Discussion
The distributions of X under \(H_{0}\) and \(H_{1}\) are given by :
| \(x\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(H_{0}\) | 0.25 | 0.25 | 0.25 | 0.25 |
| \(H_{1}\) | 0.17 | 0.17 | 0.17 | 0.32 |
The most powerful test at level \(\alpha = 0.05\) is given by :
Answer: C
The most powerful test rejects \(H_{0}\) with probability 0.2 if X = 4.
By the Neyman-Pearson lemma, reject where \(P_{1}(x)/P_{0}(x)\) is largest. The ratio is \(0.17/0.25 = 0.68\) at \(x = 1, 2, 3\) and \(0.32/0.25 = 1.28\) at \(x = 4\), so \(x = 4\) comes first. Rejecting at \(x = 4\) always would give size 0.25, too large, so randomise there:
\[ 0.25\gamma = 0.05 \Rightarrow \gamma = 0.2 \]
The non-randomised test in A has size 0.25, not 0.05. Rejecting with probability 0.25 or 0.32 gives sizes 0.0625 and 0.08, also above the level.
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