In one ethnic group, the mean height for women is 5 feet 8 inches, with a standard deviation of 2 inches. In a group of 686 women from this group, how many would you expect to be more than six feet tall? (You'll need to interpret the diagram that illustrates the empirical rule.) Round to the nearest whole number.
The number of women from the group expected to be more than six feet tall is
Answer
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 \(\boxed{34}\)
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 \(\boxed{34}\)
