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The paired t-test reduces two related samples to a single set of differences, then tests whether their mean is zero. Pairing removes between-subject variation, which usually gives much greater power than an independent-samples test.
Paired t
t = d̄ / (s_d/√n), where d are the within-pair differences
The differences are 2, 3, 1, 3, 3 and 3, with mean 2.5 and t ≈ 8.0623 on 5 df — highly significant.
Whenever each observation in one group is naturally matched to one in the other, such as before-and-after measurements on the same subjects.