Design of Experiments Guide > Prospective Sample Size and Power > Reliability Demonstration Calculator > Statistical Details for the Reliability Demonstration Calculator
Publication date: 03/23/2021

Statistical Details for the Reliability Demonstration Calculator

The reliability demonstration depends on the assumed failure time distribution with scale parameter σ. The reliability standard, or probability of survival at time t and location μ is stated as follows:

Equation shown here

where μ is solved for using the following:

Equation shown here.

To calculate sample size and the size of the test, the probability of survival at time t is posed as a hypothesis test:

Equation shown here

where p* is the standard probability of survival at time t*.

We want to test the hypothesis at the α level or as follows:

α = Pr(k or few failures | H0 true).

Since the test is of n independent units, the number of failures has a binomial (n, p) distribution where p is the probability of a unit failing before time t. Therefore, we can express α as a function of t and n:

Equation shown here

where μ* and σ* are from the assumed reliability standard.

Properties of the binomial and beta distributions result in being able to solve for t using:

Equation shown here

For n, Brent’s method is used to find the root of:

Equation shown here


B1(α; n k, k + 1) is the α quantile of the Beta(n k; k + 1) distribution

and Φ() is the cumulative distribution function of the assumed failure time distribution.

For more information about calculations in JMP, see Barker (2011, Section 5).

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