What test could I use to test $\mu_1$ and $\mu_2$ instead of $\bar{x}$ and $\mu$?

Question

A single sample $t$ test is only against a sample and a population correct? Can it test against a population and itself or two populations? ie $\mu_1$ vs. $\mu_2$....if not what tests would you use $\mu_1$ and $\mu_2$? it seems like a dependent t test could do that....I could be wrong about everything though. Thanks in advance.

Answer 1

A single sample $t$ test is not "against a sample and a population". It is a test looking for evidence against some hypothesis regarding some parameter.

For example, you might have the hypothesis that a population mean is $28$. That is, the hypothesis that $\mu=28$. Your sample value of $\bar{x}$ might provide evidence against this. Or your sample might be consistent with this hypothesis, in which case you have no evidence of anything.

Or you might have the hypothesis that a population proportion is $0.45$. That is, the hypothesis that $p=0.45$. Your sample value of $\hat{p}$ might provide evidence against this. Or your sample might be consistent with this hypothesis, in which case you have no evidence of anything.

Or you might have the hypothesis that a one population's mean is the same as another population's mean. That is, the hypothesis that $\mu_1-\mu_2=0$. Your sample value of $\bar{x}_1-\bar{x}_2$ might provide evidence against this. Or your sample might be consistent with this hypothesis, in which case you have no evidence of anything.