1. **Problem:** Derive a confidence interval for the mean $\mu$ when the variance $\sigma^2$ is unknown. 2. **Formula and Explanation:** When $\sigma^2$ is unknown, we use the Student's t-distribution. The confidence interval for $\mu$ is given by: $$\bar{X} \pm t_{\alpha/2, n-1} \frac{S}{\sqrt{n}}$$ where $\bar{X}$ is the sample mean, $S$ is the sample standard deviation, $n$ is the sample size, and $t_{\alpha/2, n-1}$ is the critical value from the t-distribution with $n-1$ degrees of freedom. 3. **Intermediate Work:** - Calculate sample mean: $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$ - Calculate sample variance: $S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$ - Use $t$-distribution because $\sigma^2$ is unknown and $S$ estimates it. 4. **Final Confidence Interval:** $$\left( \bar{X} - t_{\alpha/2, n-1} \frac{S}{\sqrt{n}}, \quad \bar{X} + t_{\alpha/2, n-1} \frac{S}{\sqrt{n}} \right)$$ This interval estimates $\mu$ with $(1-\alpha)100\%$ confidence.