Standard error: SE = $\sqrt{\frac{{(s_1)}^2}{n_1}+\frac{{(s_2)}^2}{n_2}}$

Test statistic (t-score): t_c = $\frac{({\overline{x}}_1-{\overline{x}}_2)-{\delta }_0}{\sqrt{\frac{{(s_1)}^2}{n_1}+\frac{{(s_2)}^2}{n_2}}}$

Degrees of freedom:
$df= \frac{{\left(\frac{{(s_1)}^2}{n_1}+ \frac{{(s_2)}^2}{n_2}\right)}^2}{\left(\frac{1}{n_1-1}\right){\left(\frac{{(s_1)}^2}{n_1}\right)}^2+\left(\frac{1}{n_2-1}\right){\left(\frac{{(s_2)}^2}{n_2}\right)}^2}$

where:

$s_1$ and $s_2$ are the sample standard deviations, and $n_1$ and $n_2$ are the sample sizes.

${\overline{x}}_1$ and ${\overline{x}}_2$ are the sample means.

Cohen’s d is the measure of effect size:

$d=\frac{{\overline{x}}_1-{\overline{x}}_2}{s_{pooled}}$
where $s_{pooled}=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}$

where $S\stackrel{2}{p}$ is the pooled variance given by the formula:

Pooled Proportion: p_c = $\frac{x_A+x_B}{n_A+n_B}$

Test Statistic (z-score): $Z_c=\frac{(p^{\prime }{}_A-p^{\prime }{}_B)}{\sqrt{p_c(1-p_c)\left(\frac{1}{n_A}+\frac{1}{n_B}\right)}}$

where

$p_A^{\text{'}}$ and $p_B^{\text{'}}$ are the sample proportions, $p_A$ and $p_B$are the population proportions,

P_c is the pooled proportion, and n_A and n_B are the sample sizes.

Test Statistic (z-score):

$Z_c=\frac{({\bar{x}}_1-{\bar{x}}_2)-{\delta }_0}{\sqrt{\frac{{({\sigma }_1)}^2}{n_1}+\frac{{({\sigma }_2)}^2}{n_2}}}$

where:
${\sigma }_1$ and ${\sigma }_2$ are the known population standard deviations. n₁ and n₂ are the sample sizes. ${\bar{x}}_1$ and ${\bar{x}}_2$ are the sample means. μ₁ and μ₂ are the population means.

Test Statistic (t-score): t_c = $\frac{{\bar{x}}_d-{\mu }_d}{\left(\frac{s_d}{\sqrt{n}}\right)}$

where:

${\bar{x}}_d$ is the mean of the sample differences. μ_d is the mean of the population differences. s_d is the sample standard deviation of the differences. n is the sample size.