## 3 Bootstrap Method  
There are methods that simplify the procedure when assessing process capability using numerical indicators. One of such methods is bootstrap analysis based on the so-called bootstrap samples [44], which can be used when the sample size is not very large (show Figure 3). The purpose of this analysis is the possible verification of previously obtained results using factor methods.  
Bootstrap methods have been known for over 20 years, but only in recent years they have been widely used, primarily in stochastic simulation models. The basis of this method is the assumption that the future is similar to the past. Therefore, instead of studying the past and trying to describe it using theoretical distributions, and then simulating the future using the selected distributions, you can generate simulation input data directly from historical data [45; 46]. As a consequence, this means that since the observed sample of real data contains all the necessary information about the studied population, this sample can be treated as a population.  
Bootstrap analysis consists of a draw with returning individual results from a random sample (from the original data) and then creating a new sample from the drawn samples for the study [47]. Thus, it is a method of estimating the distribution of estimation errors, using multiple random draws  
Figure 3: Scheme of bootstrap analysis. Source: own study.  
Figure 2: Graphic illustration of process capability. Source: own study.  
of observations with returns from the original sample. The draws take into account all possible combinations of elements from the sample, based on real data [48]. It allows checking the distribution of parameters in relation to the initial sample [49]. The bootstrap method is useful when the form of the distribution of the variable in the population is not known and when the quality or amount of information collected does not allow the use of classic statistical methods [50]. Due to the fact that it does not make assumptions about the distribution in the population, it is included in the non-parametric methods.  
Let \(x_{jl}\), \(x_{j2}\), \(x_{j3}\),... \(x_{ji}\) denote the values of the process variable \(X_{i}\) of the production process under investigation. From a given set of \(x_{ji}\) values, we draw with returning \(n\) measurement values and in this way we get new bootstrap samples \({B_{j}}^{\star 1}\), \({B_{j}}^{\star 2}\), \({B_{j}}^{\star 3}\),... \({B_{j}}^{\star B}\).  
Each bootstrap sample consists of exactly the same number of \"\(n\)\" elements as the number of values tested [30; 51; 52; 53].  
The number of bootstrap draws cannot be less than \(n^{n}\). Based on empirical research, it has been shown that a sufficient number of draws for conducting tests is \(B=1000\) measurements [54]. After obtaining the set number of bootstrap attempts, based on them, inference is calculated by calculating the appropriate statistics \(\phi\). The empirical distribution obtained in this way is used to make inferences about the parameter \(\theta\)[48; 55].  
The main advantage of this is that the process variable distribution is not studied but empirically constructed based on measuring a large number of samples of the process variable value [56]. The advantage of this is also the ability to assess process capabilities based on abnormal distributions, characterized by high skewness, flattening, drift, etc.  
To sum up the above, the following analogy is crucial for the use of the bootstrap method in statistical inference: the bootstrap sample is for the sample drawn what the sample drawn for the entire population.