Why can variance be estimated from a sample taken from an alternative hypothesis? This is a new question going off this earlier question of mine. Let's say we have the following scenario: 

A neurologist is testing the effect of a drug on response time by injecting 100 rats with a unit dose of the drug, subjecting each to neurological stimulus, and recording its response time. The neurologist knows that the mean response time for rats not injected with the drug is 1.2 seconds. The mean of the 100 injected rats' response times is 1.05 seconds with a sample standard deviation of 0.5 seconds. Do you think that the drug has an effect on response time?

When we do this, we assume the sample standard deviation is equal to our population standard deviation. However, from what I see, this 1.05 sample is not necessarily a sample from our population. This 1.05 measurement is a sample taken from a population that might have a standard deviation that's completely different from the null hypothesis population. For example, what if this drug made all the rats have a response time clustered very closely around 1.05. This means the standard deviation of the population who took the drug would be less than the standard deviation of the population who didn't take the drug.
Because we're not necessarily sampling from the same population distribution, how can we use the sample distribution to estimate another population distribution? 
The only explanation I can think of is that we have to assume our drug has no effect, assuming the null hypothesis, and then it would be sampling from the distribution under the null hypotehsis. I get that. But if the null hypothesis is FALSE and there really IS a difference, you'd be sampling from a different population (distribution under the alternative hypothesis), but we still ASSUME it's from the distribution under the null hypothesis, and that's what I can't understand.
 A: There are all sorts of quandaries in your question(s).
(1) There are Type I and type II errors. You can, as you point out, get a false positive or a false negative. 
(2) You don't have a Std Dev for the control group.
(3) I get queasy looking at a value of 1.05 seconds with a standard deviation of 0.5 seconds. The rats can't have a negative reaction time so the reaction data is probably skewed towards higher reaction time which makes using a Normal distribution borderline suspect. 
(4) If the 1.2 seconds for the control group is "absolute" (has been measured over a million rats) then that number is solid. Since you have 100 rats (which is really a large sample) you'd essentially multiple your measured std. dev. by a fudge factor to try to prevent accepting a false result. As sample size gets bigger the factor gets smaller. It would be very near 1 for a sample size of 100. This is the Student's T test. (You're testing the difference of the means compared to the standard deviation, assuming a normal distribution.) 
(5) If had the standard deviation of the control group, then you could do a F-Test to compare the standard deviations. (A 100 rats is a good big sample for such a test.)
(6) You could test the raw data against the expected values of a normal distribution by converting the raw data to a histogram. 
Does that help?
