Discussion
A random variable X follows N(\(\mu\), 1) distribution. A conventional test based on sample mean is used to test the hypothesis \(H_{0} : \mu = \mu_{0}\) against \(H_{1} : \mu = \mu_{1} \gt \mu_{0}\). Consider the following statements :
(i) If \(H_{0}\) is rejected at \(\alpha = 0.05\), it is also rejected at \(\alpha = 0.10\).
(ii) If \(H_{0}\) is accepted at \(\alpha = 0.05\), it is also accepted at \(\alpha = 0.01\).
Then :
Answer: C
(i) and (ii) both are correct.
For this one-sided test the rejection region is \(\bar{X} \gt \mu_{0} + z_{\alpha}/\sqrt{n}\). A larger \(\alpha\) gives a smaller critical value \(z_{\alpha}\), so the rejection regions are nested: the region at \(\alpha = 0.05\) lies inside the region at \(\alpha = 0.10\), and the region at \(\alpha = 0.01\) lies inside the one at 0.05.
(i) A sample rejected at 0.05 falls in the larger 0.10 region, so it is rejected there too. (ii) A sample accepted at 0.05 is outside the 0.05 region, hence outside the smaller 0.01 region, so it is accepted at 0.01. Equivalently, H0 is rejected at level \(\alpha\) exactly when the p-value is at most \(\alpha\).
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