Final answer:
The independent t-test can be robust to violations of normality, particularly with large sample sizes, due to the Central Limit Theorem. For small samples or significant deviations from normality, the Aspin-Welch t-test is a more robust alternative. Verification of equal variances is important, and if not met, the F test or adjusted t-tests should be used.
Step-by-step explanation:
The independent t-test is a statistical method used to compare the means of two independent groups. In terms of robustness to the normality assumption, the independent t-test can tolerate deviations from normality, especially when the sample sizes are large. This is because of the Central Limit Theorem, which suggests that the distribution of the sample means will approximate a normal distribution regardless of the population distribution when the sample size is sufficiently large.
However, when sample sizes are small and the distributions are not normal, the t-test can become less reliable. It is at this point where alternative tests like the Aspin-Welch t-test might be recommended, as it is designed to compare means from populations with unknown and possibly unequal variances, addressing some concerns of robustness.
For the assumption of equal variances (homogeneity of variances), statistical tests such as the F test for equality of two variances can be used to determine if this assumption holds before performing an ANOVA or a t-test. If the variances are significantly different, alternative methods or modifications to the standard t-test, such as using adjusted degrees of freedom as in the Aspin-Welch method, may be necessary. In any real-world application, verifying assumptions is crucial, though in educational settings, these assumptions may be relaxed for the purposes of learning and practicing the techniques.