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 97.5% 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 47.5% of the time. d. it takes the factory more than 27 minutes to manufacture a pair of running shoes about 50% of the time.

Explanation:

1. First, calculate the z - score:

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

1. Then, use the standard normal distribution table:

• The standard normal distribution table gives the cumulative probability \(P(Z\lt z)\). For \(z = 2\), from the standard normal table, \(P(Z\lt2)=0.9772\).
• The probability that \(Z\gt2\) is \(P(Z\gt2)=1 - P(Z\lt2)\).
• So, \(P(Z\gt2)=1 - 0.9772 = 0.0228\approx2.3\%\).

Answer:

C. It takes the factory more than 27 minutes to manufacture a pair of running shoes about 2.3% of the time.