**4. Subject and Research Method**

The paper presents a method for assessing the manufacturing process capability of an shafts with grooves, a typical production process whose quality assessment is made on the basis of capability analysis and is carried out during the final product control. Based on the analysis of many industrial products it can be concluded that one half of all machine parts are rotational parts: shafts (over 40%), discs, sleeves, thin wall cylinders, rings, etc. [57].

In the case of production, there are two categories of processes when the capability analysis using the bootstrap method seems to be practically the best solution. These categories include processes in which:


The production process of an shafts with grooves belongs to the second of the mentioned categories of processes.

In the present case, due to the small size of the general population of products *n* = 10, the analysis of the process capability will be carried out using the bootstrap method, with the population size being the same as the sample size.

The shafts with grooves is manufactured by a manufacturing plant that provides complete reinforcement for defense purposes. The shafts is a necessary component of this armament.

The production of the shafts with the groove was chosen for the following reasons:


• Due to the number of shafts produced, it is not possible to assess the shape of the process variable distribution, so the assumption about the normality of the distribution of quality characteristics is rejected.

The results analyzed were obtained on the basis of measurements made using a CMM (coordinate-measuring machine). The geometry of this product defines 19 parameters (18 measurement parameters and total length), with parameter 12 selected as the process variable for analyzing the process capability.

To assess the process capability, a research procedure consisting of five activities was used (see Figure 4):


**Figure 4.** Diagram of the algorithm for assessing the quality of processes using the classic, percentile and bootstrap methods. Source: own study.

The principle of process capability assessment should be to take into account confidence intervals with a lower limit *CP MIN*. This indicates that with some probability (the confidence interval was calculated at *P* = 95% in the work), the capacity of the analyzed process will not be lower than the lower limit of the confidence interval [58,59].

The detailed scope of research for individual activities is presented in Figure 5. Due to the large number of results (especially charts), the results are presented only for the final stage of the procedure, i.e., the assessment of the process capability *CP*.

**Figure 5.** Sequence of actions in the process capability assessment procedure and presentation of individual results. Source: own study.
