test statistic?

**jerryc** · June 22nd 2009, 09:05 AM

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 $N(\mu_{X},\sigma^{2}_{X})$ and $N(\mu_{Y},\sigma^{2}_{Y})$ . $\sigma^{2}_{Y}>\sigma^{2}_{X}$ . Use the modification of Z to test the hypothesis $H_{0}$ : $\mu_{X}-\mu_{Y}=0$ against the alternative hypothesis $H_{1}$ : $\mu_{X}-\mu_{Y}<0$ .

How do you define the test statistic and a critical region that has a significance level of $\alpha=0.025$ ?

thank you very much

**matheagle** · June 22nd 2009, 04:27 PM

1 Do you know the population variances?
If so the rejection region is $(-\infty ,-1.96)$

and the test statistic is ${\bar X-\bar Y\over \sqrt{ {\sigma^{2}_{Y}\over n_Y}+{\sigma^{2}_{X}\over n_X}}}$ .

2 If you don't know $\sigma^{2}_{Y}$ and $\sigma^{2}_{X}$ , then it's a t test.
BUT now I need to know if you're allowed to assume that $\sigma^{2}_{Y}=\sigma^{2}_{X}$ or not.

3 HOWEVER, if the sample sizes are large then you can approximate this via the Central Limit Theorem and use 1.

**jerryc** · June 22nd 2009, 04:30 PM

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

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