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

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

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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 $$ p $$.
The 90% confidence interval is wider than the 99% confidence interval.

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

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