Instant feedback. Score updates automatically. Assumes a normal population.
1) For a normal population, a two-sided CI for \( \sigma^2 \) uses chi-square with df \(n-1\).
2) The \((1-\alpha)\) CI for \( \sigma^2 \) is \( \big( \frac{(n-1)S^2}{\chi^2_{1-\alpha/2,n-1}},\; \frac{(n-1)S^2}{\chi^2_{\alpha/2,n-1}} \big) \).
3) To get a CI for \( \sigma \) from a CI for \( \sigma^2 \), square the endpoints.
4) The chi-square CI for \( \sigma \) remains valid even if the population is not normal.
5) Increasing \(n\) generally makes the CI for \( \sigma \) narrower (more precise).
6) Using \(S\) in place of \(S^2\) in the chi-square formula gives the correct CI for \( \sigma \).
7) A two-sided 95% CI uses the quantiles \( \chi^2_{0.05,n-1} \) and \( \chi^2_{0.95,n-1} \).
8) The CI for \( \sigma^2 \) is symmetric around \( S^2 \).
9) If all data are multiplied by a constant \(c>0\), the CI for \( \sigma \) multiplies both endpoints by \(c\).
10) Using df \(=n\) instead of \(n-1\) makes the CI exactly correct for small samples.