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: "... we should be aware that, if the observed value of $\hat{p}$is closer to $0.5$than is our educated guess, the margin of error will be larger than desired." Explain why.

One-Proportion Plus-Four z-Interval Procedure. To obtain a plus four $z$-interval for a population proportion, we first add two successes and two failures to our data (hence, the term "plus four") and then apply Procedure $11.1$on page $454$to the new data. In other words, in place of $\hat{p}$(which is $x/n$), we use $\tilde{p}=(x+2)/(n+4)$. Consequently, for a confidence level of $1-\alpha$, the endpoints of the plus-four $z$-interval are

$\tilde{p}\pm za/2·\sqrt{\tilde{p}(1-\tilde{p})/(n+4)}$

As a rule of thumb, the one-proportion plus-four $z$-interval procedure should be used only with confidence levels of $90\%$ or greater and sample sizes of $10$ or more.

Short answer

The margin of error will be bigger than expected because the observed value is closer to $0.5$than our educated guess.

Step-by-step solution

Given Information

"The margin of error will be bigger than expected if the observed value is closer to $0.5$than our educated guess."

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.

To get a sample size that is appropriate for any $p$in a given range, choose the largest sample size that is possible for any $p$in the range. This is accomplished by selecting the $p$value that falls within the range and is closest to$0.5$.

The margin of error of the estimate of population proportion, $p$, is given as $-E=z_{a/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$