For a number of months, Harry Jones, manager of a downtown barbershop, has been recording the number of customers who arrive during successive 30-minute time periods. The following is Harry's record for a total of five hundred 30-minute intervals:

Number of customers arriving during a 30-minute interval

0

1

2

3

4

5

6

Frequency

117

171

125

59

22

6

0

a.What is the average number of customers arriving during a 30-minute interval? What is the average arrival rate in customers per hour?

b.Use the formula for a Poisson distribution and the average arrival rate determined in part (a) to calculate the probabilities that exactly 0, 1, 2, 3, 4, 5 and 6 customers will arrive during a 30-minute time interval. Compare these values with the results of Harry's statistical record. Does the customer-arrival pattern tend to follow a Poisson distribution?

c. Assuming that customers arrive on a Poisson distribution with an average rate as determined in (a), and that it takes Harry an average of 15 minutes to serve a customer, determine the following:

(a) The average time (in minutes) a customer must wait before being served.

(b) The average time (in minutes) a customer will spend in Harry's shop.

(c)The average number of customers waiting to be served.

(d) The average number of customers in Harry's shop.

(e) The probability there are no customers in Harry's shop.

(f)The probability there are exactly four customers in Harry's shop.