Ok stats tweeps, help me out. I've tied myself up in knots about something that is probably so simple I'm a bit embarassed to ask. Let's use a t test as an example.

With a two-sided test:

H0: t = 0
H1: t is NOT 0

So far so good....

But with a one-sided test, does H0 change? I have conflicting sources:

H0 remains t = 0
H0 becomes t is EQUAL TO OR LESS THAN ZERO

My understanding is that H0 has to be a point estimate - that's how we can draw a t distribution. So the second option above feels wrong

But when we consider what happens if we observe an unexpected effect (i.e., in the opposite direction than we are testing for), it feels weird to say that we can't reject H0: t = 0.

It feels more logical to say that we can't reject H0: t is LESS THAN OR EQUAL TO ZERO

I'm writing a lecture on one-sided and two-sided tests and want to make sure I get this right!