Question

Describe how each of the following factors affects the width of the large- sample confidence interval for \(p\) : a. The confidence level b. The sample size c. The value of \(\hat{p}\)

Short answer

The width of the large-sample confidence interval for \(p\) is influenced by the confidence level, the sample size, and the value of \(\hat{p}\). Increasing the confidence level widens the confidence interval, while increasing the sample size narrows the confidence interval. The value of \(\hat{p}\) influences the width of the interval such that as \(\hat{p}\) approaches 0 or 1 the interval gets narrower, and as \(\hat{p}\) approaches 0.5 the interval gets wider.

Step-by-step solution

Understanding the Effect of Confidence Level

A confidence interval is determined by the standard deviation and the confidence level. The confidence level is reflective of the z-score, or quantile of the Standard Normal distribution. Higher confidence levels would correspond to a higher z-score, and therefore larger/more spread confidence intervals. In essence, increasing the confidence level means you want to be more certain of your results hence a wider range.

Effect of the Sample Size

The sample size plays a role in the component of standard error in the confidence interval formula. Larger sample size decreases the standard error which thus narrows the confidence interval. Therefore, by increasing sample sizes, you are reducing the width of the confidence interval i.e it becomes narrower.

Effect of the Value of \(\hat{p}\)

The value of \(\hat{p}\) also notably affects the width of the confidence interval, as it is a component of the standard error. As \(\hat{p}\) approaches 0 or 1, the confidence interval gets narrower and as \(\hat{p}\) approaches 0.5, the interval gets wider. This is because the most uncertain scenario is when \(\hat{p}=0.5\), in which case the confidence interval will be at its maximum width.