# Methods and formulas for reliability testing in Demonstration Test Plan

## Reliability is provided

If you specify the reliability goal in terms of a time and a reliability, Minitab first calculates the parameter to be demonstrated using the formula for the specified distribution in the following table.

Next, Minitab calculates the testing time or sample size the same way as in the substantiation testing case.

Distribution (parameter) Parameter to be demonstrated
Weibull (scale)
Exponential (mean)

Smallest extreme value (location)

Normal (mean)

Logistic (mean)

Lognormal (log-location)

Loglogistic (log-location)

### Notation

TermDescription
ttime
R(t)reliability function
θscale (Weibull) or mean (exponential)
βshape parameter (Weibull)
σscale parameter (lognormal, loglogistic, logistic, normal, smallest extreme value)
μmean (normal, logistic) or log-location (lognormal, loglogistic)
Φ-1 inverse CDF for the corresponding distribution

## Pth percentile is provided

If you specify the reliability goal in terms of the Pth percentile (tp), Minitab first calculates the parameter to be demonstrated using the formula for the specified distribution in the following table.

Next, Minitab calculates the testing time or sample size the same way as in the substantiation testing case.

Distribution (parameter) Parameter to be demonstrated
Weibull (scale)
Exponential (mean)

Smallest extreme value (location)

Normal (mean)

Logistic (mean)

Lognormal (log-location)

Loglogistic (log-location)

### Notation

TermDescription
ttime
R(t)reliability function
ppercentile
αscale (Weibull)
θ mean (exponential)
μmean (normal, logistic), location (smallest extreme value), or log-location (lognormal, loglogistic)
Φ-1 inverse CDF for the corresponding distribution

## MTTF is provided

If you specify the reliability goal in terms of the MTTF, Minitab first calculates the parameter to be demonstrated using the formula for the specified distribution in the following table.

Next, Minitab calculates the testing time or sample size the same way as in the substantiation testing case.

Distribution (parameter) Parameter to be demonstrated
Weibull (scale)
Exponential (mean)

Normal (mean)

Logistic (mean)

Lognormal (log-location)

Loglogistic (log-location)

Smallest extreme value (location)

### Notation

TermDescription
αscale (Weibull)
σscale (lognormal, loglogistic, smallest extreme value distribution)
βshape (Weibull)
θ mean (exponential)
μmean (normal, logistic), location (smallest extreme value), or log-location (lognormal, loglogistic)
MTTFmean time to failure
cEuler's constant ≈ 0.5772
Γgamma function
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