79. Consider 95% CI s for two different parameters u1 and u2, and let Ai (i  1, 2) denote the event that the value of ui is included in the random interval that results in the CI. Thus P(Ai)  .95.

a. Suppose that the data on which the CI for u1 is based is independent of the data used to obtain the CI for u2 (e.g., we might have u1  m, the population mean height for American females, and u2  p, the proportion of all Kodak digital cameras that don t need warranty service). What can be said about the simultaneous (i.e., joint)

con dence level for the two intervals? That is, how con dent can we be that the rst interval contains the value of u1 and that the second contains the value of u2? [Hint: Consider P(A1A2).]

b. Now suppose the data for the rst CI is not independent of that for the second one. What now can be said about the simultaneous con dence level for both intervals? (Hint: Consider , the probability that at least one interval fails to include the value of what it is estimating. Now use the fact that

[why?] to show that the probability that both random intervals include what they are estimating is at least .90. The generalization of the bound on to the probability of a k-fold union is one version of the Bonferroni inequality.)

c. What can be said about the simultaneous condence level if the con dence level for each interval separately is 100(1  a)%? What can be said about the simultaneous con dence level if a 100(1  a)% CI is computed separately for each of k parameters u1, . . . , uk?