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| '''Weibull Distribution Example - Demonstrate MTTF'''
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| In this example, we will design a test to demonstrate <math>MTTF=75</math> hours, with a 95% confidence. We will once again assume a Weibull distribution with a shape parameter <math>\beta =1.5</math>. No failures will be allowed on this test, or <math>f=0</math>. We want to determine the number of units to test for <math>{{t}_{TEST}}=60</math> hours to demonstrate this goal.
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| The first step in this case involves determining the value of the scale parameter <math>\eta </math> from the <math>MTTF</math> equation. The equation for the <math>MTTF</math> for the Weibull distribution is:
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| <br>
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| <center><math>MTTF=\eta \cdot \Gamma (1+\frac{1}{\beta })</math></center>
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| <br>
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| where <math>\Gamma (x)</math> is the gamma function of <math>x</math>. This can be rearranged in terms of <math>\eta</math>:
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| <br>
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| <center><math>\eta =\frac{MTTF}{\Gamma (1+\tfrac{1}{\beta })}</math></center>
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| <br>
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| Since <math>MTTF</math> and <math>\beta </math> have been specified, it is a relatively simple matter to calculate <math>\eta =83.1</math>. From this point on, the procedure is the same as the reliability demonstration example. Next, the value of <math>{{R}_{TEST}}</math> is calculated as:
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| <br>
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| <center><math>{{R}_{TEST}}={{e}^{-{{({{t}_{TEST}}/\eta )}^{\beta }}}}={{e}^{-{{(60/83.1)}^{1.5}}}}=0.541=54.1%</math></center>
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| <br>
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| The last step is to substitute the appropriate values into the cumulative binomial equation. The values of <math>CL</math>, <math>{{t}_{TEST}}</math>, <math>\beta </math>, <math>f</math> and <math>\eta </math> have already been calculated or specified, so it merely remains to solve the binomial equation for <math>n</math>. The value is calculated as <math>n=4.8811,</math> or <math>n=5</math> units, since the fractional value must be rounded up to the next integer value. This example solved in Weibull++ is shown next.
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| <br>
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| [[Image:RDT Weibull Demonstrate MTTF.png|thumb|center|650px| ]] | |
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| <br>
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| The procedure for determining the required test time proceeds in the same manner, determining <math>\eta </math> from the <math>MTTF</math> equation, and following the previously described methodology to determine <math>{{t}_{TEST}}</math> from the binomial equation with Weibull distribution.
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