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 |
|
0.80 |
|
0 80 |
Now. what is the
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:
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.
(a)
To find: The probability that both interval hold at the same time and also explain the risk that at least one interval does not hold must be 36%.
Answer to Problem 17CR
Solution: The probability that both interval hold at the same time is 0.64.
Explanation of Solution
Calculation:
Since,
Probability that both interval together is
Thus the probability that both interval holds at the same time is 0.64.
Thus, the probability that at least one of item does not contain
(b)
To find: The confidence level cthat each interval should achieve to have 90% combined level of confidence.
Answer to Problem 17CR
Solution: The value of confidence c to achieve 90% combined level of confidence is 0.949.
Explanation of Solution
The value of c is 0.949.
(c)
To explain: The importance of having individual intervals at a higher level of confidence, if we both intervals to hold at 90% level of confidence and its application in complex engineering designs.
Explanation of Solution
A higher probability of success for individual components is needed, when both components must succeed in order for a particular system to be functional.
For example:
Consider an aircraft having two engines; if both engines in an aircraft must operate properly for the functionality of aircraft, then each engine must have a higher probability than the combined pair of engines. If the probability of success for the first engine is 0.975 and the probability of success for the second engine is also 0.975, then the probability of success for both of the engines is given as:
Hence we can see that for getting the 95% success probability for both of the engines, we have to use 97.5% probability of success for the individual engines.
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Chapter 8 Solutions
Understanding Basic Statistics
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,