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Separate a limit statement from a cutoff
The classical CLT describes convergence of a standardized mean under a fixed independent common-distribution model with finite variance. It does not specify that every model becomes close to normal at the same finite n.
Inspect a rare-event population
For independent zero-one values with event probability 0.001, a thirty-observation sample contains no events with probability 0.999^30, about 97.0431%. The sample mean is concentrated at zero and at a small set of nonnegative outcomes.
Ask what affects the approximation
| Feature | Question |
|---|---|
| Dependence | Are the relevant observations independent? |
| Distribution | Are variance and tail behavior compatible with this CLT? |
| Sample size | Is n large enough for this specific model and accuracy need? |
| Statistic | Does the theorem concern this mean or another statistic? |
Use a suitable method for the actual task
A rare-count task can be evaluated from its exact model in this exercise. Other tasks may need a justified asymptotic, resampling or design-based procedure. Increasing n does not automatically repair frame omissions, selection bias or nonstationary observations.