Chapter 8, Problem 13CR

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

(a) Suppose we have two independent random variables x₁ and x₂ with

respective population means µ₁ and µ₂. Let us say that we use sample data to construct two 80% confidence intervals.

Confidence Interval	Confidence Level
$A_1<{\mu }_1<B_1$	0.80
$A_2<{\mu }_2<B_2$	0 80

Now. what is the probability that both intervals hold at the same time? Use methods of Section 5.2 to show that

$P\left(A_1<{\mu }_1<B_{1}and A_2<{\mu }_2<B_2\right)=0.64$

Hint: You are combining independent events. If the confidence 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% confidence (i.e., only 10%risk level). How much confidence c should each interval have to achieve this combined level of confidence? (Assume that each interval has the same confidence level c.)

Hint: $P(A_1<{\mu }_1<B_{1}and{A}_2< {\mu }_2<B_2)=0.90$

$$ $P(A_1<{\mu }_1<B_1\text{ ) }\times P(A_2< {\mu }_2<B_2)=0.90$

$c\times c=0.90$

Now solve for c.

(c) If we want both intervals to hold at the 90% level of confidence, then the individual intervals must hold at a higher level of confidence. 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.