Question

The mean life of a pair of shoes is 40 months with a standard deviation of 8 months. If the life of the shoes is normally distributed, how many pairs of shoes out of one million will need replacement before 47 months?

Answer 1

Using the normal distribution, it is found that 809,200 pairs of shoes will need replacement before 47 months.

Normal Probability Distribution

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

• It measures how many standard deviations the measure is from the mean.

• After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.

In this problem, the mean and the standard deviation are given by, respectively:

$\mu = 40, \sigma = 8$.

The proportion that will need replacement before 47 months is the p-value of Z when X = 47, hence:

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{47 - 40}{8}$

$Z = 0.875$

$Z = 0.875$ has a p-value of 0.8092.

Out of one million:

0.8092 x 1,000,000 = 809,200.

809,200 pairs of shoes will need replacement before 47 months.

More can be learned about the normal distribution at https://brainly.com/question/24663213