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 6.0 minutes and a standard deviation of 1.5 minutes. For a randomly received emergency call, find the following probabilities. (Round your answers to four decimal places.)

A button hyperlink to the SALT program that reads: Use SALT.

(a) the response time is between 3 and 7 minutes.
(b) the response time is less than 3 minutes
(c the response time is more than 7 minutes

Answer 1

(a) 0.8186

(b) 0.0227

(c) 0.1587

Explanation:

To find the probabilities, we can use the cumulative distribution function (CDF) of the normal distribution.

Let X be the police response time. Then, X ~ N(6, 1.5^2).

(a) P(3 < X < 7) = P(X < 7) - P(X < 3)

Using the SALT program or a standard normal distribution table, we get:

P(X < 7) = P(Z < (7 - 6) / 1.5) = P(Z < 1) = 0.8413

P(X < 3) = P(Z < (3 - 6) / 1.5) = P(Z < -2) = 0.0227

So, P(3 < X < 7) = 0.8413 - 0.0227To find the probabilities, we can use the cumulative distribution function (CDF) of the normal distribution.

Let X be the police response time. Then, X ~ N(6, 1.5^2).

(a) P(3 < X < 7) = P(X < 7) - P(X < 3)

Using the SALT program or a standard normal distribution table, we get:

P(X < 7) = P(Z < (7 - 6) / 1.5) = P(Z < 1) = 0.8413

P(X < 3) = P(Z < (3 - 6) / 1.5) = P(Z < -2) = 0.0227

So, P(3 < X < 7) = 0.8413 - 0.0227 To find the probabilities, we can use the cumulative distribution function (CDF) of the normal distribution.

Let X be the police response time. Then, X ~ N(6, 1.5^2).

(a) P(3 < X < 7) = P(X < 7) - P(X < 3)

Using the SALT program or a standard normal distribution table, we get:

P(X < 7) = P(Z < (7 - 6) / 1.5) = P(Z < 1) = 0.8413

P(X < 3) = P(Z < (3 - 6) / 1.5) = P(Z < -2) = 0.0227

So, P(3 < X < 7) = 0.8413 - 0.0227 = 0.8186

(b) P(X < 3) = 0.0227

(c) P(X > 7) = 1 - P(X < 7) = 1 - 0.8413 = 0.1587