Grouped per Configuration - Standard Gompertz Model

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This example appears in the Reliability growth reference.

A new design is put through a reliability growth test. The requirement is that after the ninth stage the design will exhibit an 85% reliability with a 90% confidence level. Given the data in the following table, do the following:

1. Estimate the parameters of the standard Gompertz model.
2. What is the initial reliability at [math]\displaystyle{ T=0\,\! }[/math] ?
3. Determine the reliability at the end of the ninth stage and check to see whether the goal has been met.

Grouped per Configuration Data

Stage	Number of Units	Number of Failures
1	10	5
2	8	3
3	9	3
4	9	2
5	10	2
6	10	1
7	10	1
8	10	1
9	10	1

Solution

1. The data is entered in cumulative format and the estimated standard Gompertz parameters are shown in the following figure.
   
2. The initial reliability at [math]\displaystyle{ T=0\,\! }[/math] is equal to:

   [math]\displaystyle{ \begin{align} {{R}_{T=0}}= & a\cdot b \\ = & 0.9497\cdot 0.5249 \\ = & 0.4985 \end{align}\,\! }[/math]

3. The reliability at the ninth stage can be calculated using the Quick Calculation Pad (QCP) as shown in the figure below.
   

   The estimated reliability at the end of the ninth stage is equal to 91.92%. However, the lower limit at the 90% 1-sided confidence bound is equal to 82.15%. Therefore, the required goal of 85% reliability at a 90% confidence level has not been met.