$Why is the t-test for two independent samples $\ t^* = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}$?$

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$Why is the t-test for two independent samples $\ t^* = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}$?$

I'm going through a biometrics course and I'd like an explanation as to why $\ t^* = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}$ for two independent samples instead of just memorizing the formula.

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Mathe

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$Why is the t-test for two independent samples $\ t^* = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}$?$

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