The test used is the two sample
$t$-test. The test statistic
$t$ is defined by;
where
${s}^{2}=\frac{\left({n}_{x}-1\right){s}_{x}^{2}+\left({n}_{y}-1\right){s}_{y}^{2}}{{n}_{x}+{n}_{y}-2}$ is the pooled variance of the two samples.
The test of
${H}_{0}$ is carried out against one of three possible alternatives:
- ${H}_{1}:{\mu}_{x}\ne {\mu}_{y}$; the significance level, $p=P\left(t\ge \left|{t}_{\mathrm{obs}}\right|\right)$, i.e., a two tailed probability.
- ${H}_{1}:{\mu}_{x}>{\mu}_{y}$; the significance level, $p=P\left(t\ge {t}_{\mathrm{obs}}\right)$, i.e., an upper tail probability.
- ${H}_{1}:{\mu}_{x}<{\mu}_{y}$; the significance level, $p=P\left(t\le {t}_{\mathrm{obs}}\right)$, i.e., a lower tail probability.