Question

Discuss how each of the following factors affects the width of the confidence interval for \(\pi\) : a. The confidence level b. The sample size c. The value of \(p\)

Short answer

The confidence level and the width of the confidence interval are directly proportional — as one increases, so does the other. Sample size is inversely proportional to the width of the confidence interval — as sample size increases, the interval decreases. The value of \(p\) affects the interval width, but not linearly — the width is largest when \(p = 0.5\) and gets narrower as \(p\) approaches 0 or 1.

Step-by-step solution

Effect of Confidence Level

The confidence level is correlated with the width of the confidence interval. As the confidence level increases, the z-score also increases. This is because a higher confidence level means the interval needs to capture the true population parameter, \(\pi\), more frequently, so the interval must be wider. Thus, the confidence level is directly proportional to the width of the confidence interval — as one increases, so does the other.

Effect of Sample Size

The sample size has an inverse relationship with the width of the confidence interval. As the sample size increases, the standard error decreases as it is calculated as \( \sqrt{pq/n} \), where \( n \) is the sample size, and \( p \) and \( q \) are the estimated proportions of the population. Because the standard error is in the denominator, a larger sample size means a smaller standard error, and thus a narrower confidence interval.

Effect of Value of \(p\)

The value of \(p\) affects the width of the confidence interval but the relationship is not linear. The standard error is at its maximum when \(p = 0.5\). As \(p\) moves away from 0.5 in either direction (towards 0 or 1), the standard error decreases and so does the width of the confidence interval.