Failed Uniformly Most Powerful Test

Question

Suppose $X_1,...,X_n$ is a random sample from $N(\theta,1)$. We want to test $H_0: \theta = \theta'$ vs $H_1: \theta \ne \theta'$.

I know that since there are 2 critical region for $H_1$, there is no UMPT for testing $H_0: \theta = \theta'$ vs $H_1: \theta \ne \theta'$

How do I find the best critical region for this hypotheses? What method can I use?

Thanks

Answer 1

As per the fact that we are sampling from a Gaussian population, for this two-sided test there is a very simple procedure: the "Confidence Interval Method"

so you reject $H_0$ iff the Statistic Test $\bar{X}_n$ is out of the interval and, on the countrary, you will not reject it if the Statistic test is in.

So, as an example, you will not reject the Hypotesis that $\theta=\theta_0$ if and only if

$$ \bbox[5px,border:2px solid black] { \theta_0-\frac{z}{\sqrt{n}}\leq \bar{X}_n \leq \theta_0+\frac{z}{\sqrt{n}} \qquad } $$

Of course I set

$z=z_{(1-\frac{\alpha}{2})} $