Forecasting Ticket Sales at Cineplex

The Cineplex is a large cinema complex in Ottawa. The complex has 18 screens and a total of 4620 seats. Ticket sales vary from day to day but are generally higher on the weekends (Friday and Saturday nights have the highest attendance). Also,children's matinees increase attendance on Sunday. A couple of years ago Cineplex initiated a half-price policy on Tuesdays to help boost box-office sales on an otherwise slow night.

The Cineplex's co-owner and manager, Erin, wants to develop a predictive model for daily ticket sales at the Cineplex, since this will help her to plan staffing schedules, order quantities for the concessions in the four-level building (there is one concession on each floor), and deal with other aspects of running a large multi-screen movie complex.

Erin has decided that when the attendance is anticipated to be less than 2500 she will open only two of the four concessions (on the first and third floors), and she will reduce the staff by four persons. The co-owner of the Cineplex, David, points out that popular movies usually run simultaneously on two or more screens and recommends that on slow nights those films be shown on only one screen, thereby cutting staff even further.

Ina strategic planning meeting David mentions, "all of this is highly speculative, we can't makeany meaningful decisions until we analyze the data and see what the ticket sales look like onvarious days of the week." Erin agrees and replies, "I manage theday-to-day operations, so wehave all the data on file". I will produce a spreadsheet showing attendance for the last month or soand we can use that to develop a planfor the future".

David says, "We can try to construct a statistical model using regression analysis to detect the weekly pattern of ticket sales, but we would need to include some explanatory variables - what other data can you include in the spreadsheet?"

David and Erin spend the next half hour brainstorming the problem of which explanatory variables to include in a regression model. After serious considerations, they decide that a useful regression model would involve a variable that differentiates between weekdays (Monday to Thursday) and weekends (Friday, Saturday, Sunday). Therefore, they decide to include the Binary variable:

{1 if day=Monday, Tuesday, Wednesday, or Thursday 0 if day=Friday, Saturday, or Sunday

The other major factor affecting attendance is climate. In fact, the data were collected during five winter weeks when the temperature was quite variable. Therefore, Erin includes the average daily temperature in her spreadsheet. The tables below show the daily number of ticket sales for the 34 days for which the ticket sales were recorded, and the average daily temperature for the same period.

Ticket Sales

Day

Week 1

Week 2

Week 3

Week4

Week 5

Monday

2120

2096

2246

2256

2421

Tuesday

2870

2768

2872

2879

3041

Wednesday

2230

2185

2224

2320

2076

Thursday

2459

2543

2446

2654

2580

Friday

3120

3200

3180

3280

3175

Saturday

3564

3480

3560

3568

3762

Sunday

3342

3312

3427

3342

Temperature (Celsius)

Day

Week 1

Week 2

Week 3

Week4

Week 5

Monday

-8

3

2

-6

6

Tuesday

-2

-1

2

-4

2

Wednesday

0

-5

1

-7

-3

Thursday

-4

-3

-4

-2

-5

Friday

1

7

3

4

0

Saturday

4

10

7

7

6

Sunday

5

8

6

5

Before doing any modeling, David consults with her sister Julie, who is a third-year student in business school, to get her opinion on the problem. He explains the problem to her and asks if he is on the right track. Shereplies, "I think I would start with a multiplicative decomposition to try to establish some seasonal indices for different days of the week-that would help to develop a good forecasting modeling that explicitly incorporates the daily pattern of ticket sales in the forecast. Then you can run the regression model that is pretty straightforward in Excel. I would also suggest to build a third model using Exponential smoothing method. Then we can compare the accuracy of these models and select the best one !".

1. The next day's ticket sales is forecasted to be 3375. Construct a 95% confidence interval for the forecasted value of 3375. (3 points)
2. Construct a 95% confidence interval for weekday predictor and interpret what it means. (2 points)
3. If David wants to use Exponential smoothing, what is the optimal value of smoothing constant ()that would minimize the Mean Absolute % Error? Use a graph that demonstrates the relationship between value of smoothing constant ()and Mean Absolute % Error. (3 points)
4. Forecast ticket sales for the next day using Exponential smoothing method, with the optimal value of smoothing constant ()you found in previous part. (2 points)
5. Compare the three models you developed based on Mean Absolute % Error, Mean Absolute Error, and Root Mean Square Error. Which of the methods analyzed here would you use? Explain your answer. (7 points)