I'm using a one sample t-test to compare individual case specimens to a know sample. The sample is skewed data which contains the absolute value of differences between left/right continuous data. The t-statistic being calculated here is t=case1 - mean (sample) /sd (sample). Given that the sample size is very large (more than 4000) does the central limit come into play? As far as I understand it, the sample means approach normality, which I've tested by randomly sampling from the sample and plotting the means of those samples. Also, if the mean is approximately normal, is it still valid to use a 2-tail test? A friend of mine is arguing that we need to use a one tail since the sample is skewed in one direction.
Edited: I've run the same test using bootstrapping and transformation for normality both of which achieved very similar results to using the data as is.