The small-sample confidence interval for comparing two proportions is a simple adjustment of the large-sample one. Recall that for a small-sample confidence interval for a single proportion, we used the ordinary formula after adding four observations; two of each type (see the end of Section 8.4). In the two-sample case we also add four observations, two of each type, and then use the ordinary formula, by adding one observation of each type to each sample. 19 When the proportions are near 0 or near 1, results are more sensible than with the ordinary formula. Suppose there are no successes in either group, with n1 = n2 = 10.
a. With the ordinary formula, show that
(i) p̂1 = p̂2 = 0,
(ii) se = 0, and
(iii) The 95% confidence interval for p1 - p2 is 0 { 0, or (0, 0). Obviously, it is too optimistic to predict that the true difference is exactly equal to 0.
b. Find the 95% confidence interval using the small sample method described here. Are the results more plausible?