The Empirical Rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation (σ) of the mean (μ).
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In this case, we know the following parameters:
- Mean, \( \mu = 10 \)
- Standard deviation, \( \sigma = 1.5 \)
To find the probability that a man will have a shoe size between 13 and 14.5, we first need to determine how many standard deviations these shoe sizes are from the mean:
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Calculate the z-scores for the values 13 and 14.5:
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For a shoe size of 13: \[ z = \frac{X - \mu}{\sigma} = \frac{13 - 10}{1.5} = \frac{3}{1.5} = 2 \]
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For a shoe size of 14.5: \[ z = \frac{X - \mu}{\sigma} = \frac{14.5 - 10}{1.5} = \frac{4.5}{1.5} = 3 \]
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Interpret the z-scores:
The z-score of 2 corresponds to being 2 standard deviations above the mean, and the z-score of 3 corresponds to being 3 standard deviations above the mean.
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Use the Empirical Rule:
- Approximately 95% of the data falls between \( \mu - 2\sigma \) and \( \mu + 2\sigma \), which is \( 10 - 3 = 7 \) and \( 10 + 3 = 13 \).
- Approximately 99.7% of the data falls between \( \mu - 3\sigma \) and \( \mu + 3\sigma \), which is \( 10 - 4.5 = 5.5 \) and \( 10 + 4.5 = 14.5 \).
To find the probability of having a shoe size between 13 and 14.5, we need to consider the area between z-scores 2 (representing shoe size 13) and 3 (representing shoe size 14.5):
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The area between z = 2 and z = 3 can be found using standard normal distribution tables or software:
- The cumulative probability of z = 2 is approximately 0.9772 (or 97.72%).
- The cumulative probability of z = 3 is approximately 0.9987 (or 99.87%).
- Calculate the probability between z = 2 and z = 3: \[ P(2 < Z < 3) = P(Z < 3) - P(Z < 2) \approx 0.9987 - 0.9772 = 0.0215 \]
Thus, the probability that a man will have a shoe size between 13 and 14.5 is approximately 0.0215, or 2.15%.