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.(1 point)
%

Answer 1

To apply the Empirical Rule (also known as the 68-95-99.7 rule) to the normal distribution, we need to understand how many standard deviations away from the mean our values are.

Given:
- Mean (μ) = 10
- Standard deviation (σ) = 1.5

First, we calculate how many standard deviations away the values 13 and 14.5 are from the mean.

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

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

So, we have:
- Size 13 is 2 standard deviations above the mean.
- Size 14.5 is 3 standard deviations above the mean.

According to the Empirical Rule:
- Approximately 68% of the data falls within 1 standard deviation (between μ - σ and μ + σ).
- Approximately 95% of the data falls within 2 standard deviations (between μ - 2σ and μ + 2σ).
- Approximately 99.7% of the data falls within 3 standard deviations (between μ - 3σ and μ + 3σ).

In this case, the range of 13 to 14.5 covers:
- From size 13 (at 2σ) to 14.5 (at 3σ), which is part of the 99.7% confidence interval (full range from 7 to 13).

Now, to find the probability of a shoe size falling between these two points:
- From the Empirical Rule, we know that 95% lies between 8 and 12, while 99.7% lies between 7 and 13. We're looking at the section between 2σ and 3σ.

The proportion of the distribution between 2σ and 3σ (13 to 14.5) is:

\[
\text{Area from 2σ to 3σ} = \frac{1}{2} \times (99.7\% - 95\%) = \frac{1}{2} \times 4.7\% \approx 2.35\%
\]

Thus, the probability that a man will have a shoe size between 13 and 14.5 is approximately **2.35%**.