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 7.2 minutes and a standard deviation of 2.1 minutes. For a randomly received emergency call, find the probability that the response time is between 3 and 9 minutes. (Round your answer to four decimal places.)

Answer 1

Answer:

0.7823 = 78.23% probability that the response time is between 3 and 9 minutes.

Step-by-step explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Normal distribution with a mean of 7.2 minutes and a standard deviation of 2.1 minutes.

This means that $\mu = 7.2, \sigma = 2.1$

Probability that the response time is between 3 and 9 minutes.

This is the pvalue of Z when X = 9 subtracted by the pvalue of Z when X = 3. So

X = 9

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{9 - 7.2}{2.1}$

$Z = 0.86$

$Z = 0.86$ has a pvalue of 0.8051

X = 3

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3 - 7.2}{2.1}$

$Z = -2$

$Z = -2$ has a pvalue of 0.0228

0.8051 - 0.0228 = 0.7823

0.7823 = 78.23% probability that the response time is between 3 and 9 minutes.