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 $\displaystyle \alpha=0.025$?

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