The total cholesterol levels of a sample of men aged 35-44 are normally distributed with a mean of 209 milligrams per deciliter and a standard deviation of 36.7 milligrams per deciliter.

(a) What percent of the men have a total cholesterol level less than 216 milligrams per deciliter of blood?

(b) If 258 men in the 35-44 age group are randomly selected, about how many would you expect to have a total cholesterol level greater than 264 milligrams per deciliter of blood?

#### Solution

Let $X$ denote the total cholesterol level of men aged 35-44. Given that $\mu = 209$ and $\sigma = 36.7$.

$X\sim N(209, 36.7^2)$.

a) The probability that a total cholesterol level less than 216 milligrams per deciliter of blood is

` $$ \begin{aligned} P(X < 216) &=P\bigg(\frac{X-\mu}{\sigma} < \frac{216-209}{36.7}\bigg)\\ &=P\bigg(Z < 0.191\bigg)\\ &= P(Z < 0.191)\\ &=0.5756 \end{aligned} $$ `

b) The probability that a total cholesterol level greater than 264 milligrams per deciliter of blood is

` $$ \begin{aligned} P(X > 264)&=1-P(X < 264)\\ &=1-P\bigg(\frac{X-\mu}{\sigma} < \frac{264-209}{36.7}\bigg)\\ &= 1-P(Z< 1.499)\\ &=1-0.933\\ &=0.067 \end{aligned} $$ `

Thus out of 258 men in the 35-44 age group, the number of men to have a total cholesterol level greater than 264 milligrams per deciliter of blood is $258*0.067 = 17$.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators