## 3 Results
It can be seen that for sample sizes of around 60 and the listed scenario-dependent values of the constant c, the average power equals or exceeds 0.7. Examples of situations when the mean difference equals the scenario-specific constant are for scenario 1... \([k_{1},\theta_{1}]=[2,3]\) and \([k_{2},\theta_{2}]=[4,6.5]\) or \([k_{1},\theta_{1}]=[4,2]\) and \([k_{2},\theta_{2}]=[5,5.6]\); for scenario 2... \([k_{1},\theta_{1}]=[12,14]\) and \([k_{2},\theta_{2}]=[13,15.3]\) or \([k_{1},\theta_{1}]=[16,11]\) and \([k_{2},\theta_{2}]=[16,12.9]\); for scenario 3... \([k_{1},\ \theta_{1}]=[21,25]\) and \([k_{2},\ \theta_{2}]=[21.5,26.2]\); and for scenario 6... \([k_{1},\ \theta_{1}]=[23,21],\ [k_{2},\ \theta_{2}]=[23,22.3]\). To conclude the results so far, a few remarks are due. First, when the shape parameter exceeds thirty, the normality approximation of the gamma distribution is generally expected \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Size n** & **Average Alpha** & **2.5\%; 97.5\% Percentiles** \\ \hline
30 & 0.064 & 0.057; 0.070 \\
40 & 0.060 & 0.053; 0.067 \\
50 & 0.058 & 0.051; 0.065 \\
60 & 0.057 & 0.050; 0.063 \\
70 & 0.056 & 0.049; 0.062 \\
80 & 0.055 & 0.049; 0.061 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Type I error probability alpha for the test (lower \(d=-1.95\) upper \(d=1.97\)): average alpha (averaged across \(k_{1},k_{2},\theta_{1}\) from scenario 6) and percentiles for alpha. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Size n** & **Avg. Power (sc1; sc2; sc3)** & **Avg. Power (sc4; sc5; sc6)** \\ \hline
30 & 0.59; 0.56; 0.56 & 0.60; 0.68; 0.59 \\
40 & 0.64; 0.62; 0.61 & 0.65; 0.72; 0.65 \\
50 & 0.68; 0.66; 0.66 & 0.69; 0.76; 0.69 \\
60 & 0.72; 0.70; 0.70 & 0.72; 0.79; 0.73 \\
70 & 0.74; 0.73; 0.73 & 0.75; 0.82; 0.76 \\
80 & 0.77; 0.76; 0.76 & 0.77; 0.84; 0.79 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Average test power per scenario “sc” and constant c: sc1\(\sim\)c = 20, sc2\(\sim\)c = 30, sc3\(\sim\)c = 38, sc4\(\sim\)c = 14, sc5\(\sim\)c = 16, sc6\(\sim\)c = 28. The average taken across within-scenario combinations of \(k_{1},k_{2},\theta_{1}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Size n** & **Average Alpha** & **2.5\%; 97.5\% Percentiles** \\ \hline
30 & 0.070 & 0.060; 0.092 \\
40 & 0.066 & 0.056; 0.084 \\
50 & 0.063 & 0.054; 0.078 \\
60 & 0.061 & 0.053; 0.074 \\
70 & 0.060 & 0.052 0.071 \\
80 & 0.059 & 0.051; 0.069 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Type I error probability alpha for the test (lower \(d=-1.91\) upper \(d=2.01\)): average alpha (averaged across \(k_{1},k_{2},\theta_{1}\) from scenario 5) and percentiles for alpha. to perform. Therefore, the scenarios when both the shape parameters exceed thirty were not analyzed. Normality-based testing could be used instead. Secondly, in situations when the first scale parameter is above thirty, a change in the physical units in which the variables are observed to a higher order of magnitude squeezes this parameter for the transformed variable, so the results of the analysis expressed in different units may be applicable after the transformation as well. Thirdly, the sample sizes used range from thirty to eighty. Sample sizes of one hundred and more were not analyzed due to the possibility of a central limit-based normality approximation in the testing [17]. To check this possibility, an experiment was run: The Shapiro-Wilk test [18; 19] was carried out for the sample average difference normality using all the considered combinations of \(k\), theta, and the combinations set up so that \(H_{0}\) held. For each combination, the \(p\)-value of the test was stored. Figure 1 shows the distribution of the \(p\)-values, a majority of them leading to accepting normality. Specifically, 95 percent of them are 0.011 or higher. In other words, for samples of \(n=100\), the test in a majority of \(k\)-theta cases accepted the normality of the sample average difference at the 1% sig.