
test statistic?
X and Y stand for the size of the male and femal of a specific species of moth. Assume the distributions of X and Y are $\displaystyle N(\mu_{X},\sigma^{2}_{X})$ and $\displaystyle N(\mu_{Y},\sigma^{2}_{Y})$. $\displaystyle \sigma^{2}_{Y}>\sigma^{2}_{X} $. Use the modification of Z to test the hypothesis $\displaystyle H_{0}$: $\displaystyle \mu_{X}\mu_{Y}=0$ against the alternative hypothesis $\displaystyle H_{1}$: $\displaystyle \mu_{X}\mu_{Y}<0$.
How do you define the test statistic and a critical region that has a significance level of $\displaystyle \alpha=0.025$?
thank you very much

1 Do you know the population variances?
If so the rejection region is $\displaystyle (\infty ,1.96)$
and the test statistic is $\displaystyle {\bar X\bar Y\over \sqrt{ {\sigma^{2}_{Y}\over n_Y}+{\sigma^{2}_{X}\over n_X}}}$.
2 If you don't know $\displaystyle \sigma^{2}_{Y}$ and $\displaystyle \sigma^{2}_{X}$, then it's a t test.
BUT now I need to know if you're allowed to assume that $\displaystyle \sigma^{2}_{Y}=\sigma^{2}_{X}$ or not.
3 HOWEVER, if the sample sizes are large then you can approximate this via the Central Limit Theorem and use 1.

That is all the info I have (and part of why I am so stuck)