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.
34%
13.5%
2.35%
To find the probability that a man will have a shoe size between 13 and 14.5, we need to calculate the z-scores for both shoe sizes and then use the Empirical Rule to find the probability within that range.
First, let's find the z-scores for shoe sizes 13 and 14.5:
Z-score for shoe size 13:
Z = (13 - 10) / 1.5 = 2/1.5 = 1.33
Z-score for shoe size 14.5:
Z = (14.5 - 10) / 1.5 = 4.5/1.5 = 3
Now, we can use the Empirical Rule to find the probability that a man will have a shoe size between 13 and 14.5:
- From the Empirical Rule, we know that approximately 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
- The z-scores we calculated for shoe sizes 13 and 14.5 fall within 1 and 2 standard deviations from the mean.
So, the probability that a man will have a shoe size between 13 and 14.5 is approximately 95%.
Therefore, the correct answer is 95%.