Question

In discussing the sample size required for obtaining a confidence interval with a prescribed confidence level and margin of error, we made the following statement: "If we have in mind a likely range for the observed value of $\hat{p}$, then, in light of Fig. $11.1$, we should take as our educated guess for $\hat{p}$the value in the range closest to $0.5$"Explain why.

Short answer

As a result, in order to get a sample size appropriate for every $p$in a given range, the largest sample that may arise for any $p$in the range should be chosen. This is accomplished by selecting the $p$value that falls within the range and is closest to $0.5$.

Step-by-step solution

Given Information

"If we have a likely range in mind for the observed value of $\hat{p}$, then, in light of Fig. $11.1$, we should select the value in the range closest to $0.5$as our informed prediction for$\hat{p}$."

Explanation

The sample size $p(1-p)$is proportional to the quantity.

When $p=0.5$, the quantity $p(1-p)$has a maximum value of $0.25$.

As $p$approaches $0.5$from either above or below $0.5$, the sample size grows.