Let's solve the problem using the properties of the normal distribution.

Given:

• Mean ($\mu$) = 178 mg/dL
• Standard deviation ($\sigma$) = 32 mg/dL

Part a) Proportion of 18-year-olds with serum cholesterol values between 141 mg/dL and 196 mg/dL

We need to find $P(141 < X < 196)$, where $X$ is the serum cholesterol level.

1. Calculate the Z-scores for 141 mg/dL and 196 mg/dL: $Z_1 = \frac{141 - 178}{32}$ $Z_2 = \frac{196 - 178}{32}$

2. Find the probabilities corresponding to these Z-scores using the standard normal distribution.

3. The proportion of the population within this range is the difference between the probabilities for $Z_2$ and $Z_1$.

Part b) Proportion of groups where the mean cholesterol value is between 141 mg/dL and 196 mg/dL

Here, we need to find the proportion of groups where the mean cholesterol level of five 18-year-olds falls between 141 mg/dL and 196 mg/dL.

1. Standard Error of the Mean ($\sigma_{\bar{X}}$): $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{32}{\sqrt{5}}$

2. Calculate the Z-scores for the sample mean (141 mg/dL and 196 mg/dL): $Z_1 = \frac{141 - 178}{\sigma_{\bar{X}}}$ $Z_2 = \frac{196 - 178}{\sigma_{\bar{X}}}$

3. Find the proportion as in Part a by determining the difference between the probabilities corresponding to $Z_2$ and $Z_1$.

Let's compute these values.Here are the results:

Part a) Proportion of 18-year-olds with serum cholesterol values between 141 mg/dL and 196 mg/dL

The proportion of 18-year-olds who have serum cholesterol levels between 141 mg/dL and 196 mg/dL is approximately 0.589 (or 58.9%).

Part b) Proportion of groups where the mean cholesterol value is between 141 mg/dL and 196 mg/dL

The proportion of groups (where each group consists of five 18-year-olds) with a mean cholesterol level between 141 mg/dL and 196 mg/dL is approximately 0.891 (or 89.1%).

Summary:

• Part a: About 58.9% of 18-year-olds have serum cholesterol levels between 141 mg/dL and 196 mg/dL.
• Part b: About 89.1% of groups of five 18-year-olds will have a mean cholesterol level within this range.

Would you like any further clarification or have additional questions?