# Does the null hypothesis need be a statement of no difference?

I'm just trying to self-learn the basics of statistics. And I don't understand completely the null vs alternative hypothesis.

What I don't understand is: can the null hypothesis be a statement with directionality, or does it always have to be a statement of no difference?

For example, say we have groups 1 and 2:

$$H_0: \mu_1 = \mu 2$$ $$H_1: \mu_1 \neq \mu 2$$

vs a different statement

$$H_0: \mu_1 \leq \mu 2$$ $$H_1: \mu_1 \gt \mu 2$$

Are both statements valid, and so $$H_0$$ can have directionality ?

I have discussed this with ChatGPT and it gets into obvious contradictions.

The null can certainly be one-directional. It can also be something else altogether. You can, e.g. have $$H_0: \mu_1 - \mu_2 = 3$$ And with tests of equivalence, you can reverse the usual roles of null and alternative hypotheses. See equivalence_testing

The purpose of a null hypothesis is to provide a specific thing to test. So in most circumstances, it will usually be an equality (rather than an inequality) or a statement that is similar to an equality (such as "is normally distributed").

The more typical way to express your one-sided test would be $$H_0: \mu_1=\mu_2$$ and $$H_1: \mu_1>\mu_2$$ - this makes clear that you will be testing $$\mu_1=\mu_2$$ for testing purposes, and then establishes that you only care about "failing" in the direction of $$\mu_1>\mu_2$$ (that is, you don't care if $$\mu_1<\mu_2$$).

However, the equality need not be "no difference". As noted by Peter Flom, you could have statements such as $$H_0:\mu_1-\mu_2=3$$ (which would then be paired by something like $$H_1:\mu_1-\mu_2>3$$ or $$\neq3$$). Any such statement involving distributions could conceivably be a null hypothesis - for example, you could have $$H_0:\mu_1\mu_2=1$$ (not sure what the test would be for that one, off the top of my head...).

I mentioned "in most circumstances"... this is because there is what is called a "composite null" - this occurs where multiple possible values are considered. For more on this detail, see here.

The best way to think about it is that the null hypothesis should be specific, and the alternative is usually more general, or a natural "fall-back" if the null is incorrect.