27. Confidence intervals of fixed length for a normal mean. (i) In the two-stage procedure ill defined in part (iii) of the preceding problem, let the number c be determined for any given L > 0 and 0< y < 1 by (no-I(y) dy = y, -L/2~ where ( " 0- 1 denotes the density of the r-distribution with no - 1 degrees of freedom . Then the intervals 1:7-1aj X; ± LI2 are confidence intervals for of length L and with confidence coefficient y. (ii) Let c be defined as in (i), and let the sampling procedure be il2 as defined in part (iv) of Problem 26. The intervals X±LI2 are then confidence intervals of length L for with confidence coefficient y, while the expected number of observations required is slightly lower than under ill.

[(i): The probability that the intervals cover equals n \ L .L a;( X; - n L) ,-\ P€.o - 21C s IC s 21C = y. (ii): The probability that the intervals cover equals f,l f,lL} {f,lIX-~1 L} P€.o S s 2S P€.o S s 21C =y.] [5.16