2.4 Checking the assumptions

The \(t\) test assumes independent errors, normality and, for the pooled form, equal variances. Independence is secured by randomisation, not checked after the fact. Normality is checked with a normal probability plot: the ordered residuals \(y_{ij} - \bar y_i\) are plotted against the expected normal order statistics, approximately \(\Phi ^{-1}\bigl ((k - 0.5)/N\bigr )\) for the \(k\)th smallest of \(N\), and should lie close to a straight line. Moderate departures from normality matter little, because the \(t\) test is robust to them, as the randomisation argument shows.

Equality of variances can be tested formally.

To test \(H_0 : \sigma _1^2 = \sigma _2^2\), compute \(F_0 = s_1^2/s_2^2\) and reject if \(F_0 > F_{\alpha /2,\,n_1-1,\,n_2-1}\) or \(F_0 < F_{1-\alpha /2,\,n_1-1,\,n_2-1}\). A \(100(1 - \alpha )\%\) confidence interval for \(\sigma _1^2/\sigma _2^2\) is \[ \frac {s_1^2/s_2^2}{F_{\alpha /2,\,n_1-1,\,n_2-1}} \le \frac {\sigma _1^2}{\sigma _2^2} \le \frac {s_1^2/s_2^2}{F_{1-\alpha /2,\,n_1-1,\,n_2-1}} , \] and for a single variance, since \((n - 1)S^2/\sigma ^2 \sim \chi ^2_{n-1}\), \[ \frac {(n - 1)s^2}{\chi ^2_{\alpha /2,\,n-1}} \le \sigma ^2 \le \frac {(n - 1)s^2}{\chi ^2_{1-\alpha /2,\,n-1}} . \]