约定
- 假设中形如 $\mu\sim \mu_{0}$ 以及拒绝域中形如 $Z\sim Z(…)$ 表示
| 零假设 $H_{0}$ | 备择假设 $H_{1}$ | 拒绝域 |
|---|---|---|
| $\mu\leq \mu_{0}$ | $\mu>\mu_{0}$ | $Z\geq Z_{\alpha}$ |
| $\mu\geq \mu_{0}$ | $\mu<\mu_{0}$ | $Z\leq Z_{1-\alpha}$ |
| $\mu=\mu_{0}$ | $\mu\neq \mu_{0}$ | $Z\geq Z_{\alpha/2} \lor Z\leq Z_{1-\alpha/2}$ |
注意取等只在 零假设$H_{0}$ 和 拒绝域
| 假设 | 条件 | 枢轴变量 | 拒绝域 |
|---|---|---|---|
| $\mu\sim \mu_{0}$ | $\sigma^{2}$ 已知 | $\frac{\bar{X}-\mu}{\sigma/\sqrt{ n }}\sim N(0,1)$ | $\frac{\bar{X}-\mu_{0}}{\sigma/\sqrt{ n }}\sim N(0,1)$ |
| $\mu \sim \mu_{0}$ | $\sigma^{2}$ 未知 | $\frac{\bar{X}-\mu}{S/\sqrt{ n }}\sim t(n-1)$ | $\frac{\bar{X}-\mu_{0}}{S/\sqrt{ n }}\sim t(n-1)$ |
| $\sigma^{2}\sim\sigma_{0}^{2}$ | $\mu$ 未知 | $\frac{(n-1)S^{2}}{\sigma^{2}}\sim \chi^{2}(n-1)$ | $\frac{(n-1)S^{2}}{\sigma_{0}^{2}}\sim \chi^{2}(n-1)$ |
| $\mu_{1} - \mu_{2}\sim \delta$ | $\sigma_{1}^{2},\sigma_{2}^{2}$ 已知 | $\frac{(\bar{X}{1}-\bar{X}{2}) - (\mu_{1}-\mu_{2})}{ \sqrt{ \frac{\sigma_{1}^{2}}{n_{1}^{2}}+\frac{\sigma_{2}^{2}}{n_{2}^{2}}}}\sim N(0,1)$ | $\frac{(\bar{X}{1}-\bar{X}{2})-\delta}{ \sqrt{ \frac{\sigma_{1}^{2}}{n_{1}^{2}}+\frac{\sigma_{2}^{2}}{n_{2}^{2}}}}\sim N(0,1)$ |
| $\mu_{1} - \mu_{2}\sim \delta$ | $\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}$ 未知 | $\frac{(\bar{X}{1}-\bar{X}{2})-(\mu_{1}-\mu_{2})}{S_{W} \sqrt{ \frac{1}{n_{1}^{2}}+\frac{1}{n_{2}^{2}}}}\sim t(n_{1}+n_{2}-2)$ $S_{W}^{2}=\frac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2}$ |
$\frac{(\bar{X}{1}-\bar{X}{2})-\delta}{S_{W} \sqrt{ \frac{1}{n_{1}^{2}}+\frac{1}{n_{2}^{2}}}}\sim t(n_{1}+n_{2}-2)$ $S_{W}^{2}=\frac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2}$ |
| $\sigma_{1}^{2}\sim\sigma_{2}^{2}$ (等价于 $\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\sim1$) |
$\mu_{1},\mu_{2}$ 未知 | $\frac{S_{1}^{2}/\sigma_{1}^{2}}{S_{2}^{2}/\sigma_{2}^{2}}\sim F(n_{1}-1,n_{2}-1)$ | $\frac{S_{1}^{2}}{S_{2}^{2}}\sim F(n_{1}-1,n_{2}-1)$ |
| $\mu_{D}\sim 0$ | 成对数据 | $\frac{\bar{D}-\mu_{D}}{S_{D}/\sqrt{ n }}\sim t(n-1)$ | $\frac{\bar{D}-0}{S_{D}/\sqrt{ n }}\sim t(n-1)$ |