## 2 Materials and Methods
The problem here would just be shifted from the unknown \(p\) to the unknown \(d\) even if \(var\) was known. This situation can be helped, however, if it turns out that the multiple \(d\) happens to manifest a certain stability across various gamma-distribution set-ups and/or various sample sizes. If this property is present, the knowledge of what distributions the samples came from is not so relevant anymore since \(d\) is more or less the same regardless of the distributions. In this fortuitous instance, the only remaining problem is then the unknown population variance \(var\), which must be estimated. This logic leads to the suggestion of using the formula \(p=d\cdot\widehat{var}^{1/2}\) for testing purposes as long as \(d\) does have the property just described. Finding the \(d\)s for various situations and examining their stability ends the second part of the analysis. Since \(\widehat{var}\) is used instead of \(var\) in the percentile-generating formula, the potential of this method should be verified as well even if \(d\) is relatively stable. This constitutes the final stage of the analysis. The verification takes the form of analyzing the error rates of the method [15] and comparing it with other procedures. Regarding the technical details, using the common symbols \(\Gamma(k_{1},\ \theta_{1})\) and \(\Gamma(k_{2},\ \theta_{2})\) for the two gamma distributions worked with, the distribution of the variable \(\overline{X}-\overline{Y}\) is analyzed under \(H_{0}\) : \(E(X)=E(Y)\) or equivalently [1]:  
\[H_{0}:k_{1}\theta_{1}=k_{2}\theta_{2}. \tag{5}\]  
The following distribution set-ups are analyzed with the distribution parameter combinations grouped into several \"scenarios\":  
\[\text{scenario 1}-k_{1},k_{2}\in K=\{1,2,\dots,10\},\ \theta_{1}\in M=\{1,2, \dots,30\}, \tag{6}\]  
\[\text{scenario 2}-k_{1},k_{2}\in L=\{11,\ 12,\dots,\ 20\},\ \theta_{1}\in M, \tag{7}\]  
\[\text{scenario 3}-k_{1},k_{2}\in N=\{21,\ 22,\dots,\ 30\},\ \theta_{1}\in M, \tag{8}\]  
\[\text{scenario 4}-k_{1}\in K,\ k_{2}\in L,\ \theta_{1}\in M, \tag{9}\]  
\[\text{scenario 5}-k_{1}\in K,\ k_{2}\in N,\ \theta_{1}\in M, \tag{10}\]  
\[\text{scenario 6}-k_{1}\in L,\ k_{2}\in N,\ \theta_{1}\in M. \tag{11}\]  
All parameter combinations work with all sample sizes from the set \(n\in\{30,\ 40,\...,\ 80\}\). Both samples always have the same size. This is usually not a problem to achieve in practice as long as a controlled experiment is possible. Since the focus is on the percentiles of \(\overline{X}-\overline{Y}\) under \(H_{0}\), a specific sample size is chosen and scenario-based \(k_{1},k_{2},\theta_{1}\) parameters are set up, determining also \(\theta_{2}=k_{1}\theta_{1}/k_{2}\) so that the null hypothesis holds whereupon thousands of realizations of the statistic \(\overline{X}-\overline{Y}\) are generated from two independent random samples drawn from the set-up gamma distributions. This experiment suggests to an acceptable level of detail the distribution of the difference under \(H_{0}\). Once the distribution is detected,its 2.5% and 97.5% \(p\) percentiles are calculated and hence also the multiples \(d=p\cdot var^{-1/2}\) where  
\[var=var\big{(}\overline{X}\big{)}+var\big{(}\overline{Y}\big{)}=n^{-1}\Big{(}k_ {1}\theta_{1}^{2}+k_{2}\theta_{2}^{2}\Big{)} \tag{12}\]  
is calculated directly since its form is known within the given parameter setting [1]. At the end of these calculations, one should rather talk about entire distributions of the identified \(ds\), one \(d\) having been calculated for the 2.5% percentile and the other for the 97.5% percentile, because each \(d\) is generally different for each parameter set-up and sample size. Therefore, the findings presented in this text are in the form of \"the average lower \(d\)\" and \"the average upper \(d\)\", each average calculated for a given scenario and sample size. The average results from the aggregation of the within-scenario results.