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What are the restrictions on $H_0$ and $H_1$ in a hypothesis test?

Specifically, if you were doing a hypothesis test on the mean of a normal distribution, $\mu$, could you have something along the lines of: $H_0: \mu=a$ vs. $H_1: \mu=b$ where $a,b \in \mathbb{R}$, or does it have to be more like $H_0: \mu=a$ vs. $H_1: \mu>a$ or $H_0: \mu=a$ vs. $H_1: \mu \neq a$?

I don't think it makes sense to have $H_1: \mu = b$, but I can't really explain why other than the confidence interval you choose is usually dependent on the alternative hypothesis in some way (e.g. two vs one tailed). It just doesn't make intuitive sense to be to have a very specific alternative hypothesis, since then rejecting $H_0$ really doesn't tell you anything about $H_1$ either . It seems more meaningful to have $H_1: \mu \neq a$ or $H_1: \mu> a$ since your confidence interval directly reflects these hypotheses.

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In practice I agree that it may not be all that scientifically interesting to test something of the form $$ H_0 : \theta = \theta_0 \hspace{3mm}\text{ v.s. }\hspace{3mm} H_1: \theta = \theta_1 $$ for a parameter $\theta \in \mathbb R$, but in theory there's nothing wrong with this. We simply compare the evidence for $\theta = \theta_0$ with the evidence for $\theta = \theta_1$. The Neyman-Pearson lemma, which is a foundational result in hypothesis testing, is exactly for this situation.

More generally, if $\theta \in \Theta$ we often partition $\Theta = \Theta_0 \cup \Theta_1$ and the Karlin-Rubin theorem allows us to construct uniformly most powerful tests for comparing $H_0: \theta \in \Theta_0$ with $H_1: \theta \in \Theta_1$.

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