Robustness

Real data never satisfy a condition perfectly, so what matters is whether a small violation ruins the answer. The t procedures are robust against mild non-normality, and the larger n is, the more skewness they tolerate: the AP course accepts $n \geq 30$ as enough for the sampling distribution of $\bar{x}$ (x-bar, the sample mean) to be approximately normal, and asks for data free from strong skewness and outliers when n is smaller. For example, a one-sample t-interval from 40 mildly right-skewed measurements is trustworthy, while the same interval from 8 measurements with a clear outlier is not. Robustness has a hard limit: no procedure is robust against a biased sampling method, because bias moves the center rather than the shape.