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1
Question:

A researcher selects a random sample of 100 men age 18-24 and determines their mean serum total cholesterol is 180 mg/dL with a standard deviation of 40 mg/dL.  According to national statistics, cholesterol levels for the population of men age 18-24 follow a normal (gaussian) distribution.  Based on this information, approximately 50% of the men in the sample will have which of the following serum total cholesterol levels?

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Explanation:

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A normal distribution refers to a symmetric, bell-shaped distribution pattern with a fixed proportion of observations lying within specific distances from the mean.  For a normal distribution, mean equals median.  The median divides a distribution of data in half, so for a normal distribution about 50% of observations lie below the mean and 50% of observations lie above the mean.

In this case, a researcher took a random sample of men age 18-24 and estimated their mean serum total cholesterol at 180 mg/dL with a standard deviation of 40 mg/dL.  Therefore, approximately 50% of men age 18-24 in the sample will have serum total cholesterol levels <180 mg/dL.

(Choices A, B, C, and E)  According to the 68-95-99.7 rule for normal distributions, the distribution of serum total cholesterol levels should have the following pattern:

  • 68% of serum total cholesterol values will be within 1 SD from the mean, calculated as being between [180 − (1 × 40)] = 140 mg/dL and [180 + (1 × 40)] = 220 mg/dL, and (100% − 68%) = 32% will be <140 mg/dL or >220 mg/dL.

  • 95% of serum total cholesterol values will be within 2 SDs from the mean, calculated as being between [180 − (2 × 40)] = 100 mg/dL and [180 + (2 × 40)] = 260 mg/dL, and (100% − 95%) = 5% will be <100 mg/dL or >260 mg/dL.

  • 99.7% of serum total cholesterol will be within 3 SDs from the mean, calculated as being between [180 − (3 × 40)] = 60 mg/dL and [180 + (3 × 40)] = 300 mg/dL, and (100% − 99.7%) = 0.3% will be <60 mg/dL or >300 mg/dL.

Educational objective:
In a normal distribution, the mean is equal to the median; 50% of observations lie below the mean and 50% of observations lie above the mean.