## 5 Conclusions  
This paper presented a simple method that can be used to test the statistical significance of the difference between two independent gamma-distributed random variables in a small-to-medium-sized data setting. There are many real-life situations when such a technique might be useful provided the objective is to discern situations or scenarios from one another, the scenarios being evaluated or described by a positive or nonnegative measure. Some interesting applications were mentioned in the first part of this text. The method was designed based on simulations that were also used for its validation, a standard procedure in statistics, and its comparison with other methods. The simulation-based approach was adopted because the usual techniques of deriving the precise distribution of a test criterion under H\({}_{0}\): \"No difference in the expected values\" are too complex and incomparable to the presented technique in terms of the difficulty. The analysis was made possible through a code written in the R statistical software development environment, version 2022.07.2 Build 576, using suitable API functions for working with the gamma distribution, as well as API graphical tools (see Appendix A of this text). The presented conclusions concern small- and medium-sized samples of 30 to 80 datapoints coming independently from gamma distributions. For larger samples (at least a hundred datapoints), it seems that \(\overline{X}-\overline{Y}\) is approximately normally distributed under H\({}_{0}\) in a majority of considered cases, as was confirmed with the Shapiro-Wilk test. This suggests that an approximate normality-based test could be applied for such sample sizes with the same objective in mind instead of the proposed procedure. The normality approach to testing can also be used once the gamma distribution shape parameters are, say, 30 or higher. The resulting method has a very good type I error probability within the considered situations, while its power averages above 0.7 for sample sizes of at least 60 for reasonably pronounced differences in the expected values (see Tables 8 and 9). The test power of the method was compared to the methodology proposed in [12] that claims to be as good as other methods, if not better. The method proposed here seems to possess comparable power unless the means differ little. This was suggested by the comparisons that could be made, at least approximately, for the parameter scenarios and sample sizes used in [12] were not always close to what was considered in this paper. The analyses suggest that in many cases, it is possible to use the method designed in the paper, the major advantage of it being its simplicity (see page 6). Table 15 summarizes the recommendations. In practice, the true scenario behind the parameters can be identified with parameter estimates. Once the scenario estimate is given, the proper row in Table 15 is selected and the \(d\)s identified, leading to the percentiles and the test implementation. Generally speaking, if the practitioner wants to keep the procedure even simpler, it is conceivable to use a constant \(d\) between \(1.95\) and \(2\), whatever the parameter scenario. The results obtained in this way should still possess reasonable statistical properties. In the current context, it is also possible to think of the Vysochanskij-Pettunin inequality, which implies that under H\({}_{0}\), the difference in the sample averages will fall outside the interval \(\left(-3 var^{1/2},\;3 var^{1/2}\right)\) with a probability of at most \(0.05\). The result is valid for unimodal distributions with finite variance. This instead suggests using a multiple of three to get at the proper percentiles for the testing. But of course, the result concerns the unknown population variance \(var\) and not its estimate that must be used in practice. And even if \(var\) was known, the \"at most\" theoretical conclusion additionally suggests that the multiple of three is probably too high.