Question

For each of the following choices, explain which would result in a narrower large-sample confidence interval for \(p\) : a. \(95 \%\) confidence level or \(99 \%\) confidence level b. \(n=200\) or \(n=500\)

Short answer

a) The \(95 \%\) confidence level will result in a narrower large-sample confidence interval for \(p\). b) A sample size of n=500 will result in a narrower large-sample confidence interval for \(p\).

Step-by-step solution

Understanding confidence intervals

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data. The width of a confidence interval depends on the chosen confidence level and the sample variance. A higher confidence level will result in a wider confidence interval.

Analyzing the confidence levels

Given the confidence levels of \(95 \%\) or \(99 \%\) in scenario a), it can be understood that the \(99 \%\) confidence level will provide a wider range as compared to the \(95 \%\) confidence level. This is because a \(99 \%\) confidence level will be more precise and therefore will provide a larger interval range. So, the \(95 \%\) confidence level will result in a narrower large-sample confidence interval for \(p\).

Understanding the Sample Size

In statistics, when sample size increases, it provides more accurate and reliable results. It reduces the margin of error and narrows the confidence interval. This is all because increasing sample size tends to lead to a more precise estimate of the population parameter.

Analyzing the Sample Sizes

Given the two sample sizes of n=200 or n=500 in scenario b), it is clear that a larger sample size will provide more accurate results and hence a narrower confidence interval. So, in this case, n=500 will result in a narrower large-sample confidence interval for \(p\).