9. Two-sided t-test. (i) Let Xl" ' " X" be a sample from N( t 0 2). For testing = 0 against =1' 0, there exists a UMP invariant test with respect to the group XI = eX" c =1' O. given by the two-sided r-test (17) of Chapter 5. (ii) Let X1 . . .. ,Xm and YI , . .. ,y" be samples from N(t02) and N(1/.02) respectively. For testing 1/ = against 1/ =1' there exists a UMP invariant test with respect to the group XI = aX, +

b. Jj' = aJj +

b, a =1' 0, given by the two-sided r-test (30) of Chapter 5. [(i): Sufficiencyand invariance reduce the problem to Itl, which in the notation of Section 4 has the probability density P8 (t) + h ( - t) for t > O. The ratio of this density for 8 = 81 to its value for 8 = 0 is proportional to /(f(e 8tl' + e-8" ')gr2(u) du, which is an increasing function of t 2 and hence of Itl.]