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 19 minutes and a standard deviation of 3.3 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% of the customers. a. Determine an estimate of an advertised maximum wait time so that 1% of the customers would receive a free meal. Round to one decimal place. minutes

Can this be explained in excel format? It would help me better understand it.

Answer 1

Let X be the time in minutes that a customer has to wiat. X is normally distributed with mean minutes and standard deviation minutes

We want to give away free meals to at the most 1% of the customers. That is, if we get 1000 customers in a day, we want to give free meals to not more than 10 customers. We want to find the value of wait time X such that not more than 10 customers have a wait time more than X minutes

Let us generate 1000 wait times from the normal distribution with mean and standard deviation in excel using datat--->data analysis-->random number geneartion

sort the genearted wait time in descending order (largest to smallest) and get the following

In this simulated data, the top10 customers have a wait time greater than or equal to 26.6569 minutes. That is if the restaurant sets the maximum wait time as 26.7 (rounding to 1 decimal place ) minutes then we know that there would be 10 customers, or 1% of the total customers who would get a free meal.

Now we will find the actual theoretical value for this X. When we say that only 1% of the customers need to have a wait time greater than x minutes, it means that the probability that a randomly chosen customer will have a wait time greater than x minutes is 0.01

Using symbols

We standard dize X using the mean and standard deviation and express this a Z score

Now we will use standard normal table or excel function =NORM.INV(0.99,0,1) and get the for Z=2.33 we have P(Z<2.33) = 0.99

Now we equate 2.33 to

the probability that a randomly chosen customer would have to wait more than 26.7 minutes is 0.01

That means if the restaurant sets the maximum wait time to be 26.7 minutes then not more than 1% customers would receive a free meal