30.4 The test statistic: Two independent means

The observed sample mean difference (observations), relative to what was expected, is found by computing the test statistic, in this case, a \(t\)-score. The software output (jamovi: Fig. 30.1; SPSS: Fig. 30.2) can be used, but the \(t\)-score can also be computed manually:

\[\begin{align*} t &= \frac{\text{sample statistic} - \text{assumed population parameter, from $H_0$}} {\text{standard error for sample statistic}}\\ &= \frac{ (\bar{x}_P - \bar{x}_C) - (\mu_P - \mu_C)} {\text{s.e.}(\bar{x}_P - \bar{x}_C)}\\ &= \frac{51.594 - 0}{19.612} = 2.631, \end{align*}\] as in the software output.