I have seen tons of situation where the null hypothesis reads along the lines of $\mu = \mu_0$, but what if I wish to test is $\mu \leq \mu_0$? Or more complicated null hypothesis?

For a more concrete example, let's say I have data which I believe is from $N(\mu,1)$, where $\mu$ is unknown. I want to test the null hypothesis that $\mu \leq 0$. How might I do that?

  • $\begingroup$ The concrete example is called a one-tailed test. $\endgroup$ – whuber Jun 18 '14 at 15:27
  • $\begingroup$ @whuber I think the OP is asking about something like $H_0: \mu_0 \epsilon (-\infty, c]$. $\endgroup$ – shadowtalker Jun 18 '14 at 15:29
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    $\begingroup$ @ssde That's right (reading your "$\mu_0$" as "$\mu$", which is the only symbol involved in the concrete case): that's a classic one-tailed test, assuming that the alternative hypothesis is $\mu\in(c,\infty)$. $\endgroup$ – whuber Jun 18 '14 at 15:30
  • $\begingroup$ @whuber wouldn't that be $H_0: \mu = c $ and $H_A: \mu < c$? Or is that equivalent to $H_0: \mu \geq c $ and $H_A: \mu < c$? $\endgroup$ – shadowtalker Jun 18 '14 at 15:32
  • $\begingroup$ @ssde I think you might have written the reverse of what you meant. In the normal-theory case posited here, testing $H_0: \mu\le 0$ (which is "the null hypothesis that $\mu\le 0$") against $H_A: \mu\gt 0$ turns out to be the same as testing $\mu=0$ against $\mu\gt 0$ because this is an MLR family. $\endgroup$ – whuber Jun 18 '14 at 17:33

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