Paired t test or 2 sample t test?

Lim:  Perhaps the bread baking example is not clear enough.  Here is another example that I think is easier to understand.
We are interested in evaluating the difference in wear characteristics between two materials used for athletic shoes.  We could do this a couple of different ways – 1) subject samples of each of the materials to exactly the same wear situations and then compare the measured wear between samples of A and B, this would be a 2 sample-t.  2) make shoes with one sole of material A and one of material B (swapping sides for different sample pairs) and have these “pairs” of shoes worn by different people, comparing the difference in wear between samples of A and B, this would be a paired t-test (the pairing would be through the individual who wore a pair of shoes with each material).
Pairing means that there is a relationship between the two items being evaluated, thus they are dependant.  It has nothing to do with the p value or confidence interval (although the pairings provide a different and direct relationship between the items that must be factored in).

As a practical aside, once you have your data collected, you can crank it through both the paired and 2 sample t and see if the two tests offer different “opinions”.  If you accept/fail to accept the null in both tests,  you have what you need…and dont forget to run your power and sample size test (especially if the test “fails”) to quantify its ability to detect a meaningful difference should it occur (ie power).  Good luck.

Box, Hunter, and Hunter page 100.It is the “classic” shoe example you eluded to in your post. Explained
well with examples of the right and wrong analysis.