表 7.1 正态总体参数的区间估计

总体数目	待估计参数	用到的样本函数及其分布	区间估计
一个总体	$\mu ({\sigma }^2={\sigma }_0^2\text{已知})$	$\frac{\bar{X}-\mu }{\sqrt{{\sigma }_0^2/n}}\sim N(0,1)$	$\left[\bar{X}\pm u_{1-\alpha /2}\sqrt{\frac{{\sigma }_0^2}{n}}\right]$
$X\sim N\left(\mu ,{\sigma }^2\right)$	$\mu ({\sigma }^2\text{未知})$	$\frac{\bar{X}-\mu }{S_n/\sqrt{n}}\sim t(n-1)$	$\left[\bar{X}\pm t_{1-\alpha /2}(n-1)\frac{S_n}{\sqrt{n}}\right]$
$\bar{X}=\frac{1}{n}{\sum }_{i=1}^n{X}_i$	${\sigma }^2(\mu ={\mu }_0\text{已知})$	${\sum }_{i=1}^n\frac{(X_i-{\mu }_0{)}^2}{{\sigma }^2}\sim {\chi }^2(n)$	$\left[\frac{\sum (X_i-{\mu }_0{)}^2}{{\chi }_{1-\alpha /2}^2(n)},\frac{\sum (X_i-{\mu }_0{)}^2}{{\chi }_{\alpha /2}^2(n)}\right]$
$S_n^2=\frac{1}{n}{\sum }_{i=1}^n(X_i-\bar{X}{)}^2$	${\sigma }^2(\mu \text{未知})$	$\frac{n{S}_n^2}{{\sigma }^2}\sim {\chi }^2(n-1)$	$\left[\frac{n{S}_n^2}{{\chi }_{1-\alpha /2}^2(n-1)},\frac{n{S}_n^2}{{\chi }_{\alpha /2}^2(n-1)}\right]$

总体数目	待估计参数	用到的样本函数及其分布	区间估计
两个总体	${\mu }_1-{\mu }_2({\sigma }_1^2,{\sigma }_2^2\text{已知})$	$\frac{(\bar{X}-\bar{Y})-({\mu }_1-{\mu }_2)}{\sqrt{{\sigma }_1^2/m+{\sigma }_2^2/n}}\sim N(0,1)$	$\left[(\bar{X}-\bar{Y})\pm u_{1-\alpha /2}\sqrt{\frac{{\sigma }_1^2}{m}+\frac{{\sigma }_2^2}{n}}\right]$
$X\sim N({\mu }_1,{\sigma }_1^2)$ $Y\sim N({\mu }_2,{\sigma }_2^2)$	${\mu }_1-{\mu }_2({\sigma }_1^2={\sigma }_2^2\text{未知})$	$\frac{(\bar{X}-\bar{Y})-({\mu }_1-{\mu }_2)}{S_w\sqrt{1/m+1/n}}\sim t(m+n-2)$	$(\bar{X}-\bar{Y})\pm t_{1-\alpha /2}(m+n-2)\cdot S_w\sqrt{\frac{1}{m}+\frac{1}{n}}$
$\bar{X}=\frac{1}{m}{\sum }_{i=1}^m{X}_i$ $\bar{Y}=\frac{1}{m}{\sum }_{i=1}^m{Y}_i$	$\frac{{\sigma }_1^2}{{\sigma }_2^2}({\mu }_1,{\mu }_2\text{已知})$	$\frac{n\sum (X_i-{\mu }_1{)}^2}{m\sum (Y_j-{\mu }_2{)}^2}\cdot \frac{{\sigma }_2^2}{{\sigma }_1^2}\sim F(m,n)$	$\left[\frac{1}{F_{1-\alpha /2}(m,n)}\cdot \frac{n\sum (X_i-{\mu }_1{)}^2}{m\sum (Y_j-{\mu }_2{)}^2},\frac{1}{F_{\alpha /2}(m,n)}\cdot \frac{n\sum (X_i-{\mu }_1{)}^2}{m\sum (Y_j-{\mu }_2{)}^2}\right]$
$S_{1m}^2=\frac{1}{m}{\sum }_{i=1}^m(X_i-\bar{X}{)}^2$ $S_{2n}^2=\frac{1}{n}{\sum }_{i=1}^n(Y_i-\bar{Y}{)}^2$	$\frac{{\sigma }_1^2}{{\sigma }_2^2}({\mu }_1,{\mu }_2\text{未知})$	$\frac{m{S}_{1m}^2}{n{S}_{2n}^2}\cdot \frac{{\sigma }_2^2}{{\sigma }_1^2}\cdot \frac{n-1}{m-1}=\frac{S_{1m}^{\ast 2}}{S_{2n}^{\ast 2}}\cdot \frac{{\sigma }_2^2}{{\sigma }_1^2}\sim F(m-1,n-1)$	$\left[\frac{1}{F_{1-\alpha /2}(m-1,n-1)}\cdot \frac{S_{1m}^{\ast 2}}{S_{2n}^{\ast 2}},\frac{1}{F_{\alpha /2}(m-1,n-1)}\cdot \frac{S_{1m}^{\ast 2}}{S_{2n}^{\ast 2}}\right]$