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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.

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  • $\begingroup$ This is not clear to me. Are you trying to compare each individual observation to the known sample? $\endgroup$
    – mdewey
    Jan 26, 2017 at 15:02
  • $\begingroup$ Yeah, where the individual comparison consists of a single absolute value of differences. $\endgroup$
    – JJL
    Jan 26, 2017 at 15:23
  • $\begingroup$ Do I understand correctly that you are comparing individual observations (specimens) to the mean of your sample using one sample t test? If so, then your approach is not valid in my opinion. What is your actual research question? $\endgroup$
    – sztal
    Jan 26, 2017 at 15:34
  • $\begingroup$ It's not so much a question as it is a method. The concept is to determine if the absolute value of differences is significantly different than what you would expect from the known sample. The actual comparison may be left data from an individual different than the right data, in which case it would come back as significantly different. $\endgroup$
    – JJL
    Jan 26, 2017 at 15:41

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The assUmption of normality is more than an assumption that the sampling distribution of the mean is normal. Therefore, the CLT does not ensure the assumption is met (or approximately met). Nonetheless, the t test is robust to violations of normality and tends to be conservative. A transformation to reduce skew can increase power so should be considered. The skew has nothing to do with the choice of one- versus two-tailed tests.

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  • $\begingroup$ How come all of the statistical literature mentions the CLT then? $\endgroup$
    – JJL
    Jan 26, 2017 at 15:26
  • $\begingroup$ Consider the F test which is just t^2. The assumption is that both the numerator and denominator are distributed as Chi square (divided by df). This will only be true if the data are normal. The CLT has nothing to do with the denominator. One can only speculate why this is missed by some stat literature. Perhaps because the t test is robust. $\endgroup$
    – David Lane
    Jan 26, 2017 at 16:10
  • $\begingroup$ Yeah, but isn't the t-test robust because of the CLT? $\endgroup$
    – JJL
    Jan 26, 2017 at 16:51
  • $\begingroup$ That's part of it but not all of it. It tends to be conservative. If only the normality of the sampling distribution of the mean were required, the nominal level and the actual level would be closer. $\endgroup$
    – David Lane
    Jan 26, 2017 at 18:41

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