= Solution
The \b[<normality assumption> concerns the model's error distribution, not necessarily the pooled raw observations]. In an independent-groups <analysis of variance>, errors around each group <mean> are modeled as following a <normal distribution>; combining groups with genuinely different <means> can produce a nonnormal pooled distribution even when this assumption holds. In a <paired t-test>, the relevant distribution is that of within-pair differences. <Linear regression> conventionally assumes normal errors conditional on the predictors for exact small-sample <Student t-tests> and <F-tests>.<The one-way ANOVA model and assumptions by National Institute of Standards and Technology>[Handbook Section 7.4.3.2, fixed-effects errors]{ref}
Inspect suitable within-group or fitted-model <regression residuals> using a <quantile-quantile plot> and examine outliers and asymmetry. A formal normality test is not a decisive automatic gatekeeper: with small samples it may miss important departures, while large samples can detect minor departures. Approximate robustness depends on the <sample size>, balance, tail behavior and intended inference. The <central limit theorem> concerns sampling distributions and does not make every raw observation normal or repair dependence. Depending on the question, a justified transformation, a better-fitting probability model, robust inference or an appropriate <bootstrap> may be preferable. The <normality assumption>, independence and equal <variance> are distinct assumptions.
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