1

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.

Community
  • 1
rb612
  • 644
  • 5
  • 16
  • 4
    This question is an exact duplicate of your earlier question http://stats.stackexchange.com/q/179132/83967 -- it is simply an attempt to reformulate it. But you are asking the exact same thing. This is very confusing. In the future, if your Q gets misunderstood, it's better to edit it instead of posting a new Q. I upvoted your previous question, but am now downvoting this one. – amoeba Oct 30 '15 at 09:52

1 Answers1

1

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?

MaxW
  • 370
  • 2
  • 9
  • Thank you for your answer! Although it's not really what I'm looking for. I understand what you're saying, but I don't think I made myself clear. Let's say the drug doesn't have an effect. Then, we're just sampling from the distribution under the null hypothesis and all works perfectly. However, let's say the drug DOES have an effect. Then, we're actually sampling from a completely different distribution and I don't see how we can use that to estimate our population distribution (variance is going to be different and means are going to be different --> completely separate distributions) – rb612 Oct 30 '15 at 02:20
  • If the drug DOES have an effect, you are still conducting your statistical test assuming the distribution under the null hypothesis, $H_0$. If the effect is large enough, then if you conduct your test under $H_0$, the resulting test statistic will be large, leading to a rejection of the null hypothesis. – StatsStudent Oct 30 '15 at 03:06
  • @3209Cigs OK, that helps. But it just seems like if the drug has an effect, using the variance taken from that sample is going to be different because you're sampling under the alternative hypothesis rather than the null. But from what it seems, the t-test is under the assumption only that the drug has no effect, and we assume that the variance then of our sample is a very good estimator of the variance of our population. – rb612 Oct 30 '15 at 05:47