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?

  • 3
    $\begingroup$ Your hypothesis does not make sense, unless you change the hypotheses to testing the difference of the population mean $\mu_x - \mu_y$ $\endgroup$
    – user277126
    Feb 13, 2022 at 6:28

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.