Let's say I have two conditions, and my sample size for the two conditions is extremely low. Let's say I only have 14 observations in the first condition and 11 in the other. I want to use the t-test to test if the mean differences are significantly different from one another.
First, I am a little confused about the normality assumption of the t-test, which might be why I am not totally getting bootstrapping. Is the assumption for the t-test that (A) the data is sampled from a normal population, or (B) that your sample distributions have Gaussian properties? If it is (B) then it isn't really an assumption, right? You can just plot a histogram of your data and see if it is normal or not. If my sample size is low though, I won't have enough data points to see if my sample distribution is normal though.
This is where I think bootstrapping comes in. I can bootstrap to see if my sample is normal, right? At first I thought that bootstrapping would always result in a normal distribution, but this isn't the case(Can Bootstrap Resampling be used to Calculate a Confidence Interval for the Variance of a Data Set? statexchange statexchange). So, one reason you would bootstrap is to be more certain of the normality of your sample data, correct?
At this point I become thoroughly confused though. If I perform a t-test in R with the t.test function and I put the bootstrapped sample vectors in as the two independent samples, my t value simply becomes insanely significant. Am I not doing the bootstrapped t-test right? I must not, because all bootstrapping is doing is just making my t value larger, wouldn't this happen in every case? Do people not perform a t-test on the bootstrapped samples?
Lastly, what is the benefit of computing confidence intervals on a bootstrap versus computing confidence intervals on our original sample? What do these confidence intervals tell me that confidence intervals on the original sample data don't?
I guess I am confused on (A) why to use a bootstrap if it will just make my t value more significant, (B) unsure of the correct way to utilize bootstrapping when running an independent sample t-test, and (C) unsure how to report the justification, execution, and results of bootstrapping in independent t-test situations.