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

Question

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.

Answer 1

Comment (too long for comment format):

First, T-F questions about statistical inference are very often problematic. However, my guess is that you're supposed to say that the alternative hypothesis determines the 'sidedness' of the test.

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.