What Are Confidence Bounds?
One of the most confusing concepts to a novice reliability engineer is estimating the precision of an estimate. This is an important concept in the field of reliability engineering, leading to the use of confidence intervals. This subject can be confusing, as confidence is "the probability of a probability." However, the use of confidence is becoming more and more common as more organizations include confidence bounds in their reliability requirements. In this article, we will try to briefly present the concept, in relatively simple terms, and based on solid common sense.
The Marble Analogy
If you put the ten marbles back in the pool and repeated this example
again, you might get six black marbles, changing your estimate to 60%
black marbles. Which of the two is correct? Both estimates are correct! As
you repeat this experiment over and over again, you might find out that
this estimate is usually between X1% and X2%, and
you can assign a percentage to the number of times your estimate falls
between these limits. For example, you notice that 90% of the time this
estimate is between X1% and X2%.
When we use two-sided confidence bounds (or intervals), we are looking at a closed interval where a certain percentage of the population is likely to lie. That is, we determine the values, or bounds, between which lies a specified percentage of the population. For example, when dealing with 90% two-sided confidence bounds of (X,Y), we are saying that 90% of the population lies between X and Y with 5% less than X and 5% greater than Y.
One-Sided Confidence Bounds
One-sided confidence bounds are essentially an open-ended version of two-sided bounds. A one-sided bound defines the point where a certain percentage of the population is either higher or lower than the defined point. This means that there are two types of one-sided bounds: upper and lower. An upper one-sided bound defines a point that a certain percentage of the population is less than. Conversely, a lower one-sided bound defines a point that a specified percentage of the population is greater than. For example, if X is a 95% upper one-sided bound, this would indicate that 95% of the population is less than X. If X is a 95% lower one-sided bound, this would indicate that 95% of the population is greater than X.
Care must be taken to differentiate between one- and two-sided confidence
bounds, as these bounds can take on identical values at different
percentage levels. For example, in the figures above, we see bounds on a
hypothetical distribution. Assuming this is the same distribution in all
of the figures, we see that X marks the spot below which 5% of the
distribution's population lies. Similarly, Y represents the point above
which 5% of the population resides. Therefore, X and Y represent the 90%
two-sided bounds, since 90% of the population lies between the two points.
However, X also represents the lower one-sided 95% confidence bound, since
95% of the population lies above that point, and Y represents the upper
one-sided 95% confidence bound, since 95% of the population is below Y. It
is important to be sure of the type of bounds you are dealing with,
particularly as both upper and lower one-sided bounds can be displayed
simultaneously in Weibull++.
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