Let say I have 6 independent draws from a normal population with mean $\displaystyle \mu_x$ and standard deviation $\displaystyle \sigma$. In addition, I have another 8 independent draws from another normal population with mean $\displaystyle \mu_y$ and standard deviation $\displaystyle \sigma$. Given the fact that the two populations variances are the same, I want to find an appropriate test for the null hypothesis $\displaystyle \mu_x=\mu_y$ and justify it.

Do I find the pooled estimate and hence use it under the equation where my t statistic is $\displaystyle t=\frac{(\overline{X}_1-\overline{X}_2)}{s\sqrt{\frac{1}{6}+\frac{1}{8}}}$ where this case my s is the pooled estimate. Is this the appropriate test that I'm concerned? And how do I justify it?