Question

the male shoe size within a community are normally distributed with a mean of $\mu = 10$ and a standard deviation of $\sigma = 1.5$. apply the empirical rule to find the probability that a man will have a shoe size between 13 and 14.5. (1 point)
99%
95%
2.35%
4.25%

Explanation:

Step1: Calculate the number of standard deviations from the mean

For \(x = 13\), \(z=\frac{x-\mu}{\sigma}=\frac{13 - 10}{1.5}=2\)
For \(x = 14.5\), \(z=\frac{x-\mu}{\sigma}=\frac{14.5 - 10}{1.5}=3\)

Step2: Apply the Empirical Rule

The Empirical Rule states that for a normal distribution:

• Approximately \(95\%\) of the data lies within \(\mu\pm2\sigma\) (i.e., \(z = \pm2\))
• Approximately \(99.7\%\) of the data lies within \(\mu\pm3\sigma\) (i.e., \(z=\pm3\))

The proportion of data between \(z = 2\) and \(z = 3\) is \(\frac{99.7\% - 95\%}{2}=2.35\%\)

Answer:

\(2.35\%\)