1. **State the problem:** We are given $\frac{\alpha}{2} = 0.025$ and need to find the critical value $z_{\frac{\alpha}{2}}$ rounded to 3 decimal places. 2. **Find the critical value:** Using technology (e.g., statistical software or z-tables), the critical value for $z_{0.025}$ is approximately $1.960$. 3. **Use the formula for minimum sample size:** The formula is $$n = \left( \frac{z_{\frac{\alpha}{2}} \cdot \sigma}{E} \right)^2$$ where: - $z_{\frac{\alpha}{2}}$ is the critical value, - $\sigma$ is the population standard deviation, - $E$ is the margin of error. 4. **Plug in the rounded critical value:** Use $z_{\frac{\alpha}{2}} = 1.960$ (rounded to 3 decimal places) in the formula. 5. **Interpretation:** The minimum sample size $n$ must be calculated using the above formula with the given $\sigma$ and $E$ values (not provided here). --- 6. **Next problem: Construct a 90% confidence interval for the mean number of years worked.** - Sample size $n=37$, sample mean $\bar{x} = 12.3$, sample standard deviation $s=4.2$. - Population standard deviation $\sigma$ unknown. - Population distribution unknown but $n > 30$. 7. **Determine appropriate method:** Since $\sigma$ is unknown and $n > 30$, we use the t-distribution critical value. 8. **Find the t critical value for 90% confidence and df = 36:** Using technology, $t_{0.05,36} \approx 1.688$. 9. **Calculate the confidence interval:** $$CI = \bar{x} \pm t_{\frac{\alpha}{2}, n-1} \cdot \frac{s}{\sqrt{n}}$$ Calculate margin of error: $$E = 1.688 \times \frac{4.2}{\sqrt{37}} = 1.688 \times 0.691 = 1.167$$ 10. **Final confidence interval:** $$12.3 \pm 1.167 = (11.13, 13.47)$$ Rounded to 2 decimal places, the 90% confidence interval is $(11.13, 13.47)$. --- **Summary:** - Critical value $z_{\frac{\alpha}{2}} = 1.960$ - Minimum sample size formula given - 90% confidence interval for mean years worked: $(11.13, 13.47)$ using t-distribution