"Take *any* statistic. Compare it to the number 3. Use that to accept or reject *any* hypothesis."

Let's take the mean among women, call it X_bar1. That's step 1 in your post.

Then, compare it to 3. That's step 2.

I can use that to reject or not *any* hypothesis, such as whether it is equal to the mean among men, i.e. X_bar2? Really?

Repeat after me: This. Is. Casella. And. Berger's. Example.

It trivially meets the definition of "test statistic," much like a constant function is still a function. It just won't have useful properties. Get it now? (Only error in your question is that hypotheses need to refer to population parameters.)

It's not an error in my question, it's an error in what you wrote (ie "any" hyohtesis). You're now saying it cannot be used to test some hypotheses (eg about the equality of two parameters). Got it. You were wrong. Not a huge deal. But what you wrote was incorrect and now you've corrected yourself.

Nope. You're just wrong. All hypotheses relate to *population* parameters, as C&B make clear and your second post appears to acknowledge. You wrote about testing a hypothesis about the sample mean, which makes no sense -- a statistic is just a number. This is a common error for beginner students so it's cool you got it wrong.

Nowhere did I say you can't test hypotheses about the equality of parameters or anything close to it. Did you hallucinate that?

Reminder: This is basic, first-semester stuff and these goofy stats bros still can't get it. I quoted the definitions from C&B, explained everything clearly 8 million times, and nothing gets into their pin-sized heads.

Morven: You clearly said initially that you can take *any* statistic and use it as a test statistic to test *any* hypothesis. You're not debating fairly. You keep bumping up points that have already been debunked, and revising your statements when faced with uncomfortable questions. You did the same last year in the thread that sparked this whole coefficients as test statistics discussion. You can't even come up with original criticisms. You keep projecting on your critics the criticisms that were made of you, such as the lack of intro stats knowledge, and you even imitate their wording. For example, Ermintrude said, "Got it. You were wrong. Not a huge deal.", and your reply was "it's cool you got it wrong." It's time you acknowledged you're the one who's wrong, as several people have told you, and moved on. For starters, read the whole C&B chapter, not just that intro, to understand how test statistics work.