Saturday, November 23, 2013

Ticket sales and treats -- creating a confidence interval for average revenues.

This theatre-treat problem allows for two types of ticket sales -- advanced ticket sales which are locked in and same-night sales which vary from night to night.    Profitability is determined more by concession stands than by ticket sales.


Situation:  

 A theatre has an outstanding contract with a group to purchase 20 tickets for the showing of a movie each Wednesday night.  

Additional sales are sold to the public immediately prior to the movie. 

Average sales on the night of the performance were 38.2.

The standard deviation of sales on the night of the performance was 3.4 over the previous 30 
Wednesday nights.

The theatre get $0.50 for each advanced ticket sale and $1.50 for each ticket sale on the night of the showing.

The theatre gets $4.00 for each treat sold at the concession stand.

75% of movie goers buy a treat.

Questions:  

What is the mean revenue per showing?

What is the standard deviation of revenue per showing?

What is the 95% confidence interval for revenue per showing? 

Answers:   

Revenue from the advanced-sales is $70.   ((20 x $0.5) + (20 x 0.75 x $4.00))  
This revenue is certain, at least under the assumptions inherent in this problem.

Average revenue from sales made on the night of the performance amount to $210.10  ((38.2 x 1.50) + (38.2 x 4)) or $5.50x38.2.

The average revenue from both advanced sales and night-of-showing sales is $280.10

The only variability in revenue is from sales on the night of the showing.  

Denote SA as the sample average of night-of-showing sales. 

This means revenue from night-of-showing sales is $5.50xSA.

The Variance of revenue is Var($5.5xSA) or 5.52x3.42, which is $349.7.

Take the square root to get the standard deviation STD(SA) is 18.7.


The 95 percent confidence interval is 

$280.10  plus or minus 1.96x3.42/(30)0.5  or 6.7

The lower bound is 273.4 and the upper bound is 286.8.


Note:   Since the only variability in this problem is the number of people who buy tickets on the night of the performance one could create a confidence interval for the number of same-day sales and multiply by $5.5.   A more sophisticated analysis would assume uncertainty about the  level of concession sales rather than assume that 75% of people buy snacks and the theatre gets $4.00 for each snack.  

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