I failed at the test in statistics recently and I'm preparing to another term. This a problem from the test. I'm translating the problem from Polish; I'm sorry if I'm getting some words wrong.

The problem:

We observe a single variable with the following pdf:

where is an unknown parameter.

(a) Find the uniformly most powerful test with the significance level of to verify the null hypothesis against the alternative hypothesis

(b) Find the power function of the test.

My attempt:

First, I want to find the most powerful test with the level of significance of 0.05 to verify the hypothesis against the alternative hypothesis where is a fixed number larger than 0. This I want to achieve by using the Neyman-Pearson lemma. Then I want to say that this test is also most powerful against which I would like to be implied by the arbitrarity of and a simple reasoning. I'm not entirely convinced about this simple reasoning. It comes from my book and althought I thought I understood it, I'm not so sure now. I will say what I mean on request, but now I'd like to give my other trouble.

The trouble:

The calculations seem more complicated than usually on such tests, which makes me think I'm doing something wrong.

The calculations:

We have:

where

The likelihood ratio is given by the following formula, for

N-P tells me I want to look for the rejection region of the following form:

where is a constant we will have to choose correctly.

We want

Here's my trouble. To solve the inequality with respect to I need to divide by whose sign I don't know, so I have to consider two cases and some unpleasant calculations seem to follow. As far as I know my professor, it's impossible, so I think I'm doing it the wrong way.

It's unfortunately my first attempt at solving a problem with a composite hypothesis.

I'll be very grateful for any help.

PS: I'm sorry about the size of the formulas, but I have no idea what to do about it. They show correctly in preview.