A Note on the Two-Sample Tests under the Normality

In this research, we consider two-sample problems for testing equality between two means. Before proposing new tests, we review briefly the two-sample t-test and Welch- Satterswaite t-test. Then under the normality assumption, we derive the likelihood ratio function without the assumption for equality between two population variances. Then we propose the likelihood ratio test with asymptotic distribution and permutation distributions for the null distribution of the likelihood ratio statistic. By decomposing the likelihood ratio statistic into two types of statistics, we propose another tests using the combination functions and the permutation principle. In order to compare efficiency among tests, we perform the simulation study and obtain the empirical powers with various continuous distributions. We comments on the assumption for the underlying distribution and a possible detour for the violation of assumption as concluding remarks.