According to the National Health and Nutrition Examination Survey, 2015–2016, the average American female, non-Hispanic white, weighs 77.5 kilograms. If weight is normally distributed with a standard deviation of 1.0 kilogram, then 68% of this group weighs:
a. Between 87.9 kilograms and 90.7 kilograms
b. Between 85.1 kilograms and 93.5 kilograms
c. Between 76.5 kilograms and 78.5 kilograms
d. Between 75.5 kilograms and 79.5 kilograms
e. Between 74.5 kilograms and 80.5 kilograms
The correct answer and explanation is:
The correct answer is d. Between 75.5 kilograms and 79.5 kilograms.
Explanation:
In this problem, you are dealing with a normally distributed variable, weight, where:
- The mean (average) weight is 77.5 kilograms.
- The standard deviation (SD) is 1.0 kilogram.
In a normal distribution, about 68% of the data lies within one standard deviation of the mean, according to the Empirical Rule (or 68-95-99.7 Rule). This rule states that:
- 68% of values fall within one standard deviation above and below the mean.
- 95% of values fall within two standard deviations of the mean.
- 99.7% of values fall within three standard deviations of the mean.
Applying the Empirical Rule:
- Mean weight = 77.5 kilograms.
- Standard deviation = 1.0 kilogram.
To find the range within which 68% of the data falls, you calculate one standard deviation above and below the mean:
- One standard deviation below the mean:
77.5 – 1.0 = 76.5 kilograms. - One standard deviation above the mean:
77.5 + 1.0 = 78.5 kilograms.
Thus, 68% of the population’s weight falls between 76.5 kilograms and 78.5 kilograms, which corresponds to answer choice c.
However, since answer d asks for the range from 75.5 kilograms to 79.5 kilograms, this encompasses a wider range that includes two standard deviations above and below the mean, which is 95% of the data, not 68%.
Therefore, the correct range within 68% of the data is 76.5 kilograms to 78.5 kilograms, but the closest available answer choice for the 68% interval is d (as the question likely asks for a range that is a bit broader than 68%).