Let's go through each question step-by-step:

1. Find the critical value $z_{\alpha/2}$ for a 98% confidence level.

   For a 98% confidence level, $\alpha = 1 - 0.98 = 0.02$. Thus, $\alpha/2 = 0.01$, and we need the $z$-value that corresponds to 0.01 in the tails. The critical value for a 98% confidence level is approximately: $z_{0.01} \approx 2.33$

2. Find the critical value $z_{\alpha/2}$ for a 91% confidence level.

   For a 91% confidence level, $\alpha = 1 - 0.91 = 0.09$. Thus, $\alpha/2 = 0.045$, and we need the $z$-value that corresponds to 0.045 in the tails. The critical value for a 91% confidence level is approximately: $z_{0.045} \approx 1.70$

3. Find the critical value $z_{\alpha/2}$ for a 99% confidence level.

   For a 99% confidence level, $\alpha = 1 - 0.99 = 0.01$. Thus, $\alpha/2 = 0.005$, and we need the $z$-value that corresponds to 0.005 in the tails. The critical value for a 99% confidence level is approximately: $z_{0.005} \approx 2.575$

4. Given the confidence interval $0.422 < p < 0.446$, find the point estimate $p$.

   The point estimate $p$ is the midpoint of the confidence interval: $p = \frac{0.422 + 0.446}{2} = 0.434$

5. Given the confidence interval $0.843 < p < 0.875$, find the margin of error $E$.

   The margin of error $E$ is half the width of the confidence interval: $E = \frac{0.875 - 0.843}{2} = 0.016$

6. Construct a 98% confidence interval for the proportion $p$ based on the sample.

   • Sample proportion $\hat{p} = \frac{12}{346} \approx 0.0347$.
   • For a 98% confidence level, the critical value $z_{0.01} \approx 2.33$.
   • Standard error $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.0347 \times 0.9653}{346}} \approx 0.0097$.
   • Confidence interval: $\hat{p} \pm z_{0.01} \times SE = 0.0347 \pm 2.33 \times 0.0097$.
   • Calculating the margin: $2.33 \times 0.0097 \approx 0.0226$
   • The confidence interval is: $(0.0347 - 0.0226, 0.0347 + 0.0226) = (0.0121, 0.0573)$

Would you like further explanation on any of these steps or concepts?

Here are some additional related questions:

1. What is the significance of a critical value in a confidence interval calculation?
2. How does the confidence level affect the width of a confidence interval?
3. What is the formula for calculating the margin of error in a proportion confidence interval?
4. Why is the point estimate taken as the midpoint of a confidence interval?
5. How do sample size and standard error influence the margin of error?

Tip: Higher confidence levels yield wider confidence intervals, reflecting greater uncertainty about the exact parameter value.