FORECASTING

PROBLEMS

1. A manufacturing company has monthly demand for one of its products as follows:

MONTH DEMAND

January 520 Develop a three-period average forecast and a three February 490 period weighted moving average forecast with weights

March 550 of 5, 3 and 2 for the most recent demand values, in that April 580 order.May 600 Indicate which forecast would seem to be most accurateJune 420 Make a forecast of september by using both approaches.July 510August 610

2. A computer software firm has experienced the following demand for its “Personal Finance” software package.

Period Units 1 562 61 Develop an exponential smoothing forecast using3 55 an alpha value of 0.404 705 666 657 728 75

3. The head of Business Department at EMU wants to forecast the number of students who will enroll in production/operations management next semester in order to determine how many sections to schedule. The department has accumulated the following enrollment data for the past 8 semesters.

Semester Students enrolled in POM

1 802 903 704 845 1006 1157 988 130

a) compute a 3-semester moving average forecast for semester 4 through 8b) Compute the exponentially smoothed forecast (alpha=0.20) for the

enrollment data.

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c) Compare two forecasts and indicate the most accurate.d) Make a forecast for the next semester (semester 9) with the most accurate

approach.

4. ABC Hardware handles spare parts for lawn-mowers. The following data were collected for one week in April when replacement for lawn-mower blades were in high demand.

Day Demand

10 1512 1613 1815 2217 2120 2321 24

Simulate a forecast using simple smoothing for the week, starting with F = 15 and alpha=0.2. Find also the forecast for the 8th day.

5. Fill in the blank places.

Quarter Quantity

2004 I 26

II 38

III 54

IV 34

__________________________ Moving Totals

2005 I 34 160

II 50 172

III 58 176

IV 38 180

2006 I ___ 190

II ___ 197.2

III ___ 204.4

IV ___ 211.6

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6. Using total moving average method to forecast the quarterly values of 2007.

Years Quarters Sales (million bottles)

2004 I 18.2

II 29.2

III 22.2

IV 17.4

2005 I 19.2

II 30.8

III 24.2

IV 18.2

2006 I 21.6

II 33.2

III 26.2

IV 20.8

7. The general manager of a building materials production plant feels the demand for

plasterboard shipments may be related to the number of construction permits issued in the municipality during the previous quarter. The manager has collected the data shown in the accompanying table.

Construction Plasterboard Permits Shipments 15 6 a. Find a regression forecasting equation 9 4 b. Determine a point estimate for plasterboard 40 16 shipments when the number of construction 20 6 permits is 30. 25 13 c. Given the data on permits and shipments, 25 9 compute the standard deviation of regression. 15 10 d. Find the prediction interval of 90%.(std.-t table) 35 16 e. Find the prediction interval of 95.5% (normal)

for the specific amount of shipments when the permits number 30. (for this part assume your regression equation has been derived from a sufficiently large sample that the prediction interval form equal to y+/-z.s may be used.)

f. Determine r and coefficient of determination and interpret them.g. Test the correlation coefficient at 5% level of significance. Is the correlation coefficient significant at the level 5%?

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h. By using correlation coefficient analysis find the regression forecasting equation, and explain why this equation is different than the one you found in (a).

8. ABC Hardware handles spare parts for lawn mowers. The following data were collected for one week in April when replacement lawn-mower-blades were in high demand. The firm also collected necessary data on the total sales dollars generated by the store. The manager of the store would like to know if the total sales are a good predictor of lawn-mower-blade sales.

Day Demand for Total sales Lawn-mowers of the store(000$)

1 10 10 2 12 13 3 13 14 4 15 16

5 20 19 6 25 20 7 24 20

a) For the above data calculate the correlation coefficient between Demand for lawn-mower blade and Total sales of the store, and interpret

the result.

b) What percentage of variation in lawn-mower blade sales can be explained by total sales of the store?

c) Test the correlation coefficient at 5% level of significance.

d) Compute the forecast of 8th day total sales of the store.

e) Using the forecast of total sales you found at (d), find the forecasted demand for lawn-mower blade sales for the same date with 90%

probability.

9. Ali and Arzu are planning to set up an ice-cream stand in Laguna/Gazimagusa. After six months of operation, the observed sales of ice-cream (in MU) and the number of Laguna visitors are Month Ice-cream sales (MU) Laguna Visitors

1 200 8002 300 9003 400 11004 600 14005 700 18006 800 2000

a) Determine a regression equation treating ice-cream sales as the dependent variable and Laguna visitors as the independent variable.

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b) if you expect the Laguna visitors to peak out at about 3000 persons next month, what would be the expected ice-cream sales?

c) express your forecast with 68.3% probability limits.

10. In a manufacturing process the assembly-line speed (meter/minute) was thought to affect the number of defective parts found during the inspection process. To test this theory, management devised a situation where the same batch (lot) of parts was inspected visually at a variety of line speeds. The following data were collected.

# of defective Line parts found speed

21 2019 2015 4016 3014 6017 40

a. Develop the estimated Regression Equation that relates line speed to the number of defective parts found.

b. Use the equation developed in part (a) to forecast the number of defective parts found for a line speed of 50 meters per minute.

c. Express your forecast within 95.5% probability limits. (Assuming n is large)

11. Sergio’s Restaurants collected the following data on the relationship between advertising and sales at a sample of five restaurants.

Advertising Sales

Expenditures (000 MU)

(000 MU)

1 191 44

6 40 10 52

14 55

a. Determine the strength of the causal relationship between advertising expenditures and sales of the restaurants and interpret the result.

b. What is the coefficient of determination? What does it mean to you?

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c. Test the correlation coefficient you found in (a) at 5% level of significance. Is the correlation coefficient significant at this level?

d. Using correlation coefficient find regression forecasting equation.

12. Year Quarter Demand (tons)

2004 I 105 II 150

III 93 IV 121

2005 I 140 II 170 III 105 IV 150

2006 I 150 II 170 III 110 IV 130

Use Moving Totals to forecast the quarterly demand for the year 2007.

13. The data shown in the accompanying table include the number of lost-time accidents for the Izmir Lumber Company over the past 7 years. Some additional calculations are included to help you answer the following questions. Manager of the company uses the number of employees (in thousands) to predict the number of accidents.

YEAR NO. OF NO. OF EMPLOYEES ACCIDENTS

(000)

1997 15 5 225 25 751998 12 20 144 400 2401999 20 15 400 225 3002000 26 18 676 324 4682001 35 17 1225 289 5952002 30 30 900 900 900

2003 37 35 1369 1225 1295

=175 =140 =4939 =3388 =3873

a. Use the normal equations to develop a linear regression equation for forecasting the number of accidents on the basis of the number of employees. State the

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equation. Use the equation to forecast the number of accidents when the number of employees is 33(000).

b. Assuming n is large, calculate the 95.5 percent confidence limits for the number of accidents when the number of employees is 33(000).

c. What is the correlation coefficient between number of employees and the number of accidents? Interpret your result.

d. What percentage of the variation in the number of accidents is explained by the employment level?

e. Is the correlation significant at the 5% level?

14. Kitchens of Tomorrow Inc. has collected the following data to learn if the number of building permits might be a useful predictor of their cabinet sales.

BUILDING CABINET PERMITS SALES

(00) (000 MU) a. Use the normal equations to derive a regression forecasting equation.

2 3 b. Compute the standard deviation of regression 5 5 c. Assume your regression has been derived from a 1 5 sufficiently large sample that the interval estimate

2 6 form equal to Y ±Z.Syx may be used. 5 7 d. Establish a 99.7% prediction interval estimate for

4 6 the specific amount of cabinet sales (000 MU)when 3 5 permits number 4.4(00). Compute the coefficient of

4 5 correlation and explain the meaning of it. 1 3 e. Test the significance of r for 10% and n=9.

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f. Use the correlation coefficient formula to derive a regression forecasting equation.

g. Is there any difference between the two equations you derived at a. and f.

15. A company wants to develop a means to forecast its carpet sales. The store manager believes

that the store’s sales are directly related to the number of new housing starts in town. The manager has gathered data from Chamber of Commerce records of monthly house

construction permits and from store records on monthly sales. These data as follows: Monthly Construction Monthly Carpet

Permits Sales (000 metres) 42 2070 4020 1624 12

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32 3218 882 4830 4436 3652 56

a. Develop a linear Regression Model for these data and forecast carpet sales if

30 construction permits for new homes are filed.

b. Calculate the standard deviation of regression.

c. State your forecast in the confidence limits of 90%.

16. Demand for hockey skates at a local sports store for the past eight weeks has been

Week Demand 1 1222 1303 984 1215 966 1527 1138 124

Use a simple exponential smoothing model with alpha=0.6. Assume the forecast for Period 1 was 120. Make a forecast for period 9.

17. A retail chain of eyewear specialist has been experimenting with sales price of contact lenses. The following data have been obtained.

Average lenses Price per per day_______ lens, MU

200 24190 26188 27180 28170 29162 30170 32

a. For the above data calculate the correlation coefficient between lens price and lens sales and interpret the result.

b. What percentage of variation in lens sales can be explained by prices. c. Test the correlation coefficient at 5% level of significance.d. What is 95% confidence interval for demand at price 28 MU. (Hint: n=7)

18. Fill in the blank places

Year Quarters Demand(tons)

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2004 I 105II 150III 95IV 120

______________________________________Moving TOTALS2005 I 150 515

II 200 565III 125 595IV 175 650

2006 I ____ 690II ____ 733.5III ____ 777IV ____ 820.5

19. Compute a forecast for the demand in each of the quarters of the following years, 2007.

Year Quarter Demand2005 1 92

2 823 844 92

2006 1 902 803 824 90

20. A company has collected the following data to learn if the number of building permits might be a useful predictor of their kitchen cabinet demand.

Building permits

x($00)

Cabinet Salesy

($00)

2 65 101 102 125 144 123 104 101 6

a). Use the normal equations to derive a regression forecasting equation.b). Compute the standard deviation of regressionc). Assume our regression equation has been derived from a

sufficiently large sample. Establish a 95.5% confidence

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limits estimate for the specific amount of cabinet sales ($000) when permits number is 4.4 (00).

d). Find the prediction interval of 90%, when permits number is 4.4 (00).

e). Determine r and interpret it.

f). Determine coefficient of determination and interpret it.

g). Test the correlation coefficient at 5% level of significance.

h). By using correlation coefficient analysis find the regression forecasting equation, and explain why this equation is different than the one

you found in (a).

21. A company wants to develop a means to forecast its carpet sales. The store manager believes that the store’s sales are directly related to the number of new housing starts in town. The manager has gathered data from Chamber of Commerce records of monthly house

construction permits and from store records on monthly sales.

Monthly Construction Monthly Carpet Permits Sales (000 metres)

42 1070 2020 824 632 1618 482 24

30 2236 1852 28

a. Develop a linear Regression Model for this data and forecast carpet sales if 30 construction permits for new homes are filed.

a. Calculate the standard deviation of regression.b. State your forecast in the confidence limits of 90%.c. Determine r and interpret itd. Determine the strength of the causal relationship between monthly sales and new

home construction using correlation.e. Test the correlation coefficient at 5% level of significance.

22. Using total moving average method to forecast the quarterly values of 2007.

Years Quarters Sales (million bottles)2004 I 91

II 146III 111

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IV 872005 I 96

II 154III 121IV 91

2006 I 108II 166III 131IV 104

23. TT Construction Company renovates old homes in Magusa. Over time, the company has

found that its MU volume of renovation work is dependent on the Magusa area payroll. The figures for TT’s revenues and the amount of money earned by wage earners in Magusa for the past six years are presented in the table below.

Years Sales Payroll(100.000MU) (100.000.000MU)

1997 2.0 11998 3.0 31999 2.5 42000 2.0 22001 2.0 12002 3.5 7

a. Using sales data above develop a regression equation.b. Find correlation coefficient and determination coefficient and interpret them. c. Test the correlation coefficient at 5% level of significance. Is the correlation

coefficient meaningful (significant) at this level? d. Using correlation coefficient, find regression equation and explain the difference

between the two regression equations in (a) and (d).e. Calculate standard deviation of the regression equation and express your forecast

within 90% probability limits, if the local chamber of commerce predicts the Magusa area payroll will be 600 million MU next year.

f. Find the forecast of Magusa Area Payroll for the year 2003.g. Find the regression equation using the forecast found in (f)h. Assuming sample is large (n>30) find the confidence intervals for 65.5% probability.

24. In a manufacturing process the assembly-line speed (meter/minute) was thought to affect the number of defective parts found during the inspection process. To

test this theory, management devised a situation where the same batch (lot) of parts was inspected visually at a variety of line speeds. The following data were collected.

# of defective Line parts found speed 22 20

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20 20 18 40 18 30

15 60 18 40

a. Develop the estimated Regression Equation that relates line speed to the number of defective parts found.

b. Use the equation developed in part (a) to forecast the number of defective parts found for a line speed of 50 meters per minute.

c. Express your forecast within 99.7 % probability limits.

(assuming n is large)

25. Room registrations in the Magusa Plaza Hotel have been recorded for the past nine years. Management would like to determine the mathematical trend of guest registration in

order to project future occupancy. This estimate would help the hotel management to determine whether a future expansion will be needed. Given the following time-series data, develop a trend equatin relating to registrations to time.

Then,

a) Forecast next year’s registrations.b) Give your next year’s forecast with 95% probability (i.e. assuming the level

of significance is equal to 5%)c) Assuming n is large (i.e. n≥30), show your confidence limits for the next

year with %95.5 probability.

Years Registrants(000)1993 171994 161995 161996 211997 201998 201999 232000 252001 24

26. Time 1 2 3 4 5 6 7 8 9 10 11 12 Demand 10 14 19 26 31 35 39 44 51 55 61 54

a) Assume F1 = 8 and α = 0.3 . Use an exponential smoothing factor to forecast demand in periods 2-13.

b) Find the mean absolute deviation of exponential smoothing. \

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27. Year Quarter Demand for

fertilizer (tons)2004 I 50

II 73 III 45 IV 60

2005 I 71 II 85 III 50 IV 61

2006 I 71 II 80 III 55 IV 65

a. Compute a three-quarter moving average forecast. Compute also the forecast error for each quarter.

b. Compute the quarterly forecasted demand for the year 2007.

28. The manager of Magusa Transport Co. wishes to forecast the number of miles driven by his trucks for the coming three years.

Years Thousands of

______ Miles driven2001 222002 24

2003 34 2004 30 2005 40 2006 50

a) Compute the forecast of miles driven for the next three years (2007, 2008 and 2009)b) Give your forecast for the year 2007 with %95 probability (i.e. assuming the level of

significance is equal to %5)c) Assuming n is large (i.e. n30), show your confidence limits for the year 2008 with

%68.3 probability.

29. November Demand 10 2011 2812 3813 5214 6215 70

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a) Use a simple 3-period moving average to demand for 13 November-15 November.

b) Find the average error for that period.

c) Assume that F1=24 and = 0.6. Use an exponential smoothing method to forecast demand in periods 11 November-15 November. Find

the average error.d) Compare the methods and state which one you prefer and why?

30. The monthly sales for Telco Batteries Inc., were as follows:

Month Sales Month Sales January 20 October 20February 21 November 21March 15 December 23April 14May 13June 16July 17August 18September 20

Forecast past sales using each of the following;a. A three-month moving average,b. a 6-month weighted average using 1,1,1,2,2,2, and 3 with the heaviest

weights applied to the most recent months.c. Exponential smoothing using an α = 0.3 and a January forecast of 20.d. Which method you prefer and why?e. using the method you chose, forecast January sales of the coming year.

31. Dr. Alev Yakar, a Magusa psychologist, specializes in treating patients who are agoraphic (afraid to leave their homes). The following table indicates how many patients Dr. Yakar has seen each year for the past 10 years.

Year No.of Patients 1997 361998 331999 402000 412001 402002 552003 602004 542005 582006 61

a.Using trend analysis, predict the number of patients Dr. Yakar will see in years 2007 and 2008.b.What is the standard error of the forecasts?c. Forecast number of patients in 2007 at 5% level of significance.

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d.Assuming sample is large (i.e. n>30), state your forecast of 2007 within 95.5%confidence interval.

32. Data collected on the yearly demand for 50-kg bags of fertilizer at Ilhandir Garden Supply are shown in the table below.

DEMAND FORFERTILIZER

YEAR (000 of BAGS) 1 42 63 44 55 106 87 78 99 1210 1411 15

a. Develop a three-year moving average to forecast sales.b. Develop a four-year moving average for demand for fertilizer.c. Estimate demand again with weighted three-year moving average in which sales in the most recent year are given a weight of 2 and sales in other two years are each given a weight of 1.

d. Three different forecasts were developed for the demand for fertilizer. These three forecasts are a three-year moving average, four-year moving average and a

weighted moving average. Which one would you use and explain why?

e. Use exponential smoothing with a smoothing constant of 0.3 to forecast the demand for fertilizer. Assume that last period’s (year’s) sales forecast for year 1 is 5

000 bags to begin the procedure.

f. Would you prefer to use the exponential smoothing model or one of the above models. Explain your choice. And according to your choice forecast the year 12.

33. Girne Manufacturing Company’s demand for electrical generators over the period 2000 - 2006 is shown in table below.

ElectricalYear Generators Sold 2000 742001 792002 802003 902004 1052005 1422006 122

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a. Develop a linear trend line by using the least squares method.b. Estimate the demand in 2007 and 2008.

c. Calculate the standard error of the past record. d. Give your forecast for the year 2008 at 5% level of significance.e. Assume n is large (n>30), give your forecast for the year 2007 within 95.5% confidence interval.

34. The following gives the number of pints of type O (Rh+) blood used at Nalbantoglu Hospital in the past 6 weeks:

Week of Pints Used August 4 360August 11 389August 18 410August 25 381September 1 368September 8 374

a. Forecast the demand for the week of September 15 using a 3-week moving average. b. Use a 3-week-weighted moving average, with weights of 1,3, and 6, using 6 for the

most recent week. Forecast demand for the week September 15. c. Compute the forecast for the above data using exponential smoothing with a forecast for

August 4 of 360 and α =0.2. Forecast the demand for the week of September15.(Show all your calculations and errors in tabular form.)

35. The manager of the Petroco Service Station wants to forecast the demand for unleaded gasoline next month so that the proper number of gallons can be ordered from the

distributor. The owner has accumulated the following data on demand for unleaded gasoline from sales during the past 10 months.

MONTH Gasoline Demanded (gallons)

November 800 a. Compute an exponentially smoothed forecast December 725 using α = 0.3 and F1 = 700. January 630 b. Compute the error of each month and find theFebruary 500 average error for the past record. March 645 c. Forecast the demand for the coming month April 690 September.May 730June 810July 1200August 980

36. Quarterly data for the failures of certain aircraft engines at a local military base during the last two years are

Quarters Engine failures 1 2002 250

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3 1754 1865 2256 2857 3058 190

a) Determine one-step-ahead forecasts for periods 4 and 8 using

three-period moving averages method.

b) Let us assume that the forecast for period 1 was 200. Also suppose that = 0.1. Determine one-step-ahead forecasts for periods 2 and 8.

c) Compare the above mentioned methods for the periods 4 and 8. Based on this comparison conclude which method is a superior method for the given series.

37. Bicycle sales at TT’s Bikes are shown below.

ActualWeek Bicycle Sales 1 82 103 94 115 106 13

a) Use 3-week moving average for forecasting week 4, week 5, week 6 and week 7.b) If

Weights Applied Period

3 last week2 2 weeks ago1 3 weeks ago

Forecast the weeks 4, 5, 6 and 7. c) Which method would you prefer and why?d) Use exponential smoothing to forecast bike sales. Assume that the forecast for Week 1 was 9 and α = 0.7.

38. The sales manager of a large apartment rental complex feels the demand for apartments may be related to the number of newspaper ads placed during the previous month. She has collected the data shown in the accompanying table.

Ads Purchased Apartments leased 15 6 9 440 1620 6

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25 1325 915 1035 16

a. Find the mathematical equation by using the least squares regression approach.

b. If the number of ads is 30, estimate the number of apartments leased.c. Given the data on ads and apartment rentals as above, compute the

standard deviation of regression (Syx).d. Compute the correlation coefficient and interpret.e. Compute the determination coefficient and interpret.f. Test the hypothesis, i.e. r = 0 , at 5% level of significance

39. Given below are 2 years of quarterly demand data for a particular model of personal computer from a local computer store.

Year Quarter Demand 2005 I 40

II 46 III 39 IV 42

2006 I 44 II 57 III 43 IV 45

a) Deseasonalize the data with a moving total and compute a linear equation for the trend in demand.

b) Using the trend you have developed, compute a forecast for the demand in each quarters of the following year.

40. Bus and subway ridership for the summer month in London, England, is believed to be tied heavily to the number of tourists visiting the city. During the past 12 years, the following data have been obtained.

YEAR NO. OF RIDERSHIP TOURISTS (in millions) (in millions)

-________ 1991 7 1.5 49 2.25 10.51992 2 1.0 4 1.00 2.01993 6 1.3 36 1.69 7.81994 4 1.5 16 2.25 6.01995 14 2.5 196 6.25 35.01996 15 2.7 225 7.29 40.51997 16 2.4 256 5.76 38.41998 12 2.0 144 4.00 24.01999 14 2.7 196 7.29 37.82000 20 4.4 400 19.36 88.02001 15 3.4 225 11.56 51.0

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2002 7 1.7 49 2.89 11.9TOTALS 132 27.1 1796 71.59 352.9

a. Use the normal equations to develop a linear regression equation for forecasting the number of ridership on the basis of the number of tourists. State the

equation. b. Use the equation to forecast the number of ridership when the number of tourists

visit London in a year is 10 million.

c. Explain the predicted ridership if there are no tourists at all. d. Assuming n is large, calculate the 95.5 percent confidence limits for the number

of ridership when the number of tourists is 10 million.

e. What is the correlation coefficient between number of ridership and the number of tourists? Interpret your result.

f. What percentage of the variation in the number of ridership is explained by the tourist level?

g. Is the correlation significant at the 5% level?

41. Sales of Volkswagen’s Beetle have grown steadily at auto dealership in Istanbul during the past 5 years (see the table below).

Year Sales 1999 4502000 4952001 5182002 5632003 584

a) The sales manager had predicted in 1998 that 1999 sales (F1) would be 410 VWS. Using exponential smoothing with a weight of α = 0.30, develop forecast for 2000 through 2004.

b) Use a 3-year moving average to forecast the sales of VW beetles in Istanbul through 2004.

c) Which method you would use, exponential smoothing with α = 0.3 or a 3-year moving average. (Use average errors)

d) According to the method you have chosen, forecast 2004 sales.

42. Year Quarter Demand (Units) 2005 I 92

II 82III 84IV 92

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2006 I 90II 80III 82IV 94

Compute a forecast for the demand in each of the quarters of the following year, 2007.

43. Following are the actual tabulated demands for an item for a nine-month period, from January through September. Your supervisor wants to test three forecasting methods to see which method was better over this period.

Month Actual DemandJanuary 110February 130March 150April 170May 160June 180July 140August 130September 140

a. Forecast April through September using a 3-month simple moving average.

b. Using a weighted moving average with weights 6, 3, 1 from recent to oldest, forecast April through September.

c. Use simple exponential smoothing to estimate April through September (α = 0.3) and assume that the forecast for March was 130.

d. Use absolute errors to decide which method produced be better forecast over the six-month period.

44. Dumlupinar Sports Club wants to develop its budget for the coming year using a forecast for football attendance. Football attendance accounts for the largest portion of its revenues, and the Vice Director Mr. T. Turgay believes attendance is directly related to the number of wins by the team. The Vice Director has accumulated total attendance figures for the last eight months.

WINS ATTENDANCE

4 3 6306 4 0106 4 1208 5 3006 4 4007 4 5605 3 9007 4 750

a) Develop a simple regression equation. b) Forecast attendance for at least 7 wins next year. c) If “ r = 0.948 “, what is the coefficient of determination. Interpret both.

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d) Test the correlation coefficient at 5 % level of significance. Is the correlation coefficient significant (meaningful) at this level?

e) Using correlation coefficient find regression equation and explain the difference between two regression equations you have

calculated.f) Calculate standard deviation of regression equation.

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