Question

Suppose that you are working for a chain restaurant and wish to design a promotion to disabuse the public of notions that the service is slow. You decide to institute a policy that any customer that waits too long will receive their meal for free. You know that the wait times for customers are normally distributed with a mean of 18 minutes and a standard deviation of 3.5 minutes. Use statistics to decide the maximum wait time you would advertise to customers so that you only give away free meals to at most 1.5% of the customers.

a. Determine an estimate of an advertised maximum wait time so that 1.5% of the customers would receive a free meal. Round to one decimal place

b. Include a graph illustrating the solution. For the graph do NOT make an empirical rule graph, just include the mean and the mark off the area that corresponds to the 1.5% who would receive the refund.

c. Write a response to the vice president explaining your prescribed maximum wait time. Structure your essay as follows:

1. An advanced explanation of the normal distribution
2. Why the normal distribution might apply to this situation
3. Describe the specific normal distribution for this situation (give the mean and standard deviation)
4. Explain how the graph created in part b. represents the waiting times of the customers.
5. Explain the answer to part a. in terms of both the customers who get a free meal and those who do not. Feel free to use the accurate answer in part a to determine a "nice" wait time to be used in the actual advertising campaign.
6. Use the answers to parts a. and b. to explain how the proposal will not result in a loss of profit for the company.

Answer 1

Solution:-

(c)

It has been oberved that the wait times for customers are normally distributed with mean of 18 minutes and standard deviation of 3.5 minutes. Hence in order to find the value of wait time which represents the 98.5% of customer wait times and above which only 1.5% customer wait time lies, we use central limit theorem. The value found is 25.3 mins. This means 1.5% of customers will face wait time more than 25.6 minutes.
If we go ahead with 25.6 minutes as maximum wait time to avail the free meals, 1.5 out of every 100 customers will get the free meal.