# solution

The government 311 hotline service is a non-emergency phone number that people can call in many U.S. and Canadian cities to find information about services, make complaints, or report problems like trash collection issues and potholes. Most cities perform analytics on call volumes to their call centers for purposes of maintaining adequate staffing and timely services. For the city of Chicago, the average number of 311 calls in the summer months is about 9.79.7 per hour for the night shift. In terms of minutes, this translates to 0.1620.162 call per minute. Suppose that the number of calls per time unit follows a Poisson distribution. This implies that the time ??X between calls is governed by the exponential distribution. Note that the mean and the standard deviation of the exponential distribution are both equal to the inverse of the rate of arrivals (in this case, rate of incoming 311 calls). Thus, the time between calls has an exponential distribution with mean ??=(1/0.162)=6.173Âµ=(1/0.162)=6.173 minutes and standard deviation ??=6.173s=6.173 minutes. You record the next 5050 times between 311 calls.

Find the probability that the mean time between 311 calls is less than 7 minutes.7 minutes.

(Use decimal notation. Use Table A. Give your answer to four decimal places.)

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