Step 1 :We are given that the mean height for women in a certain ethnic group is 5 feet 8 inches, which is equivalent to 68 inches. The standard deviation is 2 inches. We are asked to find out how many women in a group of 686 women from this group would be expected to be more than 6 feet tall, which is equivalent to 72 inches.
Step 2 :We can use the empirical rule, also known as the 68-95-99.7 rule, to solve this problem. This rule states that for a normal distribution, 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
Step 3 :In this case, a height of 72 inches is two standard deviations above the mean. According to the empirical rule, 95% of the data falls within two standard deviations of the mean, so we would expect 5% of the women to be more than six feet tall.
Step 4 :We can calculate this number by multiplying the total number of women (686) by 0.05. The calculation is as follows: \(686 \times 0.05 = 34.3\)
Step 5 :Since we can't have a fraction of a person, we round this number to the nearest whole number. So, the expected number of women over six feet tall is \(\boxed{34}\)