Not a math/stats person, but I am interested in finding out why "paired t" would be a preferred test for samples that are not "independent". I googled a bit on this topic, but didn't really find anything that proves paired-t better than Student t for paired samples. Similar questions have been asked on this forum, so I got the gist (hopefully correctly) - paired samples tested by student t will lose the power of detecting the difference (i.e. accepting the null hypothesis when one should not) ?? Any suggestion where I could find the math treatment on this issue or a more elaborated explanation ? Great many thanks.
It's not that paired t is "better" (whatever that means), it's just more appropriate. All data is correlated in some way. Sometimes it's just a chance correlation and the underlying data are theoretically independent or it's a correlation due to an underlying factor like measures taken from the same subject. The independent test treats the conditions as if they are independent whether they're correlated or not and the paired test uses correlations present in the data.
It's entirely possible for a paired test to be "worse" mathematically in terms of less powerful or having higher error rates. But that's not why it's used.