21.42 Plus four confidence intervals for a proportion. The large-sample confidence interval for a sample proportion p is easy to calculate. It is also easy to understand because it rests directly on the approximately Normal distribution of . Unfortunately, confidence levels from this interval can be inaccurate, particularly with smaller sample sizes where there are only a few successes or a few failures. The actual confidence level is usually less than the confidence level you asked for in choosing the critical value z*. That’s bad. What is worse, accuracy does not consistently get better as the sample size increases. There are “lucky’’ and “unlucky’’ combinations of the sample size and the true population proportion p.

Fortunately, there is a simple modification that is almost magically effective in improving the accuracy of the confidence interval. We call it the “plus four’’ method because all you need to do is add four imaginary observations, two successes and two failures. With the added observations, the plus four estimate of p is

The formula for the confidence interval is exactly as before, with the new sample size and number of successes. To practice using the plus four confidence interval, consider the following problem. Cocaine users commonly snort the powder up the nose through a rolled-up paper currency bill. Spain has a high rate of cocaine use, so it’s not surprising that euro paper currency in Spain often shows traces of cocaine. Researchers collected 20 euro bills in each of several Spanish cities. In Madrid, 17 out of 20 bore traces of cocaine. The researchers note that we can’t tell whether the bills had been used to snort cocaine or had been contaminated in currency-sorting machines. Use the plus four confidence interval method to estimate the proportion of all euro bills in Madrid that have traces of cocaine.