## 3 Results
level when the samples came from gamma distributions with the same expected values. It seems that for samples of at least \(n=100\), the central limit theorem [20] can be used to test \(H_{\mathrm{o}}\) instead. When it comes to comparing the method with others, a good place seems to be the methodology suggested by [12], which claims that it gives results \"as good as, if not better than, the other methods discussed in (its) literature...\". Even though a direct comparison is usually not possible in this case, since the cited paper uses mostly different parameter set-ups and smaller sample sizes, the sizes also being mostly unequal unlike the cases covered in this paper, a similar set-up can in some instances be detected. This allows us to shed some light on the functionality of the procedure suggested in this text. The authors of [12] use the delta symbol as the shape parameter (which equals \(k\) in this text) and the lambda letter as the scale parameter (here, the equivalent is the theta symbol). For the comparisons, the test power Table 1 published in [12], page 64, was selected, specifically the row where both sample sizes are equal to 25. This is the number closest to the sample sizes of 30 analyzed in this paper. The reproduction of their results is below in Table 12. Their methodology was also validated with simulations. Figure 1: The Shapiro–Wilk test \(p\)-value distribution when testing normality of \(\overline{X}-\overline{Y}\) under \(H_{0}:E(X)=E(Y)\); sample sizes of \(n=100\) used. Given that \(E(X)=k_{1}\theta_{1}\
eq E(Y)=k_{2}\theta_{2}\) in the test power simulations, the comparison of Tables 12 and 13 shows that the method of [12] is better when the differences in the means \(E(X)\), \(E(Y)\) of the two gamma distributions are rather small because the method of [12] is able to detect smaller differences in the means more often with its greater power as long as the simulation results in [12] are correct. In such instances, the method proposed here needs a somewhat larger difference in the means to reach similar test powers (for instance, the levels 0.51 and 0.73). For medium- and larger-sized differences, however, the methods seem to have the same test power--the best possible, nearing the level of 1. This is the case for population mean differences of the magnitude given by \(\theta_{2}\geq 2\) and \(k_{1}=k_{2}\geq 5\), as the figures show. As an example, related to smaller differences, for \(k_{1}=k_{2}=7.5\) and \(\theta_{2}=1.25\), the difference in the means is \(|k_{1}\theta_{1}-k_{2}\theta_{2}|=|7.5\)-\(1-7.5\)-\(1.25|=1.875\), whereas with \(\theta_{2}=1.5\), the difference is 3.75. These differences lead to similar test powers between 0.5 and 0.6 for both methods, as the tables show. This suggests that for not too small of differences in the means (a difference of at least five based on the tables), the occurrence of which can be assessed with parameter estimates, the method proposed here is quite usable, as its test power seems similar to other methods; it is very high, yet the proposed procedure is far simpler. Regarding small differences, they must be admitted to some degree more discernible for the proposed method to be more competitive. Otherwise, the only limitation of the proposed method, as already outlined in the methodology section, is that its properties relate at the moment only to the parameter scenarios considered in the analysis of this paper. This is not the case for methods based on mathematical derivation, the validity of which is universal, unless mathematical restrictions are placed on them too. However, the presented methodology coupled with the attached computer code may be used for researching other scenarios as well, including even smaller sample sizes than thirty datapoints.