# Statistical Power Analysis in Reliability Demonstration Testing: The Probability of Test Success
## 2 Probability of Test Success  
Product development, with all its calculations and the design of the product itself, should ensure both the actual fulfillment of the required functions and the reliability requirements under the corresponding field conditions [10; 34]. Without evidence, the requirement cannot be regarded as fulfilled, whereas a hypothesis about the non-fulfillment can be established. This hypothesis must be either rejected or accepted by conducting a reliability test. Therefore, a reliability demonstration test can be regarded as a hypothesis test. The reliability requirement is formulated as a required lifetime with a certain reliability and confidence level. The estimated lifetime quantile of the test, \(t_{\rm R_{r}}\), at the required reliability, \(R_{\rm r}\), has to be greater than or equal to the required life quantile, \(t_{\rm r}\). As a statistical test is only able to provide evidence towards the rejection of a hypothesis that states the absence of a certain phenomenon of interest [35; 36; 37], the null hypothesis, \(H_{0}\), represents the non-fulfillment of the reliability requirement and is to be rejected. Because the power of the test corresponds to the detection of the fulfillment of the requirement, the alternative hypothesis, \(H_{1}\), represents the reliability requirement in terms of the lifetime. Therefore, the hypotheses to be used in a reliability demonstration test are defined as the following [38; 39; 40; 41]:  
\[H_{0}:\ t_{R_{\rm r}}<t_{\rm r} \tag{1}\]  
\[H_{1}:\ t_{R_{\rm r}}\geq t_{\rm r} \tag{2}\]  
As a fulfillment of the reliability requirement cannot be assumed without evidence of a proper demonstration by means of a successful test, the very common terms of producer risk and consumer risk in quality engineering [42; 43; 44] are not applicable here in the classical sense of type I and type II error [37; 40]; these terms are only meaningful if a manufacturing issue is at hand. For example, testing of the production parameters of a production line always endeavors to avoid deviations and effects from the specified parameter space. However, the observed effect of interest in a reliability demonstration test is the lifetime, and therefore, the desired effect. The significance level at which the null hypothesis is to be rejected is determined by the permitted type I error, \(\alpha\), which is the probability for the null hypothesis to be rejected even though the null hypothesis actually holds true. The confidence level, \(C\), of a reliability test corresponds to the complement of the permitted significance level, \(\alpha\), as it describes the probability that the reliability test rightly accepts the null hypothesis when the reliability requirement (\(t_{\rm r}\), \(R_{\rm r}\), \(C_{\rm r}\)) is actually not met by the product according to the test data. Therefore, the required confidence level, \(C_{\rm r}\), is able to assess and ensure that the obtained reliability statement is indeed true. As a great variety of test plans are available to obtain a reliability statement, the problem of choosing the correct and most promising reliability test strategy arises. In order to assess the different test strategies, an additional assessment criterion is needed. In contrast to the approach of hypothesis tests, reliability tests do not generally consider the type II error, which describes the probability \(\beta\) that the null hypothesis \(H_{0}\) is wrongly accepted. The type II error depends on the sample size, the effect size (the actual value of the lifetime quantile) and failure distribution, the sampling variance, and the desired significance level. The complement of the type II error, \(\beta\), is termed the _statistical power_ of the test, and describes the probability that the test correctly rejects the null hypothesis for a certain effect size [35]. In the context of reliability test planning, this power describes the probability of a test demonstrating the reliability requirement.