总体数目待估计参数样本函数及其分布区间估计拒绝域
$(\mu = \mu_0)$
$(\sigma^2 = \sigma_0^2)$
一个总体

$X \sim N(\mu, \sigma^2) $

$\overline{X} = \dfrac{1}{n} \sum \limits_{i=1}^n X_i$

$ S_n^2 = \dfrac{1}{n} \sum \limits_{i=1}^n (X_i - \overline{X}) $

$ \sqrt{\dfrac{S_n^{*2}}{n}} = \sqrt{\dfrac{S_n^2}{n - 1}} $
$\mu (\sigma^2 = \sigma_0^2)$$\dfrac{\overline{X} - \mu}{\sqrt{\dfrac{\sigma_0^2}{n}}} \sim N(0, 1)$$\left[\overline{X} \pm u_{1 - \frac{\alpha}{2}} \sqrt{\dfrac{\sigma_0^2}{n}}\right]$ $\dfrac{|\overline{x} - \mu_0|}{\sqrt{\dfrac{\sigma_0^2}{n}}} \geq u_{1 - \frac{\alpha}{2}}$
$\mu$ ($\sigma^2$ 未知) $\dfrac{\overline{X} - \mu}{\sqrt{\dfrac{S_n^{*2}}{n}}} \sim t(n - 1)$ $\left[\overline{X} \pm t_{1 - \frac{\alpha}{2}} (n - 1) \sqrt{\dfrac{S_n^{*2}}{n}}\right]$ $\dfrac{|\overline{x} - \mu_0|}{\sqrt{\dfrac{S_n^{*2}}{n}}} \geq t_{1 - \frac{\alpha}{2}} (n - 1)$
$\sigma^2(\mu = \mu_0)$$\dfrac{\sum \limits_{i=1}^n (X_i - \mu_0)^2}{\sigma^2} \sim \chi^2(n)$$\left[\dfrac{\sum \limits_{i=1}^n (X_i - \mu_0)^2}{\chi^2_{1 - \frac{\alpha}{2}} (n)}, \, \dfrac{\sum \limits_{i=1}^n (X_i - \mu_0)^2}{\chi^2_{\frac{\alpha}{2}} (n)}\right]$$\dfrac{\sum \limits_{i=1}^n (x_i - \mu_0)^2}{\sigma_0^2} \leq \chi^2_\frac{\alpha}{2} (n) \\[10pt]
\dfrac{\sum \limits_{i=1}^n (x_i - \mu_0)^2}{\sigma_0^2} \geq \chi^2_{1 - \frac{\alpha}{2}} (n)$
$\sigma^2$ ($\mu$ 未知) $\dfrac{n S_n^2}{\sigma^2} \sim \chi^2(n - 1)$ $\left[\dfrac{n S_n^2}{\chi^2_{1 - \frac{\alpha}{2}} (n - 1)}, \, \dfrac{n S_n^2}{\chi^2_{\frac{\alpha}{2}} (n - 1)}\right]$ $\dfrac{n S_n^2}{\sigma_0^2} \leq \chi^2_\frac{\alpha}{2} (n - 1) \\[10pt]
\dfrac{n S_n^2}{\sigma_0^2} \geq \chi^2_{1 - \frac{\alpha}{2}} (n - 1)$
总体数目待估计参数 样本函数及其分布 区间估计 拒绝域
$(\mu_1 - \mu_2 = \delta)$
$\sigma_1^2 = \sigma_2^2$
两个总体

$S_w = \sqrt{ \dfrac{mS_{1m}^2 + nS_{2n}^2}{m + n - 2}}$

$ X \sim N(\mu_1, \sigma_1^2) $

$ \overline{X} = \dfrac{1}{m} \sum \limits_{i=1}^n X_i $

$ S_{1m}^2 = \dfrac{1}{m} \sum \limits_{i=1}^n (X_i - \overline{X}) $

$ Y \sim N(\mu_2, \sigma_2^2) $

$ \overline{Y} = \dfrac{1}{n} \sum \limits_{i=1}^n Y_i $

$ S_{2n}^2 = \dfrac{1}{n} \sum \limits_{i=1}^n (Y_i - \overline{Y}) $
$\mu_1 - \mu_2$
($\sigma_1^2, \sigma_2^2$ 已知)
$\dfrac{( \overline{X} - \overline{Y} ) - (\mu_1 - \mu_2)}{ \sqrt{\dfrac{\sigma_1^2}{m} + \dfrac{\sigma_2^2}{n} } } \sim N(0, 1) $ $\left[(\overline{X} - \overline{Y}) \pm u_{1 - \frac{\alpha}{2}} \sqrt{ \dfrac{\sigma_1^2}{m} + \dfrac{\sigma_2^2}{n}}\right]$ $\dfrac{|(\overline{x} - \overline{y}) - \delta|}{\sqrt{ \dfrac{\sigma_1^2}{m} + \dfrac{\sigma_2^2}{n}}} \geq u_{1 - \frac{\alpha}{2}}$
$\mu_1 - \mu_2$
($\sigma_1^2 = \sigma_2^2$ 未知)
$\dfrac{( \overline{X} - \overline{Y} ) - (\mu_1 - \mu_2)}{ S_w \sqrt{\dfrac{1}{m} + \dfrac{1}{n} } } \sim t(m + n - 2) $ $\left[(\overline{X} - \overline{Y}) \pm t_{1 - \frac{\alpha}{2}}(m + n - 2) \times S_w \sqrt{\dfrac{1}{m} + \dfrac{1}{n} } \right]$ $\dfrac{|(\overline{x} - \overline{y}) - \delta|}{ S_w \sqrt{\dfrac{1}{m} + \dfrac{1}{n} } } \geq t_{1 - \frac{\alpha}{2}} (m + n - 2) $
$\dfrac{\sigma_1^2}{\sigma_2^2}$
($\mu_1, \mu_2$ 已知)
$\dfrac{n \sigma_2^2 \sum \limits_{i=1}^m (X_i - \mu_1)^2 }{m \sigma_1^2 \sum \limits_{j=1}^n (Y_j - \mu_2)^2 } \sim F(m, n)$$\left[ \frac{1}{F_{1 - \frac{\alpha}{2}}(m, n)} \dfrac{n \sigma_2^2 \sum \limits_{i=1}^m (X_i - \mu_1)^2 }{m \sigma_1^2 \sum \limits_{j=1}^n (Y_j - \mu_2)^2 }, \\
\frac{1}{F_{\frac{\alpha}{2}}(m, n)} \dfrac{n \sigma_2^2 \sum \limits_{i=1}^m (X_i - \mu_1)^2 }{m \sigma_1^2 \sum \limits_{j=1}^n (Y_j - \mu_2)^2 } \right]$
$ \dfrac{n \sum \limits_{i=1}^m (x_i - \mu_1)^2 }{m \sum \limits_{j=1}^n (y_j - \mu_2)^2 } \leq F_{\frac{\alpha}{2}}(m, n) $或
$ \dfrac{n \sum \limits_{i=1}^m (x_i - \mu_1)^2 }{m \sum \limits_{j=1}^n (y_j - \mu_2)^2 } \geq F_{1 - \frac{\alpha}{2}}(m, n) $
$\dfrac{\sigma_1^2}{\sigma_2^2}$
($\mu_1, \mu_2$ 未知)
$\dfrac{ S_{1m}^{* 2} \sigma_2^2}{S_{2n}^{* 2} \sigma_1^2} \sim F(m - 1, n - 1)$ $\left[ \frac{1}{F_{1 - \frac{\alpha}{2}}(m - 1, n - 1)} \dfrac { S_{1m}^{* 2} }{S_{2n}^{* 2}} ,
\frac{1}{F_{\frac{\alpha}{2}}(m - 1, n - 1)} \dfrac { S_{1m}^{* 2} }{S_{2n}^{* 2}} \right]$
$ \dfrac{ S_{1m}^{* 2} \sigma_2^2}{S_{2n}^{* 2} \sigma_1^2} \leq F_{\frac{\alpha}{2}}(m - 1, n - 1) $ 或

$\dfrac{ S_{1m}^{* 2} \sigma_2^2}{S_{2n}^{* 2} \sigma_1^2} \geq F_{1 - \frac{\alpha}{2}}(m - 1, n - 1)$