When the sample size and the sample proportion \( \overline{\text { premain the same, a } 90} \) percent confidence interval for a population proportion \( p \) will be the 99 percent confidence interval for \( p \). Multiple Choice wider than equal to

To determine the relationship between a 90% confidence interval and a 99% confidence interval for a population proportion \( p \), we need to understand how confidence intervals are constructed. 1. **Confidence Level**: The confidence level indicates the degree of certainty that the interval contains the true population parameter. A higher confidence level means that we are more certain that the interval contains the true parameter. 2. **Margin of Error**: The margin of error in a confidence interval is influenced by the critical value associated with the confidence level. For a given sample size and sample proportion, the margin of error is calculated as: \[ \text{Margin of Error} = z \cdot \sqrt{\frac{\overline{p}(1 - \overline{p})}{n}} \] where \( z \) is the z-score corresponding to the desired confidence level, \( \overline{p} \) is the sample proportion, and \( n \) is the sample size. 3. **Critical Values**: - For a 90% confidence level, the critical value \( z \) is approximately 1.645. - For a 99% confidence level, the critical value \( z \) is approximately 2.576. 4. **Comparison**: - Since the critical value for the 99% confidence interval (2.576) is greater than that for the 90% confidence interval (1.645), the margin of error for the 99% confidence interval will be larger. Thus, the 99% confidence interval will be wider than the 90% confidence interval. ### Conclusion: The correct answer is that the 90% confidence interval for a population proportion \( p \) will be **wider than** the 99% confidence interval for \( p \).