Question

According to a human modeling​ project, the distribution of foot lengths of women is approximately Normal with a mean of 23.3 centimeters and a standard deviation of 1.2 centimeters. In the United​ States, a​ woman's shoe size of 6 fits feet that are 22.4 centimeters long. What percentage of women in the United States will wear a size 6 or​ smaller?

Answer 1

22.66% of women in the United States will wear a size 6 or​ smaller

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 23.3, \sigma = 1.2`

In the United​ States, a​ woman's shoe size of 6 fits feet that are 22.4 centimeters long. What percentage of women in the United States will wear a size 6 or​ smaller?

This is the pvalue of Z when X = 22.4. So

`Z = (X - \mu)/(\sigma)`

`Z = (22.4 - 23.3)/(1.2)`

`Z = -0.75`

`Z = -0.75` has a pvalue of 0.2266

22.66% of women in the United States will wear a size 6 or​ smaller