The confidence interval for a population proportion depends on the central limit theorem. A common rule of thumb is that to use the normal approximation for the sampling distribution for ˆp, you should have at least 10 “successes” and 10 “failures.” However, Agresti and Coull developed a method that can be used for smaller samples and increases the accuracy of all confidence intervals for proportions. The idea is simply to add 4 “pseudo observations” to the data set—2 successes and 2 failures. That is, if you have X successes from of n trials, use p = X +2 / n +4 instead of the usual p̂ = X/n = in the formulas for the confidence interval. A quality control engineer is inspecting defects on a newly designed pr nted circuit board. She inspects 50 boards and finds no defects. The usual estimate would beˆp = 0, but she does not believe that there will ever be a nodefects situation for this product. Use this Agresi-Coull estimate to come up with a 95% confidence interval for the true proportion of defects.