Question

The male shoe sizes within a community are normally distributed with a mean of μ=10 and a standard deviation of σ=1.5. Apply the Empirical Rule to find the probability that a man will have a shoe size between 13 and 14.5.

A. 36% B. 85% C. 2.35% D. 4.25%

Answer 1

To apply the Empirical Rule (also known as the 68-95-99.7 rule) to find the probability that a man will have a shoe size between 13 and 14.5, we will first standardize the values of 13 and 14.5 to calculate their corresponding z-scores.

The z-score is calculated using the formula:

\[
z = \frac{(X - \mu)}{\sigma}
\]

where \(X\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Here, we have:
- \(\mu = 10\)
- \(\sigma = 1.5\)

Now, let's calculate the z-scores for both shoe sizes.

1. For \(X = 13\):
\[
z = \frac{(13 - 10)}{1.5} = \frac{3}{1.5} = 2
\]

2. For \(X = 14.5\):
\[
z = \frac{(14.5 - 10)}{1.5} = \frac{4.5}{1.5} = 3
\]

Next, we need to find the probability corresponding to these z-scores:

- The area to the left of \(z = 2\) in a standard normal distribution is approximately 0.9772.
- The area to the left of \(z = 3\) is approximately 0.9987.

To find the probability that a man will have a shoe size between 13 and 14.5, we calculate:

\[
P(13 < X < 14.5) = P(Z < 3) - P(Z < 2)
\]
\[
P(13 < X < 14.5) = 0.9987 - 0.9772 = 0.0215
\]

This gives us a probability of approximately 0.0215, or 2.15%.

Comparing this to the answer choices, the closest option is:

**C. 2.35%**

So the correct answer is:

**C. 2.35%**