# Hypothesis testing with interval as null hypothesis

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?

• The concrete example is called a one-tailed test. – whuber Jun 18 '14 at 15:27
• @whuber I think the OP is asking about something like $H_0: \mu_0 \epsilon (-\infty, c]$. – shadowtalker Jun 18 '14 at 15:29
• @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 $\mu\in(c,\infty)$. – whuber Jun 18 '14 at 15:30
• @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 $H_A: \mu < c$? – shadowtalker Jun 18 '14 at 15:32
• @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. – whuber Jun 18 '14 at 17:33