Question

on average, it takes a shoe factory 21 minutes, with a standard deviation of 3 minutes, to manufacture a pair of running shoes. how often will it take the factory more than 27 minutes to manufacture a pair of running shoes?
note: assume that a normal model is appropriate for the distribution of manufacturing times.
a. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 95% of the time.
b. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 2.5% of the time.
c. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 5% of the time.
d. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 47.5% of the time.

Explanation:

Step1: Calculate z-score

The z-score formula is $z = \frac{x - \mu}{\sigma}$, where $x=27$, $\mu=21$, $\sigma=3$.
$z = \frac{27 - 21}{3} = 2$

Step2: Use empirical rule for normal distribution

For a normal distribution, about 95% of data lies within $\mu \pm 2\sigma$. So the percentage of data above $\mu + 2\sigma$ is $\frac{100\% - 95\%}{2} = 2.5\%$.

Answer:

B. It takes the factory more than 27 minutes to manufacture a pair of running shoes about 2.5% of the time.