25. On the basis of a sample X = (XI' " . , XII) of fixed size from N(~, (12) there do not exist confidence intervals for with positive confidence coefficient and of bounded length. [Consider any family of confidence intervals 6(X) ± LI2 of constant length L. Let ~I"' ~2N be such that It - ~jl > L whenever i * j. Then the sets S; = (x: 16(x) - ~;I L12} (i = 1,. . . ,2N) are mutually exclusive. Also, there exists (10 > ° such that 1 Ip(,.o{XE S;} - p(l.o{XE S;}I 2N for (1 > (10' as is seen by transforming to new variables Jj = (~ - ~1)/(1 and applying Lemmas 2 and 4 of the Appendix. Since min;P(I ' o{ XES;} s 1/2N, it follows for (1 > (10 that min; p(,.o{ XES;} s liN, and hence that infPt.o{16(X) - ~I !:.} s ( .0 2 N The confidence coefficient associated with the intervals 6( X) ± LI2 is therefore zero, and the same must be true a fortiori of any set of confidence intervals of length L.]