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

A₁ < µ₁ < B_{1…………………………………}0.80

A₂< µ₂ < B_{2…………………………………}0.80

Now, what is the probability that both intervals hold at the same time? Use methods of Section 4.2 to show that

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.)

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.

Transcribed Image Text:

c×c=0.90