Question: A restaurant manager is interested in taking a more statistical approach to predicting customer load. She begins the process by gathering data. One of the restaurant hosts or hostesses is assigned to count customers every five minutes from 7 P.M. until 8 P.M. every Saturday night for three weeks. The data are shown here. After the data are gathered, the manager computes lambda using the data from all three weeks as one data set as a basis for probability analysis. What value of lambda did she find? Assume that these customers randomly arrive and that the arrivals are Poisson distributed. Use the value of lambda computed by the manager and help the manager calculate the probabilities in parts (a) through (e) for any given five-minute interval between 7 P.M. and 8 P.M. on Saturday night.
a. What is the probability that no customers arrive during any given five-minute interval?
b. What is the probability that six or more customers arrive during any given five-minute interval?
c. What is the probability that during a 10-minute interval fewer than four customers arrive?
d. What is the probability that between three and six (inclusive) customers arrive in any 10-minute interval?
e. What is the probability that exactly eight customers arrive in any 15-minute interval?