24. Let X; = + U;, and suppose that the joint density of the U's is spherically symmetric, that is, a function of EU;2 only, /(u\, o.. , un) = q([ul). Then the null distribution of the one-sample t-statistic is independent of q and hence the same as in the normal case, namely Student's t with n - 1 degrees of freedom. [Write t as {nXI/EX] JE( X; - X)2/( n - I)L\j2 and use the fact that when = 0, the density of XI"' " XII is constant over the spheres Ex; = c and hence the conditional distribution of the variables X;IlEX] given EX] = c is uniform over the conditioning sphere and hence independent of q.] Note. This model represents one departure from the normal-theory assumption, which does not affect the level of the test. The effect of a much weaker symmetry condition more likely to arise in practice is investigated by Efron (1969).