. Refer to Exhibit 2. The forecast for period 10 isa. 10 b. 25 c. 30 d. 22.5Answer: d

Exhibit 3: Consider the following time series.Year (t) Yi

1 72 53 44 25 1

31. Refer to Exhibit 3. The slope of linear trend equation, b1, isa. -1.5b. +1.5 c. 8.3 d. -8.3Answer: a

32. Refer to Exhibit 3. The intercept, b0, isa. -1.5b. +1.5 c. 8.3 d. -8.3Answer: c

33. Refer to Exhibit 3. In which time period does the value of Yi reach zero?

a. 0.0 b. 0.181 c. 5.53 d. 4.21Answer: c

34. Refer to Exhibit 3. The forecast for period 10 isa. 6.7 b. -6.7 c. 23.3 d. 15Answer: b

PROBLEMS1. The sales records of a company over a period of seven years are shown below.

Year (t) Sales ($ 106)1 122 163 174 195 186 217 22

a. Develop a linear trend expression for the above time series.b. Forecast sales for period 10.Answers1: a. Tt = 12 + 1.464t b. $26,640,000

2. Student enrollment at a university over the past six years is given below.Year (t) Enrollment (In 1,000s)

1 6.32 7.73 8.04 8.25 8.8

6 8.0a. Develop a linear trend expression for the above time series.b. Forecast enrollment for year 10.Answers2: a. Tt = 6.633 + 0.343t b. 10,063

3. The following time series shows the sales of a clothing store over a 10-week period.Week Sales ($1,000s)

1 152 163 194 185 196 207 198 229 15

10 21a. 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 = 0.3 to compute the exponential smoothing values for the time series.d. Forecast sales for week 11.Answers3: a. 17, 18, 19, 19, 20, 19 b. 7.67

c. 15, 15, 15.3, 16.4, 16.89, 17.52, 18.26, 19.38, 18.07, 18.95 d. $19,560

4. The following time series shows the number of units of a particular product sold over the past six months.

Month Units Sold (1000s)1 82 33 44 55 126 10

a. Compute a 3-month MA (centered) for the above time series.b. Compute the MSE for the 3-month MA.c. Use = 0.2 to compute the exponential smoothing values for the time series.d. Forecast the sales volume for month 7.Answers4: a. 5, 4, 7 b. MSE = 73/3 = 24.33 c. 8, 8, 7, 6.4, 6.12, 7.296d. F7 = 7.836

5. The sales volumes of CMM, Inc., a computer firm, for the past 8 years is given below.Year(t) Sales($ 106)1 22 33 54 45 66 87 98 9

a. Develop a linear trend expression for the above time series.b. Forecast sales for period 9.Answers5: a. Tt = 0.929 + 1.071t b. $10,568,000

6. The sales records of a major auto manufacturer over the past ten years are shown below.Year (t) Number of Cars Sold (1000s)

1 1952 2003 2504 2705 3206 3807 4408 4609 500

10 500Develop a linear trend expression and project the sales (the number of cars sold) for time period t = 11.Answer6: Tt = 136 + 39.182t T11 = 567

7. The following data show the quarterly sales of Amazing Graphics, Inc. for the years 6 through 8.Year Quarter Sales

6 1 2.5

2 1.53 2.44 1.6

7 1 2.02 1.43 1.74 1.9

8 1 2.52 2.03 2.44 2.1

a. Compute the 4-quarter MA values for the above time series.b. Compute the seasonal factors for the four quarters.c. Use the seasonal factors developed to adjust the forecast for the effect of season for year 6.Answers7: a. Centered MAs: 1.94; 1.87; 1.77; 1.72; 1.82; 1.96; 2.12; 2.26b. Seasonal Factors: 1.16; 0.85; 1.09; 0.92c. Deseasonalized Sales (Year 6): 2.16; 1.76; 2.20; 1.74

8. John has collected the following information on the amount of tips he has collected from parking cars the last seven nights.

Day Tips1 182 223 174 185 286 207 12

a. Compute the 3-day MA for the time series. b. Compute the MSE for the forecasts.c. Compute the MAD for the forecasts. d. Forecast John's tips for day 7.Answers8: a. 19, 19, 21, 22, 20 b. 45.75 c. 5.25 d. 22

9. The following information has been collected on the sales of greeting cards for the past 6 weeks.Week Sales

1 1052 903 954 1105 1056 100

a. Produce exponential smoothing forecasts for the series using a smoothing constant of 0.2.b. Compute the MSE for the forecasts produced with a smoothing constant of 0.2.c. What is the forecast of sales for week 7?d. Is a smoothing constant of 0.2 or 0.3 better for the sales data? Explain.Answers9: a. 105, 105, 102, 100.6, 102.48, 102.984 b. 75.523c. 102.39 d. 0.2 is better since the MSE is smaller

10. Consider the following annual series on the number of people assisted by a county human resources department.

Year People (in 100s)1 222 243 284 245 226 247 208 269 24

10 2811 26

a. Prepare 3-year MA values to be used as forecasts for periods 4 through 11. Calculate the MSE measure of forecast accuracy for periods 4 through 11.

b. Use a smoothing constant of 0.4 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.Answers10: a. 24.667, 25.333, 24.667, 23.333, 22, 23.333, 23.333, 26, MSE = 7.667b. 22, 22.8, 24.88, 24.528, 23.5168, 23.71, 22.226, 23.7356, 23.8414, 25.505, MSE = 8.405c. The forecasts produced in Part a are better than those produced in Part b.

11. The temperature in Pune has been recorded for the past 7-days. You are given the information below.Day Temperature

1 822 803 844 835 806 797 82

a. Produce exponential smoothing forecasts for the series using a smoothing constant of 0.2.b. Compute the MSE for the forecasts produced with a smoothing constant of 0.2.c. What is the forecasted temperature for day 8?d. Is a smoothing constant of 0.2 or 0.3 better for the temperature data? Explain.Answers11: a. 82, 81.6, 82.08, 82.264, 81.8112, 81.249b. 4.033 c. 81.399d. A smoothing constant of 0.2 is better because the MSE is lower when 0.2 is used.

12. The yearly series below exhibits a long-term trend. Use the appropriate forecasting technique to produce forecasts for years 11 and 12.

Year Time Series Value1 1202 1323 1484 152

5 1606 1757 1828 1909 195

10 205Answer12: T = 115.2 + 9.218182t T11 = 216.6 T12 = 225.82

13. 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 MA. b. Plot the series and the MAs on a graph.c. Compute the seasonal-irregular component. d. Compute the seasonal factors for all 4-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.Answers13:a. 278.75, 287.5, 300, 308.75, 320, 340, 366.25, 391.25, 412.5, 428.75, 441.25, 460, 478.75, 495, 515, 528.75 c. 0.8767, 1.0087, 1.1667, 0.9717, 0.875, 0.9412, 1.1195, 1.0224, 0.9455, 0.9563, 1.1105, 0.9783, 0.9191, 1.0303, 1.0485, 1.0024 d. 1.1132, 0.9954, 0.9056, 0.9858 e. 269.498, 241.109, 265.018, 294.177, 314.409, 301.386, 309.187, 324.609, 368.308, 401.849, 430.654, 415.906, 440.172, 452.08, 485.866, 517.346, 485.088, 532.449, 574.205, 547.778 f. T = 216.2993 + 17.35763t g. 646.56, 595.42, 557.42, 623.90

14. Below you are given information on John's income for the past 7 years.Year Income ($1000s)1 15.02 16.2

3 17.14 18.15 18.86 19.27 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.Answers14: a. T = 14.3857 + 0.86429t b. 21.3, 22.2, 23.0, 23.9, 24.8

15. You are given the following information on the quarterly profits for Ajay Corporation.Year Quarter Quarterly Profits Yt

1 1 150 2 120 3 160 4 150

2 1 150 2 130 3 180 4 160

3 1 170 2 140 3 200 4 180

4 1 200 2 150 3 230 4 200

a. Find the 4-quarter centered MA.b. Compute the seasonal-irregular component.c. Compute the seasonal factors for all four quarters.d. Represent the deseasonalized series.Answers15: a. 145, 146.25, 150, 153.75, 157.5, 161.25, 165, 170, 176.25, 181.25, 186.25, 192.5b. 1.103, 1.026, 1, 0.846, 1.143, 0.992, 1.03, 0.824, 1.135, 0.993, 1.074, 0.779c. 1.04, 0.82, 1.132, 1.008d. 144.23, 146.34, 141.34, 148.81, 144.23, 158.54, 159.01, 161.28, 163.46, 170.73, 176.68, 178.57,

192.31, 182.93, 203.18, 198.41

16. Below you are given information on crime statistics for Middletown.Year Quarter Number of Crimes Committed Yt

1 1 10 2 20 3 25 4 15

2 1 10 2 30 3 35

4 253 1 20

2 40 3 35 4 15

4 1 20 2 50 3 45 4 35

The seasonal factors for these data areQuarter Seasonal Factor St

1 0.5892 1.3513 1.3354 0.726

a. Deseasonalize the series.b. Obtain an estimate 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.Answers16: a. 16.98, 14.8, 18.78, 20.66, 16.98, 22.21, 26.22, 34.44, 33.96, 29.61, 26.22, 20.66, 33.96, 37.01, 33.71, 48.21b. T = 13.5155 + 1.603765t c. 24.02, 57.26, 58.72, 33.1

17. 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 1.22 0.93 0.84 1.1T = 126.23 - 1.6t Produce forecasts for all four quarters of year 6 by using the seasonal and trend components.Answer17: 111.156, 81.927, 71.544, 96.613

18. 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 1602 1803 1904 170

9 1 2002 2103 2604 230

10 1 2102 2403 2904 260

a. Compute the 4-quarter MA 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.Answers18: a. 180.00, 188.75, 201.25, 217.50, 226.25, 231.25, 238.75, 245.25b. 0.935, 0.975, 1.1, 0.945 c. 213.90, 215.38, 236.36, 243.39