Question

question 9 of 10
which confidence level will produce the widest confidence interval, given a sample proportion of 0.4?
a. 68%
b. 95%
c. 90%
d. 99.7%

Explanation:

Step1: Recall Confidence Interval Concept

The formula for a confidence interval for a proportion is \( \hat{p} \pm z^\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \), where \( z^* \) is the critical value. A higher confidence level means a larger \( z^* \) (since we need to capture more of the distribution), which increases the margin of error (\( z^\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)) and thus widens the interval.

Step2: Compare Confidence Levels

Among the options: 68% < 90% < 95% < 99.7%. The highest confidence level here is 99.7%, so it will have the largest \( z^* \) and the widest interval.

Answer:

D. 99.7%