Question

Response time is an important statistic for measuring the effectiveness of a fire department, and is measured as the difference between the time a fire station receives a call and the time the first piece of fire equipment leaves the station. The response times for fire departments in a large city are found to have an approximately Normal distribution, with a mean of 4.5 minutes and a standard deviation of 1.2 minutes. What percentage of fire station response times are under 3 minutes?

Find the z-table here.

6.68%
10.56%
89.44%
93.32%

Answer 1

Using the normal distribution, it is found that 10.56% of fire station response times are under 3 minutes.

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

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

• It 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, which is the percentile of X.

In this problem:

• The mean is of 4.5 minutes, hence $\mu = 4.5$.

• The standard deviation is of 1.2 minutes, hence $\sigma = 1.2$.

The proportion of fire station response times are under 3 minutes is the p-value of Z when X = 3, hence:

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

$Z = \frac{3 - 4.5}{1.2}$

$Z = -1.25$

$Z = -1.25$ has a p-value of 0.1056.

0.1056 x 100% = 10.56%

10.56% of fire station response times are under 3 minutes.

To learn more about the normal distribution, you can take a look at https://brainly.com/question/24663213