## 2 Materials and Methods  
To design the required test for a number of situations, it is not necessary to know the true probability distribution of the test statistic \(\overline{X}-\overline{Y}\) under \(H_{0}:\ E(X)=E(Y)\) in full. All that is necessary is a proper percentile, or percentiles, of that distribution for accepting or rejecting the null hypothesis. This is in line with the theory of hypothesis testing, the percentiles being also known as the \"critical values\". The methodology used in this text therefore aims at running R-coded simulations that try to identify the necessary percentiles of the distribution of \(\overline{X}-\overline{Y}\) under \(H_{0}\) for various sample sizes and gamma distributions from which many realizations of the sample averages \(\overline{X},\ \overline{Y}\) must be calculated. The distributions do not necessarily have the same parameters, but their expected values are the same for the null hypothesis to hold. Further, since it is inconvenient to have a lot of percentile estimates for different set-ups of the two gamma distributions, a simple formula is also pursued, which could be used in practice to calculate back the percentiles identified by the simulations so that the testing can be performed. However, since the formula at first relies on an unknown population quantity, the suitability of such a formula must be verified once the unknown quantity is replaced with an appropriate estimate. This verification is implemented in this text and takes the form of estimating the type I error probability of the formula-based procedure, as well as its test power. Hence, the entire analysis consists of several steps. In the first step, the analysis focuses on detecting the 2.5% and 97.5% percentiles \(p_{1}\) and \(p_{2}\) of the distribution that \(\overline{X}-\overline{Y}\) follows under \(H_{0}\). This can be performed with large enough simulations that lead, for each considered set-up of the two gamma distributions and each considered sample size, to many realized values of the difference under \(H_{0}\), giving a good estimate of the \(\overline{X}-\overline{Y}\) distribution and its percentiles under the null hypothesis. Now, if it was known in practice that the found \(p_{1}\), \(p_{2}\) values are the two percentiles of interest under \(H_{0}\), the null hypothesis could be tested: If \(\overline{x}-\overline{y}\) calculated in practice from a single realization of the sample averages is below \(p_{1}\) or above \(p_{2}\), the hypothesis is rejected. In all other cases, it is accepted. The problem is that this is never known because the true gamma distributions are unknown, and so the question arises which pair \(p_{1}\), \(p_{2}\) of all the pairs found through the simulations should be actually used for the test. To answer this pressing question, one could think and proceed as follows: The normal and other distributions are known for the two-sigma rule or a similar rule that states that once the population standard deviation is multiplied by two and minus two or by a different number and these multiples are added to the population mean, two percentiles are obtained with the property so that it is almost certain that the random variable will take on a value lying between the two percentiles. These rules can be described by the following formula:\[p=d\cdot var^{1/2}, \tag{4}\]  
where \(var\) is the population variance, and \(d\) is the proper multiple. Applying this formula to the distribution of \(\overline{X}-\overline{Y}\) under \(H_{0}\), \(var\) is known by design, while the percentiles \(p\) are obtained via the simulations, so the multiple(s) \(d\) can be calculated from (4) through the simulations, as well. Of course, using Formula (4) to calculate the percentiles in practice based on such research will at the moment still not help since not knowing the gamma distributions the samples came from means not knowing which \(d\) to choose, let alone the fact that \(var\) is also unknown.