单个正态总体方差的假设检验的拒绝域

表 8.2 单个正态总体方差的假设检验的拒绝域
单个正态总体方差 $σ^{2}$ 的假设检验的拒绝域 (显著性水平为 $α$ )

序号 $H_{0}$ $H_{1}$ $μ$ 已知 $μ$ 未知
I $σ^{2} = σ_{0}^{2}$ $σ^{2} \neq = σ_{0}^{2}$ $\frac{i = 1 \sum n ( x _{i} - μ ) ^{2}}{σ _{0}^{2}} \leq χ_{\frac{α}{2}}^{2} (n)$ 或 $\frac{i = 1 \sum n ( x _{i} - μ ) ^{2}}{σ _{0}^{2}} \geq χ_{1 - α}^{2} (n)$ $\frac{i = 1 \sum n ( x _{i} - x ˉ ) ^{2}}{σ _{0}^{2}} \leq χ_{α /2}^{2} (n - 1)$ 或 $\frac{i = 1 \sum n ( x _{i} - x ˉ ) ^{2}}{σ _{0}^{2}} \geq χ_{1 - α /2}^{2} (n - 1)$
II $σ^{2} = σ_{0}^{2}$ $σ^{2} > σ_{0}^{2}$ $\frac{i = 1 \sum n ( x _{i} - μ ) ^{2}}{σ _{0}^{2}} \geq χ_{1 - α}^{2} (n)$ $\frac{i = 1 \sum n ( x _{i} - x ˉ ) ^{2}}{σ _{0}^{2}} \geq χ_{1 - α}^{2} (n - 1)$
III $σ^{2} \leq σ_{0}^{2}$ $σ^{2} > σ_{0}^{2}$ 同 II 同 II
IV $σ^{2} = σ_{0}^{2}$ $σ^{2} < σ_{0}^{2}$ $\frac{i = 1 \sum n ( x _{i} - μ ) ^{2}}{σ _{0}^{2}} \leq λ$ $\frac{i = 1 \sum n ( x _{i} - x ˉ ) ^{2}}{σ _{0}^{2}} \leq χ_{α}^{2} (n - 1)$
V $σ^{2} \geq σ_{0}^{2}$ $σ^{2} < σ_{0}^{2}$ 同 IV 同 IV
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Youliang Zhong

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单个正态总体方差的假设检验的拒绝域

表 8.2 单个正态总体方差的假设检验的拒绝域

序号	$H_{0}$	$H_{1}$	$μ$ 已知	$μ$ 未知
I	$σ^{2} = σ_{0}^{2}$	$σ^{2} \neq = σ_{0}^{2}$	$\frac{i = 1 \sum n ( x _{i} - μ ) ^{2}}{σ _{0}^{2}} \leq χ_{\frac{α}{2}}^{2} (n)$ 或 $\frac{i = 1 \sum n ( x _{i} - μ ) ^{2}}{σ _{0}^{2}} \geq χ_{1 - α}^{2} (n)$	$\frac{i = 1 \sum n ( x _{i} - x ˉ ) ^{2}}{σ _{0}^{2}} \leq χ_{α /2}^{2} (n - 1)$ 或 $\frac{i = 1 \sum n ( x _{i} - x ˉ ) ^{2}}{σ _{0}^{2}} \geq χ_{1 - α /2}^{2} (n - 1)$
II	$σ^{2} = σ_{0}^{2}$	$σ^{2} > σ_{0}^{2}$	$\frac{i = 1 \sum n ( x _{i} - μ ) ^{2}}{σ _{0}^{2}} \geq χ_{1 - α}^{2} (n)$	$\frac{i = 1 \sum n ( x _{i} - x ˉ ) ^{2}}{σ _{0}^{2}} \geq χ_{1 - α}^{2} (n - 1)$
III	$σ^{2} \leq σ_{0}^{2}$	$σ^{2} > σ_{0}^{2}$	同 II	同 II
IV	$σ^{2} = σ_{0}^{2}$	$σ^{2} < σ_{0}^{2}$	$\frac{i = 1 \sum n ( x _{i} - μ ) ^{2}}{σ _{0}^{2}} \leq λ$	$\frac{i = 1 \sum n ( x _{i} - x ˉ ) ^{2}}{σ _{0}^{2}} \leq χ_{α}^{2} (n - 1)$
V	$σ^{2} \geq σ_{0}^{2}$	$σ^{2} < σ_{0}^{2}$	同 IV	同 IV