Hypothesis Testing, simple against composite.

Question

i've got some trouble understanding this exercise from Amemiya's Introduction to statistics:

Given the density $f(x) = \frac{1}{\theta}$ from $0 < x < \theta$, and 0 elsewhere, we are to test the hypothesis: $H_0: \theta = 2$ against $H_1: \theta > 2$ by means of a single observed value of X. Consider the test which rejects $H_0$ if $X > c$. Determine c so that $\alpha = 0,25$ and draw the graph of its power function.

I'm having great trouble understanding this exercise, mostly because of the uniformly distributed density function. I know this is not hard but i'm used to a more "mechanic" approach to this kind of problems, and i haven't encountered a problem that used this kind of density function or simple against composite yet.

How would you start tackling this problem?

Thanks a lot!

Answer 1

Like all hypothesis tests, you need to get a p-value ($p_c$) under the null hypothesis:

$p_c = P(X \geq c|\theta = 2)$ Your rejection region is the value of $c$ that makes $p_c =0.25$ Then, if your observed value is $>c$ then there is only a 25% chance of that happening if the max were in fact 2, so you would reject with a Type I error rate of 25%.