[10 marks] To determine the average age of its customers, a large manufacturer of
men’s clothing took a random sample of 50 customers and found 𝑥̅ = 36. Assume that
the customer’s ages are independent normal random variables with unknown mean 𝜇
and population standard deviation 12. Find a 95% confidence interval for the mean age
𝜇 of a customer. What happens if the customer’s ages are not normal variables?

To determine the 95% confidence interval for the mean age \( \mu \) of the customers, we'll follow these steps:

### Given Data:
- Sample mean (\(\bar{x}\)): 36
- Population standard deviation (\(\sigma\)): 12
- Sample size (\(n\)): 50
- Confidence level: 95%

### Steps to Calculate the 95% Confidence Interval:

1. **Identify the Z-value for a 95% Confidence Interval:**
   For a 95% confidence level, the critical value (Z-value) from the standard normal distribution is approximately 1.96.

2. **Calculate the Standard Error of the Mean (SE):**
   \[
   SE = \frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{50}} = \frac{12}{7.071} \approx 1.697
   \]

3. **Calculate the Margin of Error (ME):**
   \[
   ME = Z \times SE = 1.96 \times 1.697 \approx 3.326
   \]

4. **Calculate the Confidence Interval:**
   \[
   \text{CI} = \bar{x} \pm ME = 36 \pm 3.326
   \]
   \[
   \text{CI} = [36 - 3.326, 36 + 3.326] = [32.674, 39.326]
   \]

### Conclusion:
The 95% confidence interval for the mean age \(\mu\) of the customers is approximately \( [32.674, 39.326] \).

### What Happens if the Customers' Ages Are Not Normal Variables?

If the customers' ages are not normally distributed, the following considerations apply:

1. **Central Limit Theorem (CLT):**
   For a large sample size (typically \( n \geq 30 \)), the CLT states that the sampling distribution of the sample mean will be approximately normal regardless of the distribution of the population. Since our sample size is 50, the CLT supports the use of the normal approximation.

2. **Non-normal Population Distribution:**
   - If the population distribution is heavily skewed or has outliers, the confidence interval based on the normal distribution may not be accurate.
   - In such cases, alternative methods such as bootstrapping (a non-parametric method) can be used to construct the confidence interval.

### Summary:

For the given data, the 95% confidence interval for the mean age \(\mu\) of the customers is approximately \( [32.674, 39.326] \). If the ages are not normally distributed, the large sample size (50) allows us to use the normal approximation due to the Central Limit Theorem. However, for heavily skewed distributions or outliers, non-parametric methods may provide more accurate results.

Do you want details on any specific part or have any questions?

### Follow-Up Questions:

1. How does the confidence interval change if the sample size increases?
2. What is the impact of a larger population standard deviation on the confidence interval?
3. Can you explain the Central Limit Theorem in more detail?
4. How would you calculate the confidence interval if the sample size were smaller than 30?
5. What are some examples of non-parametric methods for constructing confidence intervals?
6. How does the choice of confidence level (e.g., 90%, 99%) affect the interval?
7. What is the interpretation of a confidence interval in practical terms?
8. How can outliers affect the confidence interval calculation?

### Tip:

When interpreting confidence intervals, remember that the interval provides a range within which we are confident the true population parameter lies, but it does not guarantee that the parameter is within that range for every sample. Always consider the context and any potential biases in your data.

Learn how to calculate the 95% confidence interval for the mean age of customers using a normal distribution. Example problem with detailed steps, including formulas for SE and ME. Suitable for advanced undergraduate students and professionals.

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