Binomial data gathered from more than one population are often presented in a contingency table. For the case of two populations, the table might look like this:

where Population 1 is binomial(n1, p1), with S1 successes and F1 failures, and Population 2 is binomial(n2, p2), with S2 successes and F2 failures. A hypothesis that is usually of interest is
H0: p1 = p2 versus H1: p1 ‰  p.
(a) Show that a test can be based on the statistic

where 1 = S1/n1, 2 = S2/n2, and = (S1 + S2)/(n1 + n2). Also, show that as n1, n2 †’ ˆž, the distribution of T approaches X21. (This is a special case of a test known as a chi squared test of independence.)
(b) Another way of measuring departure from H0 is by calculating an expected frequency table. This table is constructed by conditioning on the marginal totals and filling in the table according to H0: p1 = p2, that is,

Using the expected frequency table, a statistic T* is computed by going through the cells of the tables and computing

Show, algebraically, that T* = T and hence that T* is asymptotically chi squared,
(c) Another statistic that could be used to test equality of p1 and p2 is

Show that, under H0, T** is asymptotically n(0, 1), and hence its square is asymptotically X21. Furthermore, show that (T**)2 ‰  T*.
(d) Under what circumstances is one statistic preferable to the other?
(e) A famous medical experiment was conducted by Joseph Lister in the late 1800s. Mortality associated with surgery was quite high, and Lister conjectured that the use of a disinfectant, carbolic acid, would help. Over a period of several years Lister performed 75 amputations with and without using carbolic acid. The data are given here:

Use these data to test whether the use of carbolic acid is associated with patient mortality.

Population 2Total FailuresFF2F-FF2 Total n n nn pi P2 Expected frequencies Total niS ni n2 Successes 1 n2 n2F ni +n2 n2 FailuresFF F2 ni n2 Total ni n=n1 + n2 (observed- expected)2 expected n S in2 Si 2n1+12 n+n2 T". = P1 P2 n1 Carbolic acid used? Yes No Patient lived? Ye 34 19 No 616