Parametric Binomial Example - Demonstrate MTTF

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This example appears in the Life Data Analysis Reference book.

In this example, we will use the parametric binomial method to design a test that will demonstrate MTTF=75\,\! hours with a 95% confidence if no failure occur during the test f=0\,\!. We will assume a Weibull distribution with a shape parameter \beta =1.5\,\!. We want to determine the number of units to test for {{t}_{TEST}}=60\,\! hours to demonstrate this goal.

The first step in this case involves determining the value of the scale parameter \eta \,\! from the MTTF\,\! equation. The equation for the MTTF\,\! for the Weibull distribution is:

MTTF=\eta \cdot \Gamma (1+\frac{1}{\beta })\,\!

where \Gamma (x)\,\! is the gamma function of x\,\!. This can be rearranged in terms of \eta\,\!:

\eta =\frac{MTTF}{\Gamma (1+\tfrac{1}{\beta })}\,\!

Since MTTF\,\! and \beta \,\! have been specified, it is a relatively simple matter to calculate \eta =83.1\,\!. From this point on, the procedure is the same as the reliability demonstration example. Next, the value of {{R}_{TEST}}\,\! is calculated as:

{{R}_{TEST}}={{e}^{-{{({{t}_{TEST}}/\eta )}^{\beta }}}}={{e}^{-{{(60/83.1)}^{1.5}}}}=0.541=54.1%\,\!

The last step is to substitute the appropriate values into the cumulative binomial equation. The values of CL\,\!, {{t}_{TEST}}\,\!, \beta \,\!, f\,\! and \eta \,\! have already been calculated or specified, so it merely remains to solve the binomial equation for n\,\!. The value is calculated as n=4.8811,\,\! or n=5\,\! units, since the fractional value must be rounded up to the next integer value. This example solved in Weibull++ is shown next.

The procedure for determining the required test time proceeds in the same manner, determining \eta \,\! from the MTTF\,\! equation, and following the previously described methodology to determine {{t}_{TEST}}\,\! from the binomial equation with Weibull distribution.

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