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 5% of the time.
b. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 95% of the time.
c. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 47.5% of the time.
d. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 2.5% of the time.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is $z=\frac{x-\mu}{\sigma}$, where $x = 27$ (the value we are interested in), $\mu=21$ (the mean) and $\sigma = 3$ (the standard deviation).
Substitute the values into the formula: $z=\frac{27 - 21}{3}=\frac{6}{3}=2$.

Step2: Use the properties of the normal distribution

In a normal distribution, about 95% of the data lies within $z=- 2$ and $z = 2$ (by the empirical rule). This means that the proportion of data outside of $z = 2$ (either above $z = 2$ or below $z=-2$) is $100\%-95\% = 5\%$. Since the normal distribution is symmetric, the proportion of data above $z = 2$ (i.e., $x>27$) is $\frac{5\%}{2}=2.5\%$.

Answer:

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