Question

For each of the following choices, explain which would result in a wider large-sample confidence interval for \(\pi\) : a. \(90 \%\) confidence level or \(95 \%\) confidence level b. \(n=100\) or \(n=400\)

Short answer

a. The \(95 \%\) confidence level will result in a wider confidence interval than the \(90 \%\) confidence level. b. For \(n=100\), the confidence interval will be wider than for \(n=400\).

Step-by-step solution

Compare the effects of confidence level on the Confidence Interval

The confidence level represents the degree of certainty you have that your confidence interval contains the true population parameter. A higher confidence level corresponds to a larger confidence interval. The reason for this is that to be more confident that you have captured the population parameter, you need to allow for more potential values, which requires a larger interval. Hence, a \(95 \%\) confidence level will result in a wider confidence interval for \(\pi\) than a \(90 \%\) confidence level.

Compare the effects of sample size on the Confidence Interval

The sample size, \(n\), also affects the width of the confidence interval. A smaller sample size results in a larger standard error, which in turn leads to a wider confidence interval as the estimate is less precise. Therefore, for a larger sample size, the confidence interval will be narrower. In this case, when \(n=100\) the confidence interval will be wider in comparison to when \(n=400\).