Discussion
To test \(H_{0} : p = p_{0}\) against the alternative \(H_{1} : p \ne p_{0}\), where \(p\) is the population proportion and \(p_{0}\) is a specified value of \(p\). Suppose \(P_{n}\) denotes the sample proportion based on a random sample of size \(n\) from the population. The test statistic for testing \(H_{0} : p = p_{0}\) against \(H_{1} : p \ne p_{0}\) is given by :
Answer: A
The test statistic is \(\dfrac{\sqrt{n}(P_{n} - p_{0})}{\sqrt{p_{0}(1 - p_{0})}}\).
Under \(H_{0}\), \(P_{n}\) has mean \(p_{0}\) and variance \(p_{0}(1 - p_{0})/n\). Standardising,
\[ Z = \frac{P_{n} - p_{0}}{\sqrt{p_{0}(1 - p_{0})/n}} \]
which is approximately N(0, 1) for large \(n\) and is compared with \(\pm z_{\alpha/2}\). This is option A after multiplying top and bottom by \(\sqrt{n}\).
Option B is not divided by the standard deviation, option C leaves out the \(\sqrt{n}\), and option D divides by the variance instead of its square root.
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