Question

note: assume that a normal model is appropriate for the distribution of manufacturing times.
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?
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 47.5% of the time.
c. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 2.5% of the time.
d. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 95% of the time.

Explanation:

Step1: Recall the properties of the normal distribution

In a normal distribution, about 95% of the data lies within \( \mu\pm2\sigma\), about 68% lies within \( \mu\pm\sigma\), and about 99.7% lies within \( \mu\pm3\sigma\). The probability that \( X > \mu + z\sigma\) can be found using the standard normal table. We know that \( P(X>x)=1 - P(X\leq x)\). For a normal distribution \( X\sim N(\mu,\sigma^{2})\), we standardize using \( Z=\frac{X - \mu}{\sigma}\).

Step2: Calculate the z - score

Given \( \mu = 21\) and \( \sigma=3\). We want to find \( P(X > 27)\). First, calculate the z - score: \( z=\frac{27 - 21}{3}=\frac{6}{3}=2\).

Step3: Use the standard normal table

We know that \( P(X>27)=1 - P(X\leq27)\). Since \( Z = 2\), from the standard normal table \( P(Z\leq2)=0.9772\). So \( P(X > 27)=1 - 0.9772 = 0.0228\approx2.5\%\)

Answer:

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