Question

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

Short answer

The confidence interval is wider for a confidence level of 95% when compared to 90%. When making a comparison based on the sample size, the confidence interval is wider for a sample size of 100 when compared to a sample size of 400.

Step-by-step solution

Analyze the Impact of Confidence Level

When comparing a 90% confidence level to a 95% confidence level, the one that results in a wider confidence interval is the 95% confidence level. The higher the confidence level, the wider the interval. This is because to increase our confidence that the interval contains the unknown population parameter, we have to consider more values, thus enlarging the interval.

Analyze the Impact of Sample Size

When comparing a sample size (n) of 100 to a sample size of 400, the one that results in a wider confidence interval is when the sample size is 100. The larger the sample size, the narrower the confidence interval. This is because with more data, we have more information about the population parameter and hence less uncertainty, which is reflected by a narrower interval.