fsml_ttest_2sample Interface

public interface fsml_ttest_2sample

The 2-sample t-test determines if two population means $\mu_1$ and $\mu_2$ are the same. The procedure can handle 2-sample t-tests for equal variances and Welch's t-tests for unequal variances.

Hypotheses:

The null hypothesis $H_{0}$ and alternative hypothesis $H_{1}$ can be written as: $H_{0}$: $\mu_1 \ = \mu_2$, and $H_{1}$: $\mu_1 \ \neq \mu_2$

Procedure:

The procedure defaults to Welch's t-test for unequal variances if eq_var is not specified. In this case, the test statstic $t$ is calculated as follows: $$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2} }}$$ where $\bar{x}_1$ and $\bar{x}_2$ are the sample means $s^2_1$ and $s^2_2$ are the sample standard deviations, and $n_1$ and $n_1$ are the sample sizes. The degrees of freedom $\nu$ is approximated with the Welch–Satterthwaite equation: $$\nu = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2} {\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}$$

If variances are assumed to be equal (eq_var = .true.), the procedure conducts a 2 sample t-test for equal variances, using the pooled standard deviation $s_p$ to calculate the t-statistic: $$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

In case of assumed equal variances, the degrees of freedom is calculated as follows: $$\nu = n_1 + n_2 - 2$$

Module Procedures