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 pˆ 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 plusfour 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 12.1 on page 570 to the new data. In other words, in place of pˆ (which is x/n), we use p˜ = (x + 2)/(n + 4). Consequently, for a confidence level of 1 − α, the endpoints of the plus-four z-interval are p˜ ± zα/2 ·

p˜(1 − 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.