In Problem 5.2 the test statistic is larger when the data are treated as paired, compared to treating them as two independent samples. This will result in a smaller p-value and it therefore seems that pairing has made the test more "powerful." It is often held that pairing always makes a test more "powerful," because of the removal of the2 term from the standard error, as noted in Problem 5.7. But in Problem 3.8 we saw that the standard error for paired samples may sometimes be greater than is obtained by treating the data as two independent samples. In that case, if the difference is great enough, a paired-sample test could be less "powerful" than its two-independent sample cousin. Can you explain this?