Have groups of students observe the number of people who enter a specific coffee shop or fast-food restaurant over the course of an hour, beginning at the same time every day, for two weeks. The hour should be a time of peak traffic at the shop or restaurant. The data collected will be the number of customers who enter the shop in each half hour of time. Thus, two data points will be collected each day. Let us assume that the random variable X, the number of people entering each half hour, follows a Poisson distribution. The students should calculate the sample mean and variance of X using the 28 data points collected.
(a) What evidence indicates that the Poisson distribution assumption may or may not be correct?
(b) Given that X is Poisson, what is the distribution of T, the time between arrivals into the shop during a half hour period? Give a numerical estimate of the parameter of that distribution.
(c) Give an estimate of the probability that the time between two arrivals is less than 15 minutes.
(d) What is the estimated probability that the time between two arrivals is more than 10 minutes?
(e) What is the estimated probability that 20 minutes after the start of data collection not one customer has appeared?