Two means: independent vs paired samples

In short

• Independent samples from normal populations with unknown but equal variances: pooled t-test with $n_1 + n_2 - 2$ df.
• The pooled variance $s_p^2$ is a df-weighted average of the two sample variances.
• Dependent (paired) samples, such as two assets over the same dates or the same firms before and after: compute each pair's difference and run a paired comparisons test with $n - 1$ df.
• On related data the paired test is more powerful, because differencing strips out variation the samples share.
• Usual null: no difference ($\mu_1 - \mu_2 = 0$ or $\mu_d = 0$). One-sided versions work just as for a single mean.