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The problem is asking to construct a confidence interval for the population mean \( \mu \) using the t-distribution. The details provided in the image are:

- Confidence level \( c = 0.95 \)
- Sample mean \( \bar{x} = 12.3 \)
- Sample standard deviation \( s = 3.0 \)
- Sample size \( n = 7 \)

### Steps to calculate the confidence interval:

1. **Degrees of freedom**: 
   \[
   df = n - 1 = 7 - 1 = 6
   \]

2. **Find the critical value**: Since the confidence level is 95%, we will look for the critical value \( t_{\alpha/2} \) for 6 degrees of freedom. Using a t-table or calculator, the critical value for \( \alpha = 0.05 \) (two-tailed) and \( df = 6 \) is approximately:
   \[
   t_{\alpha/2} \approx 2.447
   \]

3. **Margin of error**: 
   \[
   \text{Margin of error} = t_{\alpha/2} \times \frac{s}{\sqrt{n}} = 2.447 \times \frac{3.0}{\sqrt{7}} = 2.447 \times 1.134 = 2.776
   \]

4. **Confidence interval**:
   The confidence interval is given by:
   \[
   \bar{x} \pm \text{Margin of error} = 12.3 \pm 2.776
   \]
   So, the confidence interval is:
   \[
   (12.3 - 2.776, 12.3 + 2.776) = (9.5, 15.1)
   \]

Thus, the 95% confidence interval for the population mean is \( (9.5, 15.1) \).

Would you like more details on any step? Here are some related questions to explore further:

1. How is the t-distribution different from the normal distribution in constructing confidence intervals?
2. How do you find the critical value for any confidence level?
3. Why is the sample size important in confidence interval calculations?
4. What happens to the confidence interval as the sample size increases?
5. How would you adjust this method if the sample size were larger?

**Tip**: Always ensure that you use the correct degrees of freedom when looking up the t-critical value in the table.

Construct the indicated confidence interval for the population mean µ using the t-distribution. Assume the population is normally distributed. Given c = 0.95, x̄ = 12.3, s = 3.0, n = 7.

Learn how to construct a 95% confidence interval for the population mean using the t-distribution. With a sample mean of 12.3, standard deviation of 3.0, and a sample size of 7, this problem demonstrates how to apply statistical methods to estimate population parameters.

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