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. (1 point) Responses 2.35 2.35 4.5 4.5 1.2 1.2 2
To find the probability that a man will have a shoe size between 13 and 14.5, first we need to find the z-scores for both sizes.
For a shoe size of 13:
z = (13 - 10) / 1.5 = 2 / 1.5 = 1.33
For a shoe size of 14.5:
z = (14.5 - 10) / 1.5 = 4.5 / 1.5 = 3
Now, using the empirical rule:
- Approximately 68% of the data falls within one standard deviation of the mean (from z = -1 to z = 1).
- Approximately 95% of the data falls within two standard deviations of the mean (from z = -2 to z = 2).
- Approximately 99.7% of the data falls within three standard deviations of the mean (from z = -3 to z = 3).
Since the z-scores for shoe sizes 13 and 14.5 are both within 3 standard deviations of the mean, we can apply the empirical rule.
The probability of a man having a shoe size between 13 and 14.5 is approximately 99.7%.