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I've two normal distributions: $N_x(\mu_x,\sigma^2_x)$ and $N_y(\mu_y,\sigma^2_y)$. I need to test null hypothesis $\mu_x=\mu_y+1$ with significance level 5%.

I know how to test $\mu_x=\mu_y (EX=EY)$: In short test first $\sigma^2_x=\sigma^2_y$ then use statistic $T=\frac{\bar{X}_m-\bar{Y}_n}{S\sqrt{\frac{1}{m}+\frac{1}{n}}}$. This is tested against "Student" aka t distribution quantile $q_{t(m+n-2)}(\frac{\alpha}{2})$.

but I don't know how reflect the constant into this or should I use completely different test.

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    @Michael: a) Do I understand correctly that your comment refers to the test for $\mu_x=\mu_y$, too, and doesn't stand in contrast to Dilip's prescription for reducing the general case to the case $\mu_x=\mu_y$? b) By which criteria would you identify "the right model"?2012-02-06

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