The serum cholesterol levels in women aged 20 to 30 years old are normally distributed with a mean of 176.5 and a standard deviation of 15. Units are in mg/100 mL.

If 32 women aged 20 to 30 years old are randomly selected, find the probability that their mean serum cholesterol level is more than 182.

To solve this problem, we'll use the **Central Limit Theorem**. According to the Central Limit Theorem, for a large enough sample size, the sampling distribution of the sample mean will be normally distributed even if the population distribution is not. The sample mean will have the same mean as the population, and its standard deviation will be equal to the population standard deviation divided by the square root of the sample size.

### Given Information:
- Mean (\(\mu\)) = 176.5 mg/100 mL
- Population standard deviation (\(\sigma\)) = 15 mg/100 mL
- Sample size (\(n\)) = 32 women
- We are asked to find the probability that the mean cholesterol level of the sample is more than 182 mg/100 mL.

### Step 1: Find the standard error of the mean (SEM)

The standard error of the mean is calculated as:
\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{32}} = \frac{15}{5.657} \approx 2.653
\]

### Step 2: Convert the sample mean to a z-score

The z-score tells us how many standard deviations away from the population mean the sample mean is. The formula for the z-score of the sample mean is:
\[
z = \frac{\bar{x} - \mu}{\text{SEM}} = \frac{182 - 176.5}{2.653} = \frac{5.5}{2.653} \approx 2.073
\]

### Step 3: Find the probability

Now, using the z-score of \(2.073\), we look up the corresponding probability in the standard normal distribution table. The area to the left of \(z = 2.073\) is approximately 0.9819.

Since we are asked to find the probability that the mean is **more than** 182 mg/100 mL, we need the area to the right of \(z = 2.073\):
\[
P(\bar{x} > 182) = 1 - 0.9819 = 0.0181
\]

### Final Answer:
The probability that the mean serum cholesterol level of the 32 women is more than 182 mg/100 mL is approximately **0.0181**, or **1.81%**.

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Would you like any more details or have questions about this solution?

Here are five related questions you could explore:
1. How would the probability change if the sample size was increased to 50?
2. What is the probability that the mean serum cholesterol level is less than 170 mg/100 mL?
3. What is the z-score for a sample mean of 190 mg/100 mL?
4. If we randomly select one woman, what is the probability her cholesterol is over 182 mg/100 mL?
5. How does the Central Limit Theorem apply to this problem?

**Tip:** In normal distribution problems, converting values to z-scores helps in determining probabilities using the standard normal distribution table.

The serum cholesterol levels in women aged 20 to 30 years old are normally distributed with a mean of 176.5 and a standard deviation of 15. Units are in mg/100 mL. If 32 women aged 20 to 30 years old are randomly selected, find the probability that their mean serum cholesterol level is more than 182.

Discover how to calculate the probability that the mean serum cholesterol level for women aged 20-30 is more than 182 mg/100 mL using the Central Limit Theorem. This problem covers statistical concepts such as normal distribution, standard error of the mean, and z-scores. Ideal for students in Grades 10-12.

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