# Statistical Power Analysis in Reliability Demonstration Testing: The Probability of Test Success
## 5 Case Study
The \(P_{\text{bs}}\) for the uncensored EoL test with regard to the sample size \(n\) is shown in Figure 13.  
For an adequate value of \(P_{\text{bs}}=80\%\) of the EoL test, a specimen \(n\approx 50\) should be used. The values of \(P_{\text{bs}}\) in the right-censored EoL test for several sample sizes and censoring proportions can be seen in Figure 14.  
A greater censoring proportion requires a larger sample size in order to reach a value of \(P_{\text{bs}}\geq 80\%\). For a moderate censoring proportion of \(10\%\), the required sample size is \(>\)50, and for a censoring proportion of about 50%, the required sample size is \(>\)70. In order to assess the benefit of censoring as well as the costs one test would generate, the median of the total test costs can be calculated approximately using (36) and the sample size. For demonstration purposes, we assume the cost of one specimen as EUR 2,000 and the cost of testing one specimen as 400 EUR/MWh. The median of the total test costs of the uncensored EoL test calculated using the approximation can be seen in Figure 13. The total  
Figure 14: Contour of the \(P_{\text{bs}}\) of the right-censored EoL test for reliability demonstration of the battery with regard to sample size \(n\) and censoring proportion, calculated using the approximate method. Figure 13: \(P_{\text{bs}}\) of the uncensored EoL test (bottom) and total test costs (top) for reliability demonstration of the battery with regard to sample size \(n\), calculated using the approximate method. test costs and the required sample size of the censored EoL test for a value of \(P_{\text{ts}}=80\%\) are shown in Figure 15 (type II censoring) and Figure 16 (type I censoring). The censoring enables a maximum reduction in costs of about 21% for a fixed sample size. However, stronger censoring results in lower values of \(P_{\text{ts}}\), which in turn result in a higher required sample size, and thus higher test costs. The SR test without permitted failures and no lifetime ratio does need \(n=45\) specimen to survive \(t_{\text{r}}=400\,\text{MWh}\) in order to demonstrate the requirement of Table 3 (using (15)). However, when using (16), the SR test only yields a value of \(P_{\text{ts}}=37.8\,\%\); therefore, it should not be used at all. The values of \(P_{\text{ts}}\) in an SR test using a lifetime ratio can be seen in Figure 17.  
Figure 16: Total test costs and required sample size \(n\) for the type I right-censored EoL test to achieve a Probability of Test Success \(P_{\text{ts}}=80\%\) for reliability demonstration of the battery with regard to censoring proportion, calculated using the approximate method. Figure 15: Total test costs and required sample size \(n\) for the type II right-censored EoL test to achieve a Probability of Test Success \(P_{\text{ts}}=80\%\) for reliability demonstration of the battery with regard to censoring proportion, calculated using the approximate method. The required sample size for SR tests using the lifetime ratio is calculated accordingly, using (19) and (20). It can be seen that the SR test using a lifetime ratio reaches its maximum value of \(P_{\text{Is}}\) at 37.8 %. The required sample size increases exponentially for decreasing values of \(L_{\text{r}}\) and quickly drops to 1 for \(L_{R}>1\) due to the large shape parameter of the Weibull distribution. Using the information of the approximate calculations shown in Figures 13-16, the feasible region for a value of \(P_{\text{Is}}=80\) % for the EoL test in this scenario is a sample size between \(n\approx 45\) and \(n\approx 60\). Therefore, the \(P_{\text{Is}}\) of the uncensored EoL test is calculated using the general method for sample sizes between 45 and 60. Censoring the EoL test does not yield benefits here, as the cost of achieving the desired value of \(P_{\text{Is}}=80\) % increases with the increase in the censoring proportions; see Figures 15 and 16. A budget of EUR 13 million is available for reliability demonstration; therefore, the test needs to be assessed in terms of costs.