A small sample does not prevent an exact normal sampling distribution when the population itself is normal and the observations are independent.
The model and its reason
For independent observations \(X_1,\ldots,X_n\) from \(N(\mu,\sigma^2)\), \[\overline{X}\sim N\left(\mu,\frac{\sigma^2}{n}\right).\] The standard deviation of \(\overline{X}\) is \(\dfrac{\sigma}{\sqrt{n}}\). This result holds for every positive sample size; the Central Limit Theorem is unnecessary.
Bridge the idea
A random sample of four independent measurements comes from \(N(50,12^2)\). Then \(\overline{X}\sim N(50,36)\), with standard deviation 6. It is exactly normal even though \(n=4\). Do not write its variance as 6 or use the individual variance 144 for the mean.
| Information in the question | Answer about normality |
|---|---|
| Population normality is supplied | Use that fact; no additional normality assumption is needed. |
| Small sample; population shape unspecified | Normality of the population is an additional assumption if an exact normal mean model is required and reasonable. |
Exam wording: “Since the population is normal and the observations are independent, the sample mean is exactly normally distributed.” If a hypothesis test follows, check the population-variance requirement separately.
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