For the two samples t-test provide a sound justification why a standard normal as the reference distribution for the t-statistic can be used if the sample is sufficiently large.

As starting point consider two samples Y=[Y_1,...,Y_n] and X=[X_1,...,X_n]where Y_i~N(mean_1,sigma) and X_i~N(mean_2,sigma) with mean_1, mean_2, sigma unknown. The summary statistics sample mean, sample variances, pooled variance estimator and the t-statistic might be helpful.