Reliability with Confidence

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Musings on Reliability and Maintenance Topics

3 min readFeb 17, 2013

Reliability with confidence

There are rare situations when we would like to estimate the reliability at the lower confidence level after estimating the mean from a sample (often a test result). And, to make even more rare of a situation, we know the population standard deviation.

This is not the same as tolerance intervals which we’ll cover in a separate post. Confidence intervals deal with the sampling and the estimate of a specific population parameter. Tolerance intervals apply to the individual values not the parameter, the mean for example.

I heard from Professor Bill Meeker that they did a study of incandescent light bulbs and found they wear out with a normal distribution. For the purpose of this example let use the variance of the wear out time to be 625 hours2. And, let’s say we tracked 25 light bulbs in our home and found the average time to failure to be 2,000 hours.

We are interested in estimating the time till at least 95% of the bulbs are working with 90% confidence. What is the 95% level of reliability with 90% confidence?

One way to step through this type of problem is to consider 7 steps that essentially permit us to estimate the mean value, determine the lower confidence level for that estimated mean value, then calculate the point in time where the first 5% have failed (95% still working).

1. Determine the required reliability and confidence levels for the estimate.

This is often provide by management or local policy. For example, in this example problem we are asked to find the 95% reliability level of reliability with 90% confidence.

2. Determine the value of the standard deviation.

We are given the population variance or standard deviation for this type of problem. In this case population variance is 625 thus the population standard deviation is 25. While it is rare that we know the population variance it is possible given previous studies.

3. Obtain the samples size, n.

This is from the sample used to create the estimate of the time to failure average or mean. We have data from an experiment using 25 light bulbs, thus n = 25.

4. Estimate the value for the mean life, X̄

Sum the time to failure data and divide by n. Or, as given in this example, X̄ = 2,000

5. Estimate the standard error of the mean as σ / √n

The standard deviation is 25 and the square root of the sample size in this case in 5. Thus the standard error is 25 / 5 = 5. This value along with the Z value corresponding to the confidence level permits the next step.

6. Calculate the lower confidence for the mean life at the given confidence level

$$ \large\displaystyle {{\bar{X}}_{L}}=\bar{X}-Z\frac{\sigma }{\sqrt{n}}$$

where Z is the z-value from your favorite standard normal table corresponding to the confidence level desired for the estimate of the mean life. We have the standard error of the mean of 5 from above, and the z-value for the one-sided lower confidence of 90% is 1.282. Inserting the numbers and calculating we find

$$ \large\displaystyle {{\bar{X}}_{L}}=2,000–1.282\times 5=1,994$$