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Or more compactly: \[\begin H_0&\colon\ \mu_M=\mu_F \ H_a&\colon\ \mu_M\ne\mu_F \end\] Essentially, we assume \(H_0\) is true, and look for the data to force us to conclude that it isn't: \(H_a\) must be. Did we just get unlucky and sample non-representative individuals?
It's like a proof by (probabilistic) contradiction. that seems unlikely, so the only other choice is that \(H_a\) is likely. We want to ask a question about the means, and form these hypotheses: \[\begin H_0&\colon\ \mu_1=\mu_2 \ H_a&\colon\ \mu_1\ne\mu_2 \end\] .
Descriptions of statistical models usually emphasize the role of population quantities of interest, about which we wish to draw inference.
Whatever level of assumption is made, correctly calibrated inference in general requires these assumptions to be correct; i.e.
Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) selecting a statistical model of the process that generates the data and (second) deducing propositions from the model.
Any statistical inference requires some assumptions. A statistical model is a set of assumptions concerning the generation of the observed data and similar data.
Inferential statistics can be contrasted with descriptive statistics.
By analogy: if you start a proof by contradiction The T-test (and other statistical tests) make some assumptions about the underlying distributions.
If those assumptions aren't satisfied, then the math doesn't hold and the conclusion might not be meaningful. If you have non-normally distributed populations, then looking at sampling outcomes and calculating the t-statistic isn't (exactly) from a t-distribution.
Limiting results are not statements about finite samples, and indeed are irrelevant to finite samples.
However, the asymptotic theory of limiting distributions is often invoked for work with finite samples.
Note: \(H_0\) is that our data: run it through some function that preserves the stuff that's important, but reshapes it.