75. The one-sample CI for a normal mean and PI for a single observation from a normal distribution were both based on the central t distribution. A CI for a particular percentile (e.g., the 1st percentile or the 95th percentile) of a normal population distribution is based on the noncentral t distribution. A particular distribution of this type is speci ed by both df and the value of the noncentrality parameter d (d0 gives the central t distribution). The key result is that the variable has a noncentral t distribution with df  n  1 and d(z percentile) .

Let t.025, n, d and t.975, n, d denote the critical values that capture upper-tail area .025 and lower-tail area

.025, respectively, under the noncentral t curve with n df and noncentrality parameter d (when d  0, t.975  t.025, since central t distributions are symmetric about 0).

a. Use the given information to obtain a formula for a 95% con dence interval for the (100p)th percentile of a normal population distribution.

b. For d  6.58 and df  15, t.975 and t.025 are (from MINITAB) 4.1690 and 10.9684, respectively.

Use this information to obtain a 95% CI for the 5th percentile of the beer alcohol distribution considered in Example 8.11.

1n T 

X  m s/1n

 1z percentile 21n S/s x22

 11.9622 x21



2u21

 u22