To construct a 99% confidence interval for the true population proportion, we need to follow these steps.

---

Step 1: Identify the given values

• Sample proportion ($p$) = 90% = 0.9
• Sample size ($n$) = 514
• Confidence level = 99%
• Critical value ($z_{\alpha/2}$) = ?

---

Step 2: Find the critical value ($z_{\alpha/2}$)

For a 99% confidence level, the significance level $\alpha = 1 - 0.99 = 0.01$. Since it’s two-tailed, we split $\alpha$ into two equal parts: $\alpha/2 = 0.005$. We need the z-score corresponding to the upper 0.995 tail in the standard normal distribution.

Using a Z-table or a calculator:

$z_{\alpha/2} \approx 2.5758$

---

Step 3: Confirm the answer

• Critical value for a 99% confidence interval: 2.5758

---

Let me know if you'd like the full confidence interval calculation or further details! Below are 5 related questions to expand your understanding:

1. How do you interpret the meaning of a confidence interval in statistics?
2. How would the confidence interval change if we used a 95% confidence level instead?
3. How does sample size affect the width of the confidence interval?
4. What assumptions are made when calculating a confidence interval for proportions?
5. How can confidence intervals be used in practical decision-making?

Tip: Critical values differ for each confidence level; always reference the Z-table or a calculator!