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