|
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 µ1 and µ2. Let us say that we use sample data to construct two 80% con fidence intervals.
Confidence interval Confidence level
A1 < µ1 < B1.......................................0.80
A2< µ2 < B2.......................................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 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 con fidence c should each interval have to achieve this combined level of confidence? (Assume that each interval has the same con fidence level c.)
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. |
| New search. (Also 1294 free access solutions) |