Discussion
A test \(\phi(x)\) is called an unbiased test if :
Answer: C
A test is unbiased if \(E_{H_{0}}\phi(X) \le \alpha\) and \(E_{H_{1}}\phi(X) \ge \alpha\).
Unbiasedness means the test rejects at least as often when the alternative is true as when the null is: its size is at most \(\alpha\) and its power is at least \(\alpha\) at every alternative. Otherwise there would be an alternative under which the test is less likely to reject than under the null, which is perverse.
Each of options A and B states only one half (A is just the size condition). Option D demands power equal to \(\alpha\), which describes a useless test that ignores the data, not an unbiased one.
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