need help to construct UMPT (uniformly most powerful test)

Question.

Let be a random sample from a Poisson distribution with mean unknown. Construct the uniformly most powerful test for testing hypotheses

Suppose n=10, sample mean is 2.42, and =1.5. Will you reject Ho at a significance level of 5%?

Answer.

I started doing by the book and got stuck. I know it is a trivial problem, statisticians do this all the time, I need to do it once only, to understand )))

Start by writing down the likelihood function

By Neyman-Pearson lemma,

(my book uses the likelihood over the total sample space in the numerator)

So I find the ratio (I replaced with

Now then, I will reject Ho if this ratio is large. This ratio will be larger the larger is. But how large it should be (for say 95% confidence level?)

Here is where I am stuck. I don't know distribution of this ratio. How do I decide on the critical region? (which will be a most powerful region, I reckon). Do I need to know a distribution?... Do I use the assymptotic distribution of the 2 log of the likelihood ratio? do I use Poisson table? When it comes to a most powerful test, my head is a mess.

thank you !