EDIT: Whoops

Originally Posted by Volga

Would you mind showing how one would approach the second part of the question, viz "Suppose n=10, sample mean is 2.42, and =1.5. Will you reject Ho at a significance level of 5%?" I am still hoping to have this completed as this is a good practice question for the exam. I promise never to attempt constructing a UMPT in real life after I am done with this Stat Inference exam )))

How can the sample mean be 2.42? That implies $\displaystyle \sum X_i = 24.2$ which is impossible.

Anyways, $\displaystyle P_{\lambda = 1.5} (\sum X_i > 24.2) = .01$

so yes, you would. I used R to calculate the probability, since it's a tedious calculation to do exactly (essentially summing the pmf). In other words, the appropriate k that gives the size of desired test is less than 24.2.

I don't worry about you having to calculate UMP tests in real life because, in general, they don't exist Even throwing in a single nuisance parameter is typically going to ruin things for you, so most of the tests that people use in practice aren't UMP.

I used R to calculate the probability

what's that?

Because, I wouldn't know how to calculate $\displaystyle P_{\lambda = 1.5} (\sum X_i > 24.2)$.

Originally Posted by Volga

what's that?

Because, I wouldn't know how to calculate $\displaystyle P_{\lambda = 1.5} (\sum X_i > 24.2)$.

R is (free) software. Explicitly you calculate it as

$\displaystyle
P(\sum X_i > 24.2) = 1 - P(\sum X_i \le 24.2) = 1 - \sum_{x = 0}^{24} \frac{15^x e^{-15}}{x!}
$

because $\displaystyle \sum X_i$ is Poisson(15) when $\displaystyle \lambda = 1.5$.

Oh I see! I thought "R" was some approximation formula )))