It looks to me that this is roughly what I have done in my first post - up to where I said:

"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."

Since I need to answer part 2 of the question (reject or not), I do need to chose k. And I am back again to the same question, how large the k should be?

Another thing you said, may I ask to elaborate, because it is not obvous to me yet,

(under which H?)

I clearly see

in the formula above, why is it that we only make use of the distribution of

under H? (and this is where I have difficulty in general with

__universally__ most powerful tests.