Suppose that shoe sizes of American women have a bell-shaped distribution with a mean of 8.11
and a standard deviation of 1.46
. Using the empirical rule, what percentage of American women have shoe sizes that are between 5.19
and 11.03
?
First, we need to calculate the z-scores for the shoe sizes of 5.19 and 11.03:
For 5.19:
z = (5.19 - 8.11) / 1.46 = -2.00
For 11.03:
z = (11.03 - 8.11) / 1.46 = 2.00
According to the empirical rule:
- Approximately 68% of the data falls within 1 standard deviation of the mean
- Approximately 95% of the data falls within 2 standard deviations of the mean
- Approximately 99.7% of the data falls within 3 standard deviations of the mean
Since the shoe sizes of 5.19 and 11.03 are both 2 standard deviations away from the mean, we know that approximately 95% of American women fall within that range.
Therefore, the percentage of American women with shoe sizes between 5.19 and 11.03 is approximately 95%.