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A Numerical Example for Failure-Terminated Tests

The life test data given in Table 1 comes from an Exponential distribution. These T1, T2, ... Tn failure times correspond to (n = 45) devices D1 ... D45, placed on a reliability test as represented in Figure 2.

Table 1. Device Life Data
 12.411 58.526 46.684 49.022 77.084 7.400 21.491 28.637 16.263 53.533 93.241 43.911 33.771 78.954 399.071 102.947 118.077 61.894 72.435 108.561 46.252 40.479 95.291 10.291 27.668 116.729 149.432 59.067 199.458 45.771 272.005 60.266 233.254 87.592 137.149 50.668 89.601 313.879 150.011 173.580 220.413 182.737 6.171 162.792 82.273

Figure 2. Representation of the Times to Failure of the "n" Devices on Test (Click to Zoom)

Using the statistic of total test time, we obtain the following point estimator of the mean life.

Est. Mean Life = Total Test Time / Sample Size = Σ Ti / n
= 4495.75/45 = 99.9

The reciprocal of the mean life, yields the point estimate of the device failure rate:

Failure Rate = 1 / Mean Life = 1 / 99.9 = 0.01001

We first verify whether the statistical distribution of the life of the device is Exponential and whether the data came from independent observations (Reference 7). We then use appropriate statistical reliability methods to calculate the CI. In the Exponential case we use the Chi-Square (χ2) distribution and the Total Test Time (T) statistic to obtain a CI for the (true but unknown) device mean life θ (or rate λ = 1/θ). The formula to obtain an Exponential CI for the true, but unknown mean , with a confidence level 100(1 - α)%, is given by:

In this example: T = Σ Ti = 4496.75 is the Total Test Time and n = k = 45 is the (full) sample size. The test (lives) is failure terminated. Hence, the Chi-Square table value (χ2α/2,γ) has DF = 2n = 90. Confidence coefficient (1 - α) is then selected, according to whether our CI requires an 80%, 90%, 95% confidence, etc. A 95% confidence (1 - α) yields = 0.05, α/2 = 0.025 and 1 - α/2 = 0.975. Therefore, the Chi-Square table values for our example are:

The corresponding CI for the true mean life , with confidence level of 95% is:

(2 x 4496.75/118.14; 2 x 4496.75/65.65) = (76.13, 136.99)

Since the failure rate λ = 1/θ, an associated CI for the failure rate, with confidence level of 95%, can also be obtained by using the reciprocal values of the above CI for the mean:

(1/136.99, 1/76.13) = (0.0073, 0.0131)

Such a CI means that, 95% of the times that we derive it from test data, the true but unknown failure rate (λ) is between 0.0073 and 0.0131 (but 5% of the times it can be elsewhere).

Finally, because the Exponential is a one-parameter distribution, the device reliability at any given mission time T is also obtained using the mean as follows: R(T) = P{X ≥ T} = Exp{-T/θ) = Exp{-λT}.

Then, a 95% CI for the reliability at any mission time T can be obtained by using the mean or the failure rate CI upper/lower limits. For our example and for T = 100, we use the upper/lower limits of the CI for the failure rate (ρ) and obtain:

R1(T) = P{X > T} = Exp{-T/θ1} = Exp{-λ1T} = Exp{-0.0073 x 100} = 0.48

R2(T) = P{X > T} = Exp{-T/θ2) = Exp{-λ2T} = Exp{-0.0131 x 100} = 0.27

Therefore, a 95% CI for the reliability for a mission time T = 100 units is: (0.27, 0.48).

Such CI means that, 95% of the times we derive it from test data, the true but unknown reliability for such mission time is between 0.27 and 0.48.

Often, we just need a lower or upper bound on reliability. Assume we are interested in a 90% reliability lower bound for the above example and mission time T = 100. We re-estimate the failure rate bound, for the error α = 0.1, for only one side. This changes the Chi-Square table percentile. The new Chi-Square table (1 - α) percentile (corresponding to DF = 90 and α = 0.1) is now 107.57 and we use it to obtain the 90% confidence bound for the Exponential mean (or its reciprocal, the failure rate).

The 90% Lower Bound for the (unknown) mean, induces a failure rate bound:

Which, in turn, allows us to calculate a 90% Lower Bound for the reliability "R" of the device, for T = 100, as:

R(100) = P{X ≥ 100} = Exp{-100 x λ) =
Exp{-100 x 0.01196} = 0.3024

Such a Lower Bound means that, 90% of the times we derive it from test data, the true but unknown device reliability is at least 0.3024, for mission time T = 100.