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 11.6 minutes and a standard deviation of 2.3 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 7 and 13 minutes


(b) the response time is less than 7 minutes


(c) the response time is more than 13 minutes


To solve these probability problems, we'll use the **z-score formula** for a normal distribution:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \(x\) is the value we are evaluating,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.

### Given:
- Mean (\(\mu\)) = 11.6 minutes
- Standard deviation (\(\sigma\)) = 2.3 minutes

#### (a) Probability that the response time is between 7 and 13 minutes
We calculate the z-scores for both 7 and 13 minutes:

1. For \(x = 7\):
\[
z_1 = \frac{7 - 11.6}{2.3} = \frac{-4.6}{2.3} \approx -2.00
\]

2. For \(x = 13\):
\[
z_2 = \frac{13 - 11.6}{2.3} = \frac{1.4}{2.3} \approx 0.61
\]

Next, we find the cumulative probabilities corresponding to these z-scores using the standard normal distribution table or a calculator:
- \(P(z_1 = -2.00) \approx 0.0228\)
- \(P(z_2 = 0.61) \approx 0.7291\)

The probability that the response time is between 7 and 13 minutes is:
\[
P(7 \leq x \leq 13) = P(z_2) - P(z_1) = 0.7291 - 0.0228 = 0.7063
\]

#### (b) Probability that the response time is less than 7 minutes
Using \(z_1 = -2.00\) from part (a):
\[
P(x < 7) = P(z < -2.00) = 0.0228
\]

#### (c) Probability that the response time is more than 13 minutes
Using \(z_2 = 0.61\) from part (a):
\[
P(x > 13) = 1 - P(z \leq 0.61) = 1 - 0.7291 = 0.2709
\]

### Final Answers:
- (a) \(P(7 \leq x \leq 13) = 0.7063\)
- (b) \(P(x < 7) = 0.0228\)
- (c) \(P(x > 13) = 0.2709\)

Would you like a detailed explanation of the z-score process or calculations?  
Here are some related questions you might find useful:
1. How do z-scores help in probability calculations for a normal distribution?
2. Can this process apply to a different distribution type?
3. How can response times be better optimized using statistics?
4. What tools can automate these probability calculations?
5. What does the shape of a normal distribution indicate about real-world events?

**Tip:** Using software like Excel or Python can help calculate probabilities quickly when working with large datasets or recurring problems.

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 11.6 minutes and a standard deviation of 2.3 minutes. For a randomly received emergency call, find the following probabilities. (Round your answers to four decimal places.)
(a) the response time is between 7 and 13 minutes
(b) the response time is less than 7 minutes
(c) the response time is more than 13 minutes

Learn how to calculate the probabilities for police response times using the normal distribution. With a mean of 11.6 minutes and a standard deviation of 2.3 minutes, this problem explores key concepts such as z-scores and standard normal distribution tables. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation