Let and be random samples taken from independent normal distributions with means μ_{X} and μ_{Y} and variances and , respectively. Our objective is to test the difference between the two population means μ_{X} and μ_{Y}, where this difference will be denoted as . So if μ_{X} = μ_{Y}, then δ_{o} = 0; if μ_{X} > μ_{Y}, then δ_{o} > 0; and if μ_{X} < μ_{Y}, then δ_{o} < 0. In this regard, we may test the null hypothesis against any one of the following three alternative hypotheses:

Case 1 | Case 2 | Case 3 |

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