Question

a political scientist surveys a random sample of eligible voters and constructs a 95% confidence interval in order to estimate the proportion of eligible voters who favor a new policy. the interval is from 0.48 to 0.56. suppose the political scientist repeats the sampling process by drawing a second random sample of the same size from the same population. the political scientist then constructs a second 95% confidence interval based on the second random sample. given what you have learned about in chapter 21, which one of the following statements is most accurate?the second confidence interval may have a different center and possibly a different width than the first confidence interval.the second confidence interval will likely have the same width as the first confidence interval because the sample sizes and confidence levels are identical.the second confidence interval will have the same center as the first confidence interval but may differ slightly in width.the second confidence interval will have a center equal to the population parameter.the second confidence interval will have a width that is equal to the width of the first confidence interval because the margin of error is a fixed value that never changes.

Explanation:

• The formula for a confidence interval for a proportion is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion, \(z\) is the z - score (determined by the confidence level), and \(n\) is the sample size.
• The width of the confidence interval is \(2z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\). Since the confidence level (which determines \(z\)) and the sample size \(n\) are the same for both samples, the width of the confidence interval depends on \(\hat{p}\) (the sample proportion).
• The center of the confidence interval is \(\hat{p}\). Different samples (even of the same size from the same population) can have different sample proportions \(\hat{p}\).

Answer:

The second confidence interval will likely have the same width as the first confidence interval because the sample sizes and confidence levels are identical.