
 
3. The serum cholesterol levels of a population of 18-year-olds follow a normal distribution with a mean of 178 mg/dL and a standard deviation of 32 mg/dL. 

a) What proportion of the 18-year-olds have serum cholesterol values between 141 mg/dL and 196 mg/dL? 

b) Suppose we were choose at random from the population a large number of five 18-year-olds each. In what proportion of the groups would the group mean cholesterol value be between 141 mg/dL and 196 mg/dL? 

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?

Explore the normal distribution and Central Limit Theorem in analyzing cholesterol levels of 18-year-olds. Calculate the proportion of individuals with serum cholesterol levels between 141 mg/dL and 196 mg/dL and the proportion of groups with mean cholesterol levels in the same range. Suitable for undergraduate level study.

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