# Using the null hypothesis to determine one or two tailed test?

This is a bit of a noob question so bear with me. Is it not true that the null hypothesis test determines the type of test (one tailed or two tailed) you would use.

For example null hypothesis could be $$\mu = x$$. And the alternative hypothesis would be that $$\mu \ne x$$. You would have to use a two-tailed test here because the null hypothesis is a certain value, and the alternative says that it could either be greater than or less than that value.

Another example is that null hypothesis could be $$\mu \ge x$$. The alternative hypothesis would be that $$\mu < x$$. You would use a one tailed test here because you are only looking at the side that is less than.

Does my train of thought make sense? I ask because I was posed this question on a quiz, as a true and false:

The null hypothesis determines the type of test (one tailed or two tailed) to be used to verify it.

And I put true, because of my explanation above - I was marked wrong. My instructor has not clarified so I would like to know. Thank you.

• Hi: you can back out the type of test it is from the null hypothesis so what you said is not wrong. I think the answer depends on exactly what was asked and that's not clear from your question. If the question was: "what is purpose of a hypothesis test", I would say that the purpose is to statistically test some null hypothesis regarding one of the model parameters. Jul 13, 2020 at 17:53
• @mlofton How do you figure out the answer from the null hypothesis alone? The point is that the alternative determines the nature of the problem along with the null; neither one of them unambiguously determines the entire hypothesis space. Mathematically, your claim is tantamount to asserting a subset of a set determines the entire set, which isn't the case without additional assumptions.
– whuber
Jul 13, 2020 at 18:24
• @Whuber: You're absolutely right. So, the OP's answer was incorrect. My mistake. But I was interested in what the actual question was. Does it say in his post what it was because I couldn't find it. Thanks for correction. To The OP: Whuber is correct in that you can't back out the type of test it is just from the null hypothesis. That must be why the teacher marked you as incorrect. But could you tell us what the question was. Jul 14, 2020 at 20:44
• @mlofton that question was just the true or false. There was no context given - it was just the true and false statement. Jul 16, 2020 at 14:07
• gotcha. so whuber is absolutely correct. it does not determine the type of test. you need the alternative also in order to know whether it's one tailed or two tailed. thanks for clearing that up and thanks to whuber for correction. Jul 17, 2020 at 15:40

Second, the null hypothesis must always contain an $$=$$-sign. For some authors, it's always just $$=.$$ For other authors, the null hypothesis can be $$=, \le,$$ or $$\ge.$$ In most hypothesis-testing situations, the "job" of $$H_0$$ is to provide the null-distribution used for the test and to determine the significance level, critical value, and P-value. (For that, only the equality is used.)
If a book always follows the convention to use $$=, \le,$$ or $$\ge$$ in $$H_0$$ when the alternative is $$\ne, >,$$ or $$<,$$ respectively, then your argument is OK.
However, some books (especially ones with multiple authors and bad copy-editors) may use $$H_0: \mu \le \mu_0$$ with $$H_a: \mu > \mu_0$$ part of the time and $$H_0: \mu = \mu_0$$ with $$H_a: \mu > \mu_0$$ part of the time. This is bad writing style (and potentially confusing to students), but not technically "wrong." Then you have to look at the alternative hypothesis to know the 'sidedness' of the test for sure.
Note; Mainly in theoretical discussions, one may have only a single value each for $$H_0$$ and $$H_1.$$ For example, $$H_0: \mu=\mu_0$$ and $$H_1: \mu=\mu_1$$, where $$\mu_0 \ne \mu_1.$$ This situation is called "simple vs. simple" and then one might not talk about right or left-tailed tests.