Final answer:
Using the Empirical Rule and the given mean and standard deviation for American women's shoe sizes, we find that approximately 66% of American women have shoe sizes greater than 6.927.
Step-by-step explanation:
The students question pertains to the Empirical Rule, which is a statistical rule stating that for a bell-shaped distribution:
- About 68% of the data falls within one standard deviation of the mean.
- About 95% within two standard deviations.
- Over 99% within three standard deviations.
We're given that the mean shoe size of American women is 8.46, and the standard deviation is 1.54. To find the percentage of American women with shoe sizes greater than 6.927, we first determine how many standard deviations away from the mean this value is.
To do this, we calculate the difference between 6.927 and the mean, then divide by the standard deviation: (8.46 - 6.927) / 1.54 ≈ 0.996. This means 6.927 is just under one standard deviation below the mean. Using the Empirical Rule, we can say that approximately 68% of the data falls between one standard deviation below the mean and one standard deviation above the mean, so approximately half of this percentage, or 34%, would fall below the mean. Since the total percentage must add up to 100%, to find the portion greater than 6.927, we subtract this 34% from 100%, giving us 100% - 34% = 66%. Therefore, approximately 66% of American women have shoe sizes greater than 6.927.