For a small-sample mean test within current H2 Mathematics, check two separate conditions: an exact normal sample-mean model and a known population variance. A normality assumption resolves only the first condition.
With independent observations from \(N(\mu,\sigma^2)\) and known \(\sigma^2\), under \(H_0:\mu=\mu_0\), \[Z=\frac{\overline{X}-\mu_0}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1).\] This supports the H2 mean test even when \(n\) is small.
| Small-sample information | Required response |
|---|---|
| Normal population and known population variance | Use the exact normal mean model; normality is already supplied. |
| Unknown population shape and known population variance | State a normal population assumption if reasonable in context. |
| Normal population but only a sample variance | Normality alone does not justify the current H2 normal mean test. |
| Population explicitly non-normal | Do not silently replace the given population with a normal one. |
Bridge the idea
A sample has size 12. The population variance is stated as 25, but the population shape is not described. The needed distribution assumption is that the population is normal. If 25 is instead the sample variance, the same normality statement is insufficient for this H2 test setup.
Exam wording: “Assume the population is normally distributed; the population variance is known.” Do not add an assumption already supplied, and do not describe an estimated variance as known.
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