Paired vs. Pooled Inference... When is it okay to pair samples? I understand that paired tests are usually done on sampling distributions that have some sort of linkage. But is there a definitive way to differentiate when to use a paired t test vs. pooled? 
The basic example I can think of is when sampling the same group twice, before an experiment and after. But once this gets extended to multiple samples I have difficulty conceptualizing the different uses of each test.
Elementary explanations are encouraged (currently in a 2nd year university applied stats course.) 
 A: Under some circumstances, the paired analysis will have more power and precision than an unpaired analysis. In addition, some research suggests that paired analyses are more robust to unmeasured confounding. 
We'll assume that both groups have the same sample size and the same variance for simplicity. In an unpaired t-test, the numerator of the t-statistic is the difference in means ($\mu_1-\mu_2$) and the denominator is the standard error of the difference in means ($s_{\mu_1-\mu_2}$). In a paired t-test, the numerator of the t-statistic is the mean of the difference scores ($\mu_D$) and the denominator is the standard error of the mean of the difference scores ($s_{\mu_D}$). 
The difference in means is equal to the mean of the difference scores ($\mu_1-\mu_2=\mu_D$), so the two statistics have the same numerator. The standard error of the mean of the difference scores is equal to the standard error of the difference in means times the square root of 1 minus the correlation between the scores in the pairs ($s_{\mu_D}=s_{\mu_1-\mu_2}\sqrt{1-r}$). If the correlation is positive, meaning scores within pairs are correlated with each other, then $\sqrt{1-r}$ will be less than 1, so $s_{\mu_D}<s_{\mu_1-\mu_2}$. This means the t-statistic for the paired samples t-test will be larger than the t-statistic for the unpaired samples t-statistic. One detail is that the degrees of freedom for the paired t-test are half that of the unpaired t-test. Smaller degrees of freedom means the critical t-statistic is larger, which makes it harder to find a statistically significant result (all else equal).
So, to summarize, the paired t-test will have more power when the scores within pairs are positively correlated and when that positive correlation is large enough to offset the decrease in degrees of freedom (which will almost always be true unless the pairing is random). The paired t-test will have less power when the scores are negatively correlated within pairs or when the correlation within pairs is not large enough to offset the decrease in degrees of freedom.
When estimating a treatment effect, it makes sense to pair your participants. They should be paired on the basis of pre-treatment covariates that are correlated with the outcome. This will induce within-pair correlations, which make the paired t-test more powerful. A method for doing this is described in the (somewhat advanced) Zubizarreta, Paredes, & Rosenbaum (2014).

Zubizarreta, J. R., Paredes, R. D., & Rosenbaum, P. R. (2014). Matching for balance, pairing for heterogeneity in an observational study of the effectiveness of for-profit and not-for-profit high schools in Chile. The Annals of Applied Statistics, 8(1), 204–231. https://doi.org/10.1214/13-AOAS713
