Assume $X_{Ai}$’s and $X_{Bj}$’s are iid (independent and identically distributed) random variables with mean $\mu_A$, $\mu_B$ and variance $\sigma^2_A$, $\sigma^2_B$.
By the CLT (central limit theorem),
\[\bar{X}_A = \frac{1}{n_A} \sum_{i=1}^{n_A} X_{Ai} \xrightarrow{d} N(\mu_A, \frac{\sigma_A^2}{n_A}), \\ \bar{X}_B = \frac{1}{n_B} \sum_{j=1}^{n_B} X_{Bj} \xrightarrow{d} N(\mu_B, \frac{\sigma_B^2}{n_B}).\]By the additivity of normal distribution,
\[\bar{X}_A - \bar{X}_B \xrightarrow{d} N(\mu_A - \mu_B, \frac{\sigma_A^2}{n_A} + \frac{\sigma_B^2}{n_B}).\]If the population variances \(\sigma_A\) and \(\sigma_B\) are known, we can simply use $Z$-test. However, we usually do not know them and we use the sample variances \(s_A^2\) and \(s_B^2\) to estimate them, then we construct a test statistic \(T\) which follows the student’s \(t\) distribution with the degrees of freedom \(\upsilon\).
Hypothesis: $H_0: \mu_A=\mu_B \text{ v.s. } H_1: \mu_A \neq \mu_B$.
Test statistic:
\[T= \frac{\bar{X}_A - \bar{X}_B}{\sqrt{s^2_A/n_A + s^2_B/n_B}} \sim t_{(\upsilon)},\]which is the $t$ distribution with degrees of freedom
\[\upsilon = \frac{(s^{2}_{A}/n_{A} + s^{2}_{B}/n_{B})^{2}} {(s^2_{A}/n_{A})^2/(n_{A}-1) + (s^2_{B}/n_{B})^2/(n_{B}-1) }.\]Significance level: $\alpha$.
Critical region: Reject the \(H_0\) if \(\ \mid\ T\mid \ >\ t_{1- \alpha/2, \upsilon}\), where $t_{1- \alpha/2, \upsilon}$ is the critical value of the $t$ distribution with $ \upsilon$ degrees of freedom.
Note:
As the number of degrees of freedom grows, the t distribution approaches the standard normal distribution, which means if the sample size is large enough (usually $>30$), then $T \sim N(0,1)$ approximately and we can use the $Z$ test instead of the $t$ test.
If we assume $\sigma^2_A = \sigma^2_B$, then \(T = \frac{\bar{X}_{A} - \bar{X}_{B}} {s \sqrt{1/n_A + 1/n_B}} \sim t_{(\upsilon)}\), where \(s^{2} = \frac{(n_{A}-1){s^{2}_{A}} + (n_{B}-1){s^{2}_{B}}} {n_{A} + n_{B} - 2}, \upsilon=n_A+n_B-2\).
Reference: Two-Sample t-Test for Equal Means.