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Part 1
A simple random sample of size n equals 40 is drawn from a population. The sample mean is found to be x overbar equals 121.6 and the sample standard deviation is found to be s equals 12.9. Construct a​ 99% confidence interval for the population mean.

To construct a 99% confidence interval for the population mean, we'll follow these steps:

### Given:
- Sample size, \( n = 40 \)
- Sample mean, \( \bar{x} = 121.6 \)
- Sample standard deviation, \( s = 12.9 \)
- Confidence level = 99%

### Step 1: Determine the critical value (\( t \)-value)
Since the sample size is \( n = 40 \) (which is less than 30), and the population standard deviation is unknown, we'll use the \( t \)-distribution. For a 99% confidence level and \( n-1 = 39 \) degrees of freedom, the critical value \( t \) can be found using a \( t \)-distribution table or calculator.

The approximate \( t \)-value for 39 degrees of freedom at a 99% confidence level is \( t \approx 2.708 \).

### Step 2: Calculate the standard error (SE)
\[
SE = \frac{s}{\sqrt{n}} = \frac{12.9}{\sqrt{40}} \approx 2.04
\]

### Step 3: Construct the confidence interval
\[
\bar{x} \pm t \times SE
\]
\[
121.6 \pm 2.708 \times 2.04
\]
\[
121.6 \pm 5.53
\]
This results in the interval:
\[
(116.07, 127.13)
\]

### Conclusion
The 99% confidence interval for the population mean is approximately \( (116.07, 127.13) \).

Would you like more details or have any questions?

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Here are 5 related questions you might explore:
1. What is the 95% confidence interval for the same data?
2. How does the confidence interval change if the sample size increases to 100?
3. How does using the \( z \)-distribution instead of the \( t \)-distribution affect the interval?
4. What would happen if the sample standard deviation were larger?
5. How is the \( t \)-value determined for different sample sizes?

**Tip:** For small sample sizes (\( n < 30 \)), always use the \( t \)-distribution rather than the normal \( z \)-distribution, especially when the population standard deviation is unknown.

Learn how to construct a 99% confidence interval for the population mean using sample data with a detailed step-by-step solution. This problem covers statistical concepts such as confidence intervals, standard error calculation, and the use of the t-distribution. Suitable for college-level students and researchers.

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