I remember from L1 that we are likely to chose the alternative hypothesis to be what we want to prove and the null hypothesis as what we want to reject which worked perfectly in one tail cases. Not the case- really. The Null is what we assume to be true, the real state of nature. The alternative is what we are trying to find evidence to support. You’re also not “proving” anything, as that implies certainty that we never achieve. When we use a hypothesis test (CI, or other method with a measure of reliability), we’re essentially saying what is highly likely or what the evidence indicates (but doesn’t prove).
However, if I am an analyst that expect from previous experience that a slope b0 coefficient value to be exactly 2.5 and the regression results shows let’s say 1.3 as the value of the slope coefficient b0. I assume you mean the intercept as bo, which isn’t a slope.
will I still have the alternative hypothesis as b0=2.5 or in this case I should assume that the regression results of 1.2 is correct and as such offer that the null hypothesis be b0=2.5? No, you shouldn’t assume one is more “right” than the other-- it’s sampling variation (unless the results are so far out of sync with what theory or with what all other research tells us).
The alternative will never be of the format “=”-- in other words, you won’t have Bo = anything as an alternative hypothesis. You probably wouldn’t make a hypothesis test based off of one sample’s results, either. You could, though, conduct a test if you wanted, since we assume these estimators are correct, on average. So, if you say, “Hey, bo=2.5 at first, now it’s lower. Can we find evidence that Bo is less than 2.5?” Then you could test Ho : Bo greater than or equal to 2.5 with Ha: Bo <2.5.
Realistically, think of some example related to a theory (can’t think of one off the top of my head)-- if we run a regression and theory dictates that the intercept, Bo, equal 0, but we generate something else, say 2.5, then we might test Ho: Bo = 0 and Ha: Bo not equal to 0 [or perhaps Ha: Bo > 0](since our results conflict with our theory that the regression should have zero as the intercept).
Some of the examples i saw actually showed that the null hypothesis choice was for b0=2.5 and that the alternative that b0 not equal to 2.5. This would be an appropriate form (as I mentioned earlier), but it seems illogical if it isn’t linked to theory or mounds of prior research (which usually helps support a theory).