1. **Problem Statement:** Find the reliability factor (critical t-value) from the t-distribution for given confidence coefficients and sample sizes. 2. **Formula and Explanation:** The reliability factor $t_{\alpha/2, n-1}$ is the critical value from the t-distribution with $n-1$ degrees of freedom, where $\alpha = 1 - \text{confidence coefficient}$. It satisfies: $$P(-t_{\alpha/2, n-1} \leq T \leq t_{\alpha/2, n-1}) = \text{confidence coefficient}$$ 3. **Steps:** - Calculate degrees of freedom $df = n - 1$ for each sample size. - Find $\alpha = 1 - \text{confidence coefficient}$. - Find $t_{\alpha/2, df}$ from t-distribution tables or software. 4. **Calculations:** | Case | Confidence Coefficient | Sample Size $n$ | Degrees of Freedom $df$ | $\alpha$ | $\alpha/2$ | Reliability Factor $t_{\alpha/2, df}$ | |-------|------------------------|-----------------|------------------------|----------|------------|-------------------------------------| | a | 0.95 | 15 | 14 | 0.05 | 0.025 | $t_{0.025,14} \approx 2.145$ | | b | 0.99 | 24 | 23 | 0.01 | 0.005 | $t_{0.005,23} \approx 2.807$ | | c | 0.90 | 8 | 7 | 0.10 | 0.05 | $t_{0.05,7} \approx 1.895$ | | d | 0.95 | 30 | 29 | 0.05 | 0.025 | $t_{0.025,29} \approx 2.045$ | 5. **Interpretation:** These $t$ values are the multipliers used in confidence intervals when the population standard deviation is unknown and the sample size is small. **Final answers:** - a) $t = 2.145$ - b) $t = 2.807$ - c) $t = 1.895$ - d) $t = 2.045$