7.3.2 两个正态总体参数的区间估计

设

$X \sim N (μ_{1}, σ_{1}^{2})$ ,
$Y \sim N (μ_{2}, σ_{2}^{2})$ ,
$X$ 与 $Y$ 相互独立,
样本分别为 $(X_{1}, X_{2}, \dots, X_{m})$ 和 $(Y_{1}, Y_{2}, \dots, Y_{n})$ .

样本均值与样本方差为

$\overset{ˉ}{X} = \frac{1}{n} i = 1 \sum n X_{i}$ ,
$\overset{ˉ}{Y} = \frac{1}{n} j = 1 \sum n Y_{j}$ ,
$S_{1 m}^{2} = \frac{1}{m} i = 1 \sum m (X_{i} - \overset{ˉ}{X})^{2}$ ,
$S_{2 n}^{2} = \frac{1}{n} j = 1 \sum n (Y_{j} - \overset{ˉ}{Y})^{2}$ .
1. 期望差的区间估计 (已知期望)
2. 期望差的区间估计 (未知方差相同)
3. 方差比值的区间估计 (已知期望)
4. 方差比值的区间估计 (未知期望)

表 7.1 (两个总体)

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 $μ_{1} - μ_{2} (σ_{1}^{2}, σ_{2}^{2} 已知)$ $\frac{( X ˉ - Y ˉ ) - ( μ _{1} - μ _{2} )}{σ _{1}^{2} / m + σ _{2}^{2} / n} \sim N (0, 1)$ $[(\overset{ˉ}{X} - \overset{ˉ}{Y}) \pm u_{1 - α /2} \frac{σ _{1}^{2}}{m} + \frac{σ _{2}^{2}}{n}]$
$X \sim N (μ_{1}, σ_{1}^{2})$ $Y \sim N (μ_{2}, σ_{2}^{2})$ $μ_{1} - μ_{2} (σ_{1}^{2} = σ_{2}^{2} 未知)$ $\frac{( X ˉ - Y ˉ ) - ( μ _{1} - μ _{2} )}{S _{w} 1/ m + 1/ n} \sim t (m + n - 2)$ $(\overset{ˉ}{X} - \overset{ˉ}{Y}) \pm t_{1 - α /2} (m + n - 2) \cdot S_{w} \frac{1}{m} + \frac{1}{n}$
$\overset{ˉ}{X} = \frac{1}{m} \sum_{i = 1}^{m} X_{i}$ $\overset{ˉ}{Y} = \frac{1}{m} \sum_{i = 1}^{m} Y_{i}$ $\frac{σ _{1}^{2}}{σ _{2}^{2}} (μ_{1}, μ_{2} 已知)$ $\frac{n \sum ( X _{i} - μ _{1} ) ^{2}}{m \sum ( Y _{j} - μ _{2} ) ^{2}} \cdot \frac{σ _{2}^{2}}{σ _{1}^{2}} \sim F (m, n)$ $[\frac{1}{F _{1 - α /2} ( m , n )} \cdot \frac{n \sum ( X _{i} - μ _{1} ) ^{2}}{m \sum ( Y _{j} - μ _{2} ) ^{2}}, \frac{1}{F _{α /2} ( m , n )} \cdot \frac{n \sum ( X _{i} - μ _{1} ) ^{2}}{m \sum ( Y _{j} - μ _{2} ) ^{2}}]$
$S_{1 m}^{2} = \frac{1}{m} \sum_{i = 1}^{m} (X_{i} - \overset{ˉ}{X})^{2}$ $S_{2 n}^{2} = \frac{1}{n} \sum_{i = 1}^{n} (Y_{i} - \overset{ˉ}{Y})^{2}$ $\frac{σ _{1}^{2}}{σ _{2}^{2}} (μ_{1}, μ_{2} 未知)$ $\frac{m S _{1 m}^{2}}{n S _{2 n}^{2}} \cdot \frac{σ _{2}^{2}}{σ _{1}^{2}} \cdot \frac{n - 1}{m - 1} = \frac{S _{1 m}^{* 2}}{S _{2 n}^{* 2}} \cdot \frac{σ _{2}^{2}}{σ _{1}^{2}} \sim F (m - 1, n - 1)$ $[\frac{1}{F _{1 - α /2} ( m - 1 , n - 1 )} \cdot \frac{S _{1 m}^{* 2}}{S _{2 n}^{* 2}}, \frac{1}{F _{α /2} ( m - 1 , n - 1 )} \cdot \frac{S _{1 m}^{* 2}}{S _{2 n}^{* 2}}]$
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7.3.2 两个正态总体参数的区间估计

表 7.1 (两个总体)

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	$μ_{1} - μ_{2} (σ_{1}^{2}, σ_{2}^{2} 已知)$	$\frac{( X ˉ - Y ˉ ) - ( μ _{1} - μ _{2} )}{σ _{1}^{2} / m + σ _{2}^{2} / n} \sim N (0, 1)$	$[(\overset{ˉ}{X} - \overset{ˉ}{Y}) \pm u_{1 - α /2} \frac{σ _{1}^{2}}{m} + \frac{σ _{2}^{2}}{n}]$
$X \sim N (μ_{1}, σ_{1}^{2})$ $Y \sim N (μ_{2}, σ_{2}^{2})$	$μ_{1} - μ_{2} (σ_{1}^{2} = σ_{2}^{2} 未知)$	$\frac{( X ˉ - Y ˉ ) - ( μ _{1} - μ _{2} )}{S _{w} 1/ m + 1/ n} \sim t (m + n - 2)$	$(\overset{ˉ}{X} - \overset{ˉ}{Y}) \pm t_{1 - α /2} (m + n - 2) \cdot S_{w} \frac{1}{m} + \frac{1}{n}$
$\overset{ˉ}{X} = \frac{1}{m} \sum_{i = 1}^{m} X_{i}$ $\overset{ˉ}{Y} = \frac{1}{m} \sum_{i = 1}^{m} Y_{i}$	$\frac{σ _{1}^{2}}{σ _{2}^{2}} (μ_{1}, μ_{2} 已知)$	$\frac{n \sum ( X _{i} - μ _{1} ) ^{2}}{m \sum ( Y _{j} - μ _{2} ) ^{2}} \cdot \frac{σ _{2}^{2}}{σ _{1}^{2}} \sim F (m, n)$	$[\frac{1}{F _{1 - α /2} ( m , n )} \cdot \frac{n \sum ( X _{i} - μ _{1} ) ^{2}}{m \sum ( Y _{j} - μ _{2} ) ^{2}}, \frac{1}{F _{α /2} ( m , n )} \cdot \frac{n \sum ( X _{i} - μ _{1} ) ^{2}}{m \sum ( Y _{j} - μ _{2} ) ^{2}}]$
$S_{1 m}^{2} = \frac{1}{m} \sum_{i = 1}^{m} (X_{i} - \overset{ˉ}{X})^{2}$ $S_{2 n}^{2} = \frac{1}{n} \sum_{i = 1}^{n} (Y_{i} - \overset{ˉ}{Y})^{2}$	$\frac{σ _{1}^{2}}{σ _{2}^{2}} (μ_{1}, μ_{2} 未知)$	$\frac{m S _{1 m}^{2}}{n S _{2 n}^{2}} \cdot \frac{σ _{2}^{2}}{σ _{1}^{2}} \cdot \frac{n - 1}{m - 1} = \frac{S _{1 m}^{* 2}}{S _{2 n}^{* 2}} \cdot \frac{σ _{2}^{2}}{σ _{1}^{2}} \sim F (m - 1, n - 1)$	$[\frac{1}{F _{1 - α /2} ( m - 1 , n - 1 )} \cdot \frac{S _{1 m}^{* 2}}{S _{2 n}^{* 2}}, \frac{1}{F _{α /2} ( m - 1 , n - 1 )} \cdot \frac{S _{1 m}^{* 2}}{S _{2 n}^{* 2}}]$