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

Do I find the pooled estimate and hence use it under the equation where my t statistic is where this case my s is the pooled estimate. Is this the appropriate test that I'm concerned? And how do I justify it?