Let X1 and X2 be binomial random variables with respective parameters n1, p1 and n2, p2. Show that when n1 and n2 are large, an approximate level α test of H0 : p1 = p2 versus H1 : p1 = p2 is as follows:

Hint:

(a) Argue first that when n1 and n2 are large

where ∼˙ means “approximately has the distribution.”

(b) Now argue that when H0 is true and so p1 = p2, their common value can be best estimated by (X1 + X2)/(n1 + n2).

reject Ho if X +X ni+n2 |X/n - X2/12\ (+) (+) > 2a12