7. If X. ,... • x" and YI,. .. ,y" are samples from Na.02) and N(1/,-r 2 ) respectively, the problem of testing -r 2 = 0 2 against the two-sided alternatives -r 2 =1' 0 2 remains invariant under the group G generated by the transformations x,' = aX, +

b. Y;' = aY; +

c, a =1' 0, and XI = Y; , Y;' = x,. There exists a UMP invariant test under G with rejection region _ (L:(y;_y)2 L:(X,-X)2) W - max 2' 2 k . L:(x, - X) L:(y; - P) [The ratio of the probability densities of W for -r2/ 02= !:J. and -r2/ 02= 1 is proportional to [(1 + w)/(!:J. + w)]n-I + [(1 + w)/(1 + !:J.w)]n-I for w 1. The derivative of this expression is 0 for all !:J. .] Section4