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

Consider an x distribution with standard deviation σ = 24. (For each answer, enter an exact number.)

(a) If specifications for a research project require the standard error of the corresponding distribution to be 4, how large does the sample size need to be?
n =

(b) If specifications for a research project require the standard error of the corresponding distribution to be 1, how large does the sample size need to be?
n =

   

Answer 1

Answer: 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.

Solution:

Mean, μ = 10.4 minutes

Standard deviation, σ = 1.7 minutes

(a) the response time is between 7 and 12 minutes.

P(7 < X < 12) = P(7-10.4/1.7 < Z < 12-10.4/1.7)

P(7 < X < 12) = P(- 2 < Z < 0.9412)

P(7 < X < 12) = P(Z < 0.94) - P(Z < -2)

P(7 < X < 12) = 0.8264 - 0.0228 (from z table)

P(7 < X < 12) = 0.8036

Therefore, the response time between 7 and 12 minutes is 0.8036.

(b) the response time is less than 7 minutes:

P(X < 7) = P(Z < 7 - 10.4 / 1.7)

P(X < 7) = P(Z < - 2)

P(X < 7) = 0.0228 (from Z table)

Therefore, the response time less than 7 minutes is 0.0228.

(c) the response time is more than 12 minutes:

P(X > 12) = P(Z > 12 - 10.4 / 1.7)

P(X > 12) = P(Z > 0.9412)

P(X > 12) = 0.1736

Therefore, the response time more than 12 minutes is 0.1733.

​​​​​​​​​​..................................................................................................

Answer: Consider an x distribution with standard deviation σ = 24.

a) If specifications for a research project require the standard error of the corresponding distribution to be 4, how large does the sample size need to be?

Standard error = 4

Standard error = Standard deviation, σ / sqrt (n)

n = (σ / standard error)^2

n = (24/4)^2

n = 36

Therefore, sample size needed to be 36.

(b) If specifications for a research project require the standard error of the corresponding distribution to be 1.

Standard error = 1

Standard error = Standard deviation, σ / sqrt (n)

n = (σ / standard error)^2

n = (24/1)^2

n = 576

Therefore, sample size needed to be 576.

For any query ask in the comment box. If this answer is helpful to you then please upvote.