Discussion
For testing \(H_{0}\) against \(H_{1}\), if test function \(\phi(\underline{X})\) is UMP, then :
Answer: B
A UMP test is unbiased.
Compare the UMP level-\(\alpha\) test \(\phi\) with the trivial test \(\phi_{0} \equiv \alpha\), which has power exactly \(\alpha\) everywhere. Since \(\phi\) is most powerful among all level-\(\alpha\) tests, its power at every alternative is at least that of \(\phi_{0}\), namely \(\alpha\). Power at least the size under every alternative is the definition of an unbiased test.
That rules out 'not unbiased' and 'power less than its size'. \(\phi \equiv 1\) always rejects, so its size is 1, not \(\alpha\), and it is not a level-\(\alpha\) test.
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