Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two failures

The standard confidence intervals for proportions and their differences used in introductory statistics courses have poor performance, the actual coverage probability often being much lower than intended. However, simple adjustments of these intervals based on adding four pseudo observations, half of each type, perform surprisingly well even for small samples. To illustrate, for a broad variety of parameter settings with 10 observations in each sample, a nominal 95% interval for the difference of proportions has actual coverage probability below .93 in 88% of the cases with the standard interval but in only 1% with the adjusted interval; the mean distance between the nominal and actual coverage probabilities is .06 for the standard interval, but .01 for the adjusted one. In teaching with these adjusted intervals, one can bypass awkward sample size guidelines and use the same formulas with small and large samples.