# What is the null hypothesis for a 1-tailed test?

I've been given different answers to this question in different courses. Some professors say it is (using the example alternative hypothesis of $\mu > 3$):

$$H_0: \mu = 3$$ $$H_1: \mu > 3$$

Others say it is:

$$H_0: \mu \le 3$$ $$H_1: \mu > 3$$

How should I write my null hypothesis for 1-tailed tests?

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In general, a hypothesis is a statement that a restriction is true, where a restriction takes the form $\theta\in \Theta_0$ with $\Theta_0$ is a strict subset of a parameter space $\Theta$. If the null hypothesis is defined as $$H_0:\theta\in \Theta_0$$ the alternative hypothesis is its complement $$H_1:\theta\in \Theta_0^c.$$ By this formal definition, the second one is correct and the first one does not constitute a hypothesis test unless $\Theta=[3,+\infty)$, which is highly unlikely in application.