Test of the Equality of Proportions. Let \(X_{1}\) and \(X_{2}\) be independent random variables with Binomial distributions \(\operatorname{Binomial}\left(n_{i}, p_{i}ight), i=1,2\), where the \(n_{i}\) 's are assumed known. Consider testing the null hypothesis that the proportions (or probabilities) \(p_{1}\) and \(p_{2}\) are equal, i.e., the hypothesis is \(H_{0}: p_{1}=p_{2}\) versus \(H_{a}: p_{1} eq p_{2}\).

(a) Define an asymptotically valid size \(\alpha\) Wald-type test of \(H_{0}\) versus \(H_{a}\).

(b) If \(n_{1}=45, n_{2}=34, x_{1}=19\), and \(x_{2}=24\), is the hypothesis of equality of proportions rejected if \(\alpha=.10\) ?

(c) Define an asymptotically valid size \(\alpha\) GLR-type test of \(H_{0}\) versus \(H_{a}\). Repeat part (b).