Past exam of the options II course of the University of Oxford / 2004 / O / O4 / a

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- Solution a

Solution

The likelihood ratio for the two binomial probabilities is

\frac{f _{3/4} ( y )}{f _{1/4} ( y )} = 3^{2 y - 5},

strictly increasing in

y

. By the Neyman-Pearson lemma, reject in the highest upper tail. Under the null,

P_{1/4} (Y \geq 4) = \frac{5 \cdot 3 + 1}{4 ^{5}} = \frac{16}{1024} = \frac{1}{64} .

Hence reject exactly when $Y \geq 4$ , with no randomization required. Its statistical power at the stated alternative is

P_{3/4} (Y \geq 4) = \frac{5 \cdot 3 ^{4} + 3 ^{5}}{4 ^{5}} = \frac{81}{128} .

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