The male shoe sizes within a community are normally distributed with a mean of mu = 10 and a standard deviation of sigma = 1.5 Apply the Empirical Rule to find the probability that a man will have a shoe size between overline 13 and 14.5
To apply the Empirical Rule, we need to calculate the z-scores for shoe sizes of 13 and 14.5 based on the given mean and standard deviation.
For shoe size 13:
z = (13 - 10) / 1.5
z = 2 / 1.5
z = 1.33
For shoe size 14.5:
z = (14.5 - 10) / 1.5
z = 4.5 / 1.5
z = 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. Therefore, the probability of a man having a shoe size between 13 and 14.5 can be calculated using the z-scores calculated above.
To find the probability of a shoe size between 13 and 14.5, we need to find the area under the standard normal curve between z = 1.33 and z = 3.
Using a standard normal distribution table or a calculator, we can find the probabilities associated with these z-scores:
P(1.33 < Z < 3) = P(Z < 3) - P(Z < 1.33)
P(Z < 3) = 0.9987
P(Z < 1.33) = 0.9082
P(1.33 < Z < 3) = 0.9987 - 0.9082 = 0.0905
Therefore, the probability that a man will have a shoe size between 13 and 14.5 is approximately 0.0905 or 9.05%.