response time is an important statistic for measuring the effectiveness of a fire department, and is…
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 house 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 between 4 and 6 minutes? find the z - table here. 37.7% 44.3% 55.7% 62.3%
Answer
Answer:
C. 55.7%
Explanation:
Step1: Calculate z - score for 4 minutes
The z - score formula is $z=\frac{x-\mu}{\sigma}$. Here, $x = 4$, $\mu=4.5$, $\sigma = 1.2$. So $z_1=\frac{4 - 4.5}{1.2}=\frac{-0.5}{1.2}\approx - 0.42$.
Step2: Calculate z - score for 6 minutes
Using the z - score formula with $x = 6$, $\mu = 4.5$, $\sigma=1.2$. So $z_2=\frac{6 - 4.5}{1.2}=\frac{1.5}{1.2}=1.25$.
Step3: Find probabilities from z - table
From the z - table, $P(Z < - 0.42)=0.3372$ and $P(Z < 1.25)=0.8944$.
Step4: Calculate the probability between the two z - scores
$P(-0.42<Z<1.25)=P(Z < 1.25)-P(Z < - 0.42)=0.8944 - 0.3372=0.5572\approx55.7%$.