The calculations for the probability of passing a test plan depend on the distribution that models the failures. For a log-location-scale distribution, the probability is a function of the ratio of improvement. For a location-scale distribution, the probability is a function of the amount of improvement. The expression of the formulas divides neatly into two cases that depend on whether you specify the sample size or the testing time.
Sample size
confidence level satisfies the following equation:
,
the solution, 
,
of the equation has the following form:
is the inverse cumulative distribution function of the beta distribution with
the following shape parameters:
,
invert the function 
.
The inversion depends on the distribution family.
- Log-location-scale family
- Location-scale family
and the improvement:
where 
is the reliability function of the distribution model in terms of
and 
.
- Log-location-scale family
- Location-scale family
The following table gives the function of 
for the distribution family and the goal of the test:
| Reliability Goal | ||||
|---|---|---|---|---|
|
|
|
|
|
| Log-location-scale |    |
   |
   |
   |
| Reliability Goal | ||||
|
|
|
|
|
| Location-scale |    |
   |
   |
   |
Example of 
for the Weibull distribution
,
and a given sample size, the probability of passing has the following form:
where
Testing time
confidence level satisfies the following equation:
),
the solution of the equation, 
,
has the following form:
),
no closed-form solution exists. Meeker and Escobar (1998)1 give the following approximate solution:
where
Minitab finds the exact solution numerically when
.
and the improvement:
where 
is the reliability function of the distribution model in terms of
and 
.
- Log-location-scale family
- Location-scale family
The function 
has the same definitions as when the specifications for the test give the
sample size.
Example of 
for the Weibull distribution
,
and a given testing time, the probability of passing has the following form:
where
Notation
| Term | Description |
|---|---|
| N | sample size for the design when the specifications for the test provide the sample size |
| m | number of units that fail during the test |
 | significance level, such that the confidence level for the demonstration
test is 
|
 | scale parameter |
 | cumulative distribution function of the standard distribution for the selected log-location-scale or location-scale distribution |
 | inverse cumulative distribution function of the standard distribution for the selected log-location-scale or location-scale distribution |
 | location parameter for the distribution that meets the goal of the test |
 | shape parameter of the Weibull distribution |
 | testing time when the specifications for the test provide the sample size |
 | ratio of improvement for log-location-scale distributions or the amount of improvement for location-scale distributions |
 | reliability at time t that is the goal for the test |
 | percentile at percent p that is the goal for the test |
 | mean-time-to-failure that is the goal of the test |
 | testing time when the specifications for the test provide the testing time |
 | sample size when the specification for the test provide the testing time |
