I was wondering if it's possible to see what is the probability of getting a type 2 error in my investigation into whether there is a difference in length of time between cells in anaphase and telophase (I got a nonsignificant result using a 2 sample t-test which shouldn't really have happened since anaphase should be the shortest phase). I tried googling for answers but nothing came up and each example has a scenario of H1: μ>63 or whatever and none where H1: μa>μb so I am wondering if it's even possible to do such a thing. This isn't really required for my submission so I am not too bothered, just curious for future reference. I apologize if I am asking the question in the wrong way. All my stats knowledge comes from A level statistics and I'm a 1st-year zoology undergraduate so we didn't have many stats lessons on the more complex stats yet.
A Type II error occurs when we incorrectly fail to reject the null hypothesis, given that the alternative hypothesis is actually true. The crucial part of that statement is the assumption that the alternative hypothesis is true. The problem is that the alternative hypothesis $\mu > 63$ covers a wide range of possibilities. If the true $\mu$ is $63.0001$, we might think it's very easy for us to make a Type II error, but if the true $\mu$ is $1000$, that's much less likely.
Thus, for a hypothesis test (not just a t test), we can compute the probability of a Type II error as a function of the exact alternative hypothesis which is true. This is one minus the "power" of the test, which is the probability that we correctly reject the null hypothesis (for a specific alternative $\mu$ being equal to exactly something). Then, because the power varies depending on the alternative, we get a "power curve", which helps us understand how likely a Type II error is.
No, $t$ tests provide information about the null hypothesis, and not about the alternative hypothesis. I.e. you can make statements of type I errors, but not Type II errors. You would need to know the population effect size to be able to make statements of Type II errors. In the sketch below, you would need the position of the center of the red curve.
In your example, you would need a statement of the anticipated difference in time between cells in anaphase and telophase to be able to evaluate the Type II errors.