Expand Your Knowledge: Two Confi dence Intervals What happens if we want several confi dence intervals to hold at the same time (concurrently)? Do we still have the same level of confi dence we had for each individual interval?

(a) Suppose we have two independent random variables x1 and x2 with respective population means m1 and m2. Let us say that we use sample data to construct two 80% confi dence intervals.

Confi dence Interval Confi dence Level A1 m1 B1 0.80 A2 m2 B2 0.80 Now, what is the probability that both intervals hold at the same time?

Use methods of Section 5.2 to show that P A1 m1 B1 and A2 m2 B2 5 0.64 Hint: You are combining independent events. If the confi dence is 64%

that both intervals hold concurrently, explain why the risk that at least one interval does not hold (i.e., fails) must be 36%.

(b) Suppose we want both intervals to hold with 90% confi dence (i.e., only 10% risk level). How much confi dence c should each interval have to achieve this combined level of confi dence? (Assume that each interval has the same confi dence level c.)
Hint: P A1 m1 B1 and A2 m2 B2 5 0.90 P A1 m1 B1  P A2 m2 B2 5 0.90 c  c 5 0.90 Now solve for c.

(c) If we want both intervals to hold at the 90% level of confi dence, then the individual intervals must hold at a higher level of confi dence. Write a brief but detailed explanation of how this could be of importance in a large, complex engineering design such as a rocket booster or a spacecraft.