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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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Add $1$ to all the $Y$ data to get a new data set $Z$ with mean larger by $1$ but the same standard deviation. Then test $\mu_x = \mu_z$. –  Dilip Sarwate Feb 6 '12 at 17:09
    
You have $1/m - 1/n$ in the denominator where I'd have expected $1/m + 1/n$. Notice that for some values of $m$ and $n$ you're going to get $0$ in the denominator or a negative number under the radical. –  Michael Hardy Feb 6 '12 at 17:52
    
The jury is in some ways still out, on the Behrens-Fisher problem of inferences on the differences between means of normal populations with unequal variances. It can be modeled mathematically in various ways, and the math problems can be solved, but which is the right model? –  Michael Hardy Feb 6 '12 at 17:54
    
@Michael Hardy typo, you are right. It should be +. –  viktor Feb 6 '12 at 18:13
    
@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"? –  joriki Feb 6 '12 at 20:55

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