Discussion
Suppose X ~ B (1, p) distribution, \(0 \lt p \lt 1\). On the basis of a random sample of size n from the distribution of X, we want to test \(H_{0} : p = p_{0}\) against the alternative \(H_{1} : p \neq p_{0}\), where \(p_{0}\) is a specified constant. Then asymptotic null distribution of which of the following test statistics is N(0, 1) ?
Answer: C
The statistic with an asymptotic N(0, 1) null distribution is \(\sqrt{n}(\bar{X}_{n} - p_{0})/\sqrt{p_{0}(1 - p_{0})}\).
Under \(H_{0}\), \(\bar{X}_{n}\) is the mean of n Bernoulli(\(p_{0}\)) variables, with mean \(p_{0}\) and variance \(p_{0}(1 - p_{0})/n\). By the central limit theorem
\[ \begin{aligned} &\frac{\sqrt{n}(\bar{X}_{n} - p_{0})}{\sqrt{p_{0}(1 - p_{0})}} \\ &\to N(0, 1) \end{aligned} \]
The options without \(\sqrt{n}\) shrink to 0, and the one with n in place of \(\sqrt{n}\) blows up, so neither has a N(0, 1) limit. (Replacing \(p_{0}\) by \(\bar{X}_{n}\) under the root would also work, but only together with the factor \(\sqrt{n}\).)
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