这是一份warwick华威大学ST228-10的成功案例

Knowledge of the distribution of a sample summary is important in statistical applications. For example, suppose that the sample mean and sample variance of a random sample of size $n$ from a normal distribution are used to estimate the unlnown $\mu$ and $\sigma^{2}$. Let $\chi_{p}^{2}$ and $\chi_{1-p}^{2}$ be the $p^{t h}$ and $(1-p)^{t h}$ quantiles of the chi-square distribution with $(n-1)$ degrees of freedom, respectively. Then
$$
1-2 p=P\left(\chi_{p}^{2} \leq \frac{(n-1) S^{2}}{\sigma^{2}} \leq \chi_{1-p}^{2}\right)=P\left(\frac{(n-1) S^{2}}{\chi_{1-p}^{2}} \leq \sigma^{2} \leq \frac{(n-1) S^{2}}{\chi_{p}^{2}}\right)
$$
If the observed value of the sample variance is $s^{2}=8.72, n=12$ and $p=0.05$, then the interval
$$
\left[\frac{11(8.72)}{19.68}, \frac{11(8.72)}{4.57}\right]=[4.87,20.99]
$$
is an estimate of an interval containing $\sigma^{2}$ with probability $0.90$.

ST228-10 COURSE NOTES ：

The distribution of the ratio of $S_{x}^{2} / S_{y}^{2}$ to $\sigma_{x}^{2} / \sigma_{y}^{2}$ is important in statistical applications. For example, suppose that all four parameters $\left(\mu_{x}, \sigma_{x}, \mu_{y}, \sigma_{y}\right)$ are unknown. Let $f_{p}$ and $f_{1-p}$ be the $p^{t h}$ and $(1-p)^{t h}$ quantiles of the $f$ ratio distribution with $(n-1)$ and $(m-1)$ degrees of freedom, respectively. Then
$$
1-2 p=P\left(f_{p} \leq \frac{S_{x}^{2} / S_{y}^{2}}{\sigma_{x}^{2} / \sigma_{y}^{2}} \leq f_{1-p}\right)=P\left(\frac{S_{x}^{2} / S_{y}^{2}}{f_{1-p}} \leq \frac{\sigma_{x}^{2}}{\sigma_{y}^{2}} \leq \frac{S_{x}^{2} / S_{y}^{2}}{f_{p}}\right)
$$
If the observed sample variances are $s_{x}^{2}=18.75$ and $s_{y}^{2}=3.45, n=8, m=10$, and $p=0.05$, then the interval
$$
\left[\frac{18.75 / 3.45}{3.29}, \frac{18.75 / 3.45}{0.27}\right]=[1.65,20.13]
$$
is an estimate of an interval containing $\sigma_{x}^{2} / \sigma_{y}^{2}$ with probability $0.90$.