Bayesian Statistics: An Introduction, 4th Edition

5.1 Two-sample problems – both variances unknown

5.1.1 The problem of two normal samples

We now want to consider the situation in which we have independent samples from two normal distributions, namely,

which are independent of each other, and the quantity really of interest is the posterior distribution of

This problem arises in comparative situations, for example, in comparing the achievement in geometry tests of boy and girl pupils.

5.1.2 Paired comparisons

Before proceeding further, you should be warned against a possible misapplication of the model. If m = n and each of the xs is in some sense paired with one of the ys, say xi with yi, you should define

and then investigate the ws as a sample

for some ω. This is known as the method of paired comparisons. It might arise if, for example, the comparison of performance of boys and girls were restricted to pairs of twins of opposite sexes. The reason that such a situation is not to be treated as a two sample problem in the sense described at the start is that there will be an effect common to any pair of twins, so that the observations ...