Meaning of hypothesis test for "μ = 25"; isn't it impossible?

Question

I'm working through a stats textbook and have a question of the form:

You will perform a significance test of $H_0: μ=25$ based on an SRS of $n=25$. Assume $σ=5$.

I'm stuck on the 'equals' part of $μ=25$. Given that we're working in the set of reals, isn't the probability of $μ$ equaling 25 zero? (Assume a non-textbook dataset; e.g. the odds of $μ=180$ for the height of adult humans is zero).

I think what the hypothesis is trying to say is "$μ$ is within bounds dictated by your confidence level, but mean$(μ)=25$".

The alternate hypothesis is something like $H_a: μ \neq 25$, which I know is true for any non-hypothetical dataset.

So, what does $H_0: μ=25$ actually mean?

Answer 1

I don't see any reason why having $\mu=25$ is impossible. After all, $\mu$ has to equal some number. It is common in statistical circles to hear people say that the null hypothesis, $\text H_0: \mu_{\text{treatment}}=\mu_{\text{control}}$, must be false (which is a statement that I don't necessarily agree with, but that's another discussion). Your situation, however, is (superficially similar, but) different from this.

Another issue that might be confusing you is that you refer to "the probability of μ equaling 25". This is a very common misconception of classical (Frequentist) hypothesis testing. Properly speaking, there is no 'probability' of $\mu$ equaling 25, or any other number. You could say that the probability is either $1$ or $0$, but you don't know which. Nonetheless, $\mu$ isn't a random variable; it's a parameter, that is, an unknown constant. The only probability distribution that can be assigned to this is the degenerate distribution.

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So, what are you doing? When you perform a hypothesis test like what I suspect you are performing, you are checking to see what is the probability of getting a value (specifically a sample mean) as extreme or more extreme than yours, if the null hypothesis is true. The answer to that question is your p-value. After having done this, there are at least two valid ways of interpreting that p-value.

First, Fisher thought you should interpret it as a continuous measure of evidence against the null hypothesis. (Note that there is no alternative hypothesis or 'bright line' here, .04 is about the same as .06.)

Another possibility is to use the p-value as part of a decision making process, in keeping with the ideas of Neyman and Pearson. From this perspective, you are trying to differentiate between two hypotheses, and you designed your study explicitly to do so. For example, you conducted a power analysis, and set $\alpha$ and $\beta$ to levels of type I and type II error rates that you are willing to live with in the long run. After having analyzed your data and made a decision, you will either have made one of those errors or a correct decision, but you will never know which; you will only know that over all the times you engaged in this activity, you will have made a certain percentage of type I (type II) errors for those cases when the null hypothesis was actually true (false).

Answer 2

There are some details you've left out, and I am not sure what sort of answer you are looking for, but here goes...

Assuming you are performing something along the lines of a t-test, you are not looking for the probability of "mu = 25," but instead answering the question "is it reasonable to conclude that mu is some number other than (!=) 25?" So the idea that mu equals exactly 25 is not problematic in the sense that you were referring to in your question. If you conduct the test, you will either conclude that you can reject the null assumption as unreasonable or not, or you will calculate a p-value which answers a different, conditional, question ("what is the probability of seeing data as extreme as mine, in terms of its sample mean, if the data actually came from a distribution centered at 25?") which will allow you to draw a similar conclusion regarding the reasonableness of the assumed mu.

If you are asking the different question: "what is the true mean of this (given) data?" then you could build a confidence interval. There your concerns would be a consideration and handled in the interval's construction.

In classical/frequentist statistics, parameters like mu are treated as fixed, exact numbers; so asking "what is the probability of mu being 25?" is a sort of "nonsense" question in that philosophical context. A Bayesian approach would allow you to formulate the question in that way, but given the context of the question I don't think that is what you are working on. In that approach, there are ways to fix up a tiny interval around mu to get a probability.

You sound as though you are coming to an introductory stats book with a little extra background in probability already under your belt. Sometimes that makes the overly simple books harder to follow.