Constructing 95% CI for Difference in Means of 2 Normals w/ Var Unknown

[Discrete Data]

I have two samples, one with 21 measurements and one with 23 measurements, from a Normal distribution and I am constructing a 95% CI for the difference in the two means. I do not know the variances of the two Normal distributions.

I know that $\overline{X_1}-\overline{X_2}$~$N(\mu_1-\mu_2, \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}})$

But then as $\sigma_1$,$\sigma_2$ are unknown I use the sample variances, but then does it then follow that,

$\frac{(\overline{X_1}-\overline{X_2})-(\mu_1-\mu_2)}{\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}}≈N(0,1)$

Even though my samples are both less than 30 measurements?

 

[lippyka]

Yes, this is a valid way to construct a 95% CI for the difference in means. You can use the sample variances, and the resulting statistic will be approximately normally distributed with mean 0 and variance 1, regardless of the sample size.