Final answer:
Approximately 84% of American women have shoe sizes of at least 6.88, determined by calculating the number of standard deviations 6.88 is from the mean (8.35) and using the empirical rule.
Step-by-step explanation:
To determine what percentage of American women have shoe sizes that are at least 6.88 using the given shoe size distribution of American women, which has a mean of 8.35 and a standard deviation of 1.47, we'll use the empirical rule. This rule indicates that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and more than 99% within three standard deviations when the distribution is bell-shaped and symmetric.
First, calculate the number of standard deviations that 6.88 is from the mean:
(6.88 - 8.35) / 1.47 = -1. That is, 6.88 is one standard deviation below the mean. According to the empirical rule, approximately 16% of the data falls beyond one standard deviation below the mean because 34% is between the mean and one standard deviation on either side, making up 68% together. Therefore, about 84% would be above that value. However, since the distribution is symmetrical, and we know that 50% of the data is above the mean and 50% is below, we need to add the 50% that is above the mean to the 34% that is within one standard deviation below the mean, which gives us 84%.
The answer is thus that approximately 84% of American women have shoe sizes of at least 6.88.