Confidence Intervals (or Bounds)

One of the most confusing concepts to an engineer new to the field is the concept of putting a probability on a probability. In life data analysis, this concept is referred to as confidence intervals or confidence bounds. In this section we will try to briefly present the concept, in less than statistical terms, but based on solid common sense.

The Black & White Marbles

To illustrate, imagine a situation where there are millions of black and white marbles in a large swimming pool and your job is to estimate the percentage of black marbles. One way to do this (other than counting all the marbles!) is to estimate the percentage of black marbles by taking a sample and then counting the number of black marbles in the sample.

Taking a Small Sample of Marbles

First, pick out a small sample of marbles and count the black ones. Let’s say you picked out 10 marbles and counted 4 black marbles. Based on this, your estimate would be that 40% of the marbles are black.

If you put the 10 marbles back into the pool and repeated this example, you might get 5 black marbles, changing your estimate to 50% black marbles.

Which of the two estimates 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%, or maybe 90% of the time this estimate is between X1% and X2%.

Taking a Larger Sample of Marbles

If you now repeat the experiment and pick out 1,000 marbles, you may get results such as 545, 570, 530, etc. for the number of black marbles in each trial. Note that the range in this case will be much narrower than before. For example, let's say that 90% of the time, the number of black marbles will be from Y1% to Y2%, where X1% < Y1% and X2% > Y2%, thus giving you a narrower interval. For confidence intervals, the larger the sample size, the narrower the confidence intervals.

Back to Reliability

Returning to the subject at hand, your task is to determine the probability of failure or reliability of all of our units. However, until all units fail, you will never know the exact value. Your task is to estimate the reliability based on a sample, much like estimating the number of black marbles in the pool. If you perform 10 different reliability tests for your units and estimate the parameters using ALTA, you will obtain slightly different parameters for the distribution each time and thus slightly different reliability results. However, when employing confidence bounds, you obtain a range in which these values are more likely to occur X percent of the time. Remember that each parameter is an estimate of the true parameter, a true parameter that is unknown to us.

The One-Sided and Two-Sided Confidence Bounds is presented next.

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