  In this paper we proposed the alternative test to the two independent and
normally distributed samples t test based on the cross variance concept. We
present the simulation results of the power and the error rate of the special
case of the cross variance which is when the variances of the two samples are
homogeneous and the t tests.
  The simulation results show that the special case of the cross variance test
has the power and the error type I rate equal to the $t$ test. This result
suggests that the proposed test could be used as an alternative to detect
whether there is difference between means of the two independent normally
distributed samples. We give some example comparative case studies of the
special case of the cross variance and the t tests.
