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