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.
A. 34%
B. 13.5%
C. 2.35%
To apply the Empirical Rule, we need to determine how many standard deviations away from the mean a shoe size of 13 and 14.5 are.
For a shoe size of 13:
(13 - 10) / 1.5 = 2
For a shoe size of 14.5:
(14.5 - 10) / 1.5 = 3
According to the Empirical Rule, 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.
Since our shoe sizes of 13 and 14.5 are 2 and 3 standard deviations away from the mean respectively, we can use the Empirical Rule to determine the probability.
For 13:
P(X = 13) ≈ 68% - 34% = 34%
For 14.5:
P(X = 14.5) ≈ 99.7% - 95% = 4.7%
The probability that a man will have a shoe size between 13 and 14.5 is the sum of these probabilities:
P(13 < X < 14.5) = P(X = 13) + P(X = 14.5) ≈ 34% + 4.7% = 38.7%
Therefore, the correct answer is:
A. 34%