Statistical power of t-test in mildly skewed dataset I'm trying to learn when I can use t-test if data set is not normal "enough". 
Here are the things that I know, please verify:


*

*T-test is still robust when data is mildly skewed and light tailed.

*Mann-Whitney test for medians is a better choice when data has high skewness, or heavy tailed

*T-test can still be used even with high skewness, and heavy tails, as long as you have large sample sizes. But it has less statistical power than Mann-Whitney test. 


From here, they say:

It is helpful to note that as the sample size n increases, the T ratio: $$T = \frac{\bar{X} - \mu}{s/\sqrt{N}}$$ approaches an approximate normal distribution regardless of the distribution of the original data.

Question 1: How does the equation for $T$ approach normal distribution?
Question 2: What is the "statistical power" that people talk about? Does it give information about how many sample sizes that I need to have a meaningful result from t-test?
Question 3: Is there any way to determine sample size needed for t-test? Like, if you have skewness of $x$ and kurtosis of $y$, you need $n$ sample size for the result from t-test to be valid, even when your data is non-normal
 A: This has been discussed at length on this site.  The t-test is not very robust to skewness.  For example, with the log-normal distribution a sample size of 50,000 is not large enough for the t-based method to be sufficiently accurate.  The Wilcoxon signed-rank one-sample test does not test a median.  The Wilcoxon-Mann-Whitney test is a two-sample test.  The Wilcoxon tests are 0.95 as efficient as the t-based methods if normality holds, and can be arbitrarily more powerful than this parametric counterpart when normality does not hold.  I suggest reading an intro nonparametric statistics book.
A: 
How does the equation for  approach normal distribution?

Well, that it isn't quite true.  The sampling distribution for that statistic approaches a standard normal in the limit as n grows large.  That is very different.  T

What is the "statistical power" that people talk about? Does it give information about how many sample sizes that I need to have a meaningful result from t-test?

Power is the probability of correctly rejecting the null hypothesis when it is false.  If there really is a difference between groups, then a high powered test will correctly identify that there is a difference upon repetition of the experiment.
Power is an essential part of a sample size calculation.  The sample size depends on a few other things (namely the effect size and the noise in the data), but generally speaking more power requires more samples (all else constant).

Is there any way to determine sample size needed for t-test? Like, if you have skewness of  and kurtosis of , you need  sample size for the result from t-test to be valid, even when your data is non-normal

Not to my knowledge.  The equations for sample sizes lean heavily on the assumptions of the t-test, and so I don't think there are formulae which make use of higher moments.
