Suppose we have two distributions X and Y with the mean of X greater than or equal to the mean of Y. I want to test the hypothesis that $H_0: \bar{X} - \bar{Y} <= d $ v.s. $H_a: \bar{X} - \bar{Y} > d$ for a given d. Which test statistics should I use for testing such hypothesis?
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3$\begingroup$ Your hypothesis does not make sense, unless you change the hypotheses to testing the difference of the population mean $\mu_x - \mu_y$ $\endgroup$– user277126Feb 13, 2022 at 6:28
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This becomes clearer if you replace $Y$ with a transformation, $Z = Y + d$. Your hypotheses then become:
- $H_0: \bar X - \bar Z \leq 0$, or $\bar X \leq \bar Z$
- $H_a: \bar X - \bar Z \gt 0$, or $\bar X \gt \bar Z$
This can easily be tested using a one-tailed t-test, if you're assuming Normal distributions, or a one-tailed Mann-Whitney test if you're not making that assumption.
