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I have the following problem. Given a drug test, with $N$ people, $k$ have failed the test. And we assume a priori that the distribution of the test failing is a $\beta(p=1,q=6)$.

Then the bayes estimator for the quadratic loss would be: $$ \hat\theta=\frac{k+1}{N+7} $$

The question is the following. We are asked to compare the null hypothesis $H_0:\hat\theta\geq0.15$ wuth $H_1:\hat\theta<0.15$.

How should I do this? For example, which test should I use?

It's also worth noting, that upon observation, $N=176$ and $k=24$.

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