Most powerful test involving Poisson

[safina]

Homework Statement

The number of sales made by a used car salesman, per day, is a Poisson random variable with parameter $$\lambda$$. Given a random sample of the number of sales he made on n days, what is the most powerful test of the hypothesis Ho: p = 0.10 versus Ha: p = 0.25, where p is the probability he makes at least one sale (per day)?

Homework Equations

$$f\left(x;\lambda\right) = \frac{e^{-\lambda}\lambda^{x}}{x!}$$

The Attempt at a Solution

I applied the single likelihood ratio test which Rejects Ho if $$\lambda$$ $$\leq$$ k which I found equivalent in saying to reject Ho if $$\sum Xi$$ $$\leq$$ k' where k' is given by $$P\left[\sum Xi \leq k'\right] = \alpha$$
But it seems not correct since the hypotheses involve p and not the parameter $$\lambda$$. Please help me solve this problem.

 

[lanedance]

what are k & k'?

just a few ideas to get you started:
- first I'd look at how p is related to lambda
- then i would look at the likelihood of each hypothesis
- consider how to derive the power of the test