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Contents Equal Variances in Small Sampless> Different Variances in Small Samples Large sample Comparison of two independent groups The means of two independent probability variables, X and Y, each of which follows a normal distribution, can be compared by applying a hypothesis test: $$\begin{align}\bar{X}&=\frac{\sum^n_{i=1} X_i}{n_X} \sim N\left(\mu_X, \frac{\text{s}_X}{n_X}\right)\\ \bar{Y}&=\frac{\sum^n_{i=1} Y_i}{n_Y} \sim N\left(\mu_Y, \frac{\text{s}_Y}{n_Y}\right)\end{align}$$ Even if each sample group does not assume a normal distribution, approximately normal distribution is satisfied according to the central limit theorem . To compare the two groups, set the following null hypothesis for the difference between each mean: $$\text{H0} : \mu_X -\mu_Y =0$$ The hypothetical test statistics are calculated from a combined distribution of two groups. That is, the mean and standard deviation of the combined probability distributions of X and Y are calculated as...