Question

4. The emergency telephone (911) center in a large city receives an average of 132 hourly calls per day. On average, each call requires about 85 seconds for a dispatcher to receive the emergency call, determine the nature and location of the problem, and send the required individuals (police, firefighters, or ambulance) to the scene. The center is currently staffed by 3 dispatchers a shift but must have an adequate number of dispatchers on duty and it has asked a consultant to determine a proper level of staffing. The consultant assumes that the calls come at random over time, so that the Poisson probabilities apply.

1. How many calls should the emergency call center expect in a 15-minute period? Please show how you arrived at your answer.                             [2 points]
2. Define X to be the number of telephone calls to the emergency call center in a 15-minute period and assume that X has a Poisson Probability distribution. Please calculate the probability that the call center will receive 30 calls in the 15-minute period.                                                                            [1 point]
3. Please calculate the probability that the call center will receive at least 28 calls in the 15-minute period. Show your work.                                [2 points]
4. Please calculate the probability that the call center will receive fewer than 25 calls in the 15-minute period. Show your work.                        [2 points]
5. Please calculate the probability that the call center will receive at most 40 calls in the 15-minute period. Show your work.                                [1 point]
6. Please calculate the probability that the call center will receive more than 50 calls in the 15-minute period. Show your work.                        [2 points]
7. Calculate and interpret the standard deviation of X.                  [2 points]   
8. For what purpose can the emergency call center use the kind of information you have calculated above? If the call center wants to limit the mean caller waiting time to less than 3 seconds, do you think they should add a fourth dispatcher to each shift? Please show how you arrived at your answer. [Note: the caller waiting time in this problem is the time between the emergency telephone starts ringing and the time a dispatcher picks it up.]                              [4 points]                                           

Answer 1