Let be two i.i.d. normal (Gaussian) samples noted X~N(muX, sigmaX): nX and Y~N(muY,sigmaY): nY and the null Hypotheses H0: muX=muY. A nick pick: personally I do not like to read that we are searching if the two Populations X and Y have equal mean values. This is a short coming: what we are unquestionably finding is that if these two parameters values are mutually indiscernible or,alternatively, if one got sufficient evidence that is not the case. Sufficiently learned people do not understand otherwise, however . . . It can be proved that Sff is distributed as a Student with m-1 degrees of freedom.
___1___The Intra-Permutation (IP) algorithm
In order to try to use all m items X = X1, X2, . . . , Xm, . . . , finding Xn, consequently not discarding from the calculatins the items beyond m, we try to *associate* every item Y(g´) chosen at random without replacement to a item X(g) chosen similarly. In this manner we get g=INT(m*RND) + 1, g´ =INT(n*RND) + 1. and all values of X are equally potential candidates to be included in the generic sample. Therefore the CI we intend to calculate is based on the Sff set of thousands elements as is used in Monte Carlo evaluations. I think this is one way to use all X items without excluding arbitrarily n-m of them which it was justly criticize since decades ago.