Statistics Help

Question

Police response time to an emergency call is the difference between the time the call is first received by the dispatcher and the time a patrol car radios that it has arrived at the scene. Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 10.4 minutes and a standard deviation of 1.7 minutes. For a randomly received emergency call, find the following probabilities. (For each answer, enter a number. Round your answers to four decimal places.)

(a)

 the response time is between 7 and 12 minutes 

 

(b)

 the response time is less than 7 minutes 

 

(c)

 the response time is more than 12 minutes 

Answer 1

This problem relates to the bell curve for a normal distribution. This bell curve has defined probabilities based on the calculated standard deviation:

1. The first deviation away from the mean (either direction) represents 34.1% probability.
2. Between the first deviation to the second deviation away from the mean (either direction) represents 13.6% probability.
3. Between the second deviation to the third deviation away from the mean (either direction) represents 2.1% probability.
4. Beyond the third deviation away from the mean (either direction) represents an additional 0.1% probability.

(a) Response time between 7 and 12 minutes

The response time above represents two standard deviations less than the mean and one standard deviation more than the mean. This means we get probability contribution of 34.1% + 13.6% below the mean (10.4 min to 8.7 min to 7.0 min) and 34.1% probability contribution above the mean (10.4 min to 12.1 min). These probability contributions add up to 81.8% total probability of response time between 7 and 12 minutes.

(b) Response time less than 7 minutes

This response time looks beyond the second standard deviation less than the mean. The third deviation away contributes 2.1% and beyond the third deviation contributes 0.1%. Added together this is a 2.2% probability of a response time less than 7 minutes.

(c) Response time more than 12 minutes

This response time represents going beyond the first deviation above the mean. The second deviation contributes 13.6% probability, the third deviation contributes 2.1% and beyond that contributes 0.1%. Added together we get a total probability of response time more than 12 minutes at a probability of 15.8%.