need help to construct UMPT (uniformly most powerful test)

Question.

Let $\displaystyle Y_1,... Y_n$ be a random sample from a Poisson distribution with mean $\displaystyle \mu>0$ unknown. Construct the uniformly most powerful test for testing hypotheses

$\displaystyle H_0: \mu=\mu_0; H_1: \mu>\mu_0$

Suppose n=10, sample mean is 2.42, and $\displaystyle \mu_0$=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

$\displaystyle l_Y(y)=\frac{\mu^{\Sigma_{i=1}^nY_i}(e^{-\mu})^n}{\prod_{i=1}^nY_i!}$

By Neyman-Pearson lemma,

$\displaystyle P_{\mu_0}(\frac{L_Y(\mu;y)}{L_Y(\mu_0;y)}>k_{\alph a})=\alpha$

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

So I find the ratio (I replaced $\displaystyle \Sigma_{i=1}^nY_i$ with $\displaystyle n\bar{Y}$

$\displaystyle \frac{L_Y(\mu;y)}{L_Y(\mu_0;y)}=\frac{\mu^{n{\bar{ Y}}}}{\mu_0^{n{\bar{Y}}}}e^{(\mu_0-\mu)n}$

Now then, I will reject Ho if this ratio is large. This ratio will be larger the larger $\displaystyle \bar{Y}$ 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 !