Discussion
In a normal population \(N(\mu, \sigma^{2})\) with \(\sigma^{2} = 4\), in order to test the null hypothesis \(\mu = \mu_{0}\) against \(H_{1} : \mu = \mu_{1}\), where \(\mu_{1} \gt \mu_{0}\) based on a random sample of size n, the value of K such that \(\bar{X} \gt K\) provides a critical region of size \(\alpha = 0.05\) is :
Answer: B
The critical value is \(\mu_{0} + 3.290/\sqrt{n}\).
Under \(H_{0}\), \(\bar{X} \sim N(\mu_{0}, \sigma^{2}/n)\) with \(\sigma = 2\). The alternative is \(\mu_{1} \gt \mu_{0}\), so the best critical region is the upper tail, and its size must be 0.05:
\[ \begin{aligned} P_{0}(\bar{X} \gt K) &= 0.05 \\ \frac{K - \mu_{0}}{2/\sqrt{n}} &= 1.645 \\ K &= \mu_{0} + \frac{3.290}{\sqrt{n}} \end{aligned} \]
\(\mu_{0} + 1.645/\sqrt{n}\) forgets that \(\sigma = 2\), not 1. The values with 1.96 and 3.92 use the two-sided 5% point, which is wrong for a one-sided alternative.
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