A normal model is a modelling choice. When asked whether it is suitable, compare its predictions with the values that can actually occur; do not simply write “assume normality”.
Check support and shape
| Check | Reason to question the model |
|---|---|
| Possible values | The model assigns appreciable probability to impossible negative or out-of-range measurements. |
| Observed shape | Strong skewness, several distinct groups, or marked asymmetry may conflict with a single normal model. |
| Population and conditions | Different groups or changed production conditions may need separate models. |
Suppose a positive duration is modelled by \(W\sim N(2,1.5^2)\). The model gives \(P(W<0)=P\left(Z<-\dfrac{4}{3}\right)\approx0.0912\). Predicting about 9.12% negative durations is a substantial contextual objection.
Bridge the idea
Does every normal model for a positive measurement fail? No. If the probability of an impossible value is negligible in the relevant setting, the model may still be a useful approximation. Explain the size and practical significance of the discrepancy.
Exam wording: “The model predicts a substantial probability of negative durations, which are impossible. This casts doubt on its suitability.” A large sample of observations does not itself make the individual duration normally distributed.
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