第4講統計的検定

正規母集団の母分散の検定と推定

母平均が既知の場合

統計量

T = \frac{n {S_{0}}^{2}}{σ_{0}^{2}} \sim χ^{2} (n)

$\begin{equation} T = \frac{n{S_0}^{2}}{\sigma_{0}^{2}} \sim \chi^{2}(n) \end{equation}$

{S_{0}}^{2} = \frac{1}{n} \sum_{i = 1}^{n} (X_{i} - μ_{0})^{2}

$\begin{equation} {S_0}^{2}= \frac{1}{n}\sum_{i=1}^{n}(X_i - \mu_0)^2 \end{equation}$

検定

$H_{0}:~\sigma^{2} = {\sigma_{0}}^{2}~~~~H_{1}:~\sigma^{2} \ne {\sigma_{0}}^{2}$

T < χ_{1 - \frac{α}{2}}^{2} (n), T > χ_{\frac{α}{2}}^{2} (n)

$\begin{equation} T < \chi^{2}_{1-\frac{\alpha}{2}}(n),~~T > \chi^{2}_{\frac{\alpha}{2}}(n) \end{equation}$

$H_{0}:~\sigma^{2} = {\sigma_{0}}^{2}~~~~H_{1}:~\sigma^{2} > {\sigma_{0}}^{2}$

T > χ_{α}^{2} (n)

$\begin{equation} T > \chi^{2}_{\alpha}(n) \end{equation}$

$H_{0}:~\sigma^{2} = {\sigma_{0}}^{2}~~~~H_{1}:~\sigma^{2} < {\sigma_{0}}^{2}$

T < χ_{1 - α}^{2} (n)

$\begin{equation} T < \chi^{2}_{1-\alpha}(n) \end{equation}$

区間推定

\frac{n {S_{0}}^{2}}{χ_{\frac{α}{2}}^{2} (n)} \leq {σ_{0}}^{2} \leq \frac{n {S_{0}}^{2}}{χ_{1 - \frac{α}{2}}^{2} (n)}

$\begin{equation} \frac{n{S_0}^{2}}{\chi^{2}_{\frac{\alpha}{2}}(n)} \le {\sigma_0}^2 \le \frac{n{S_0}^{2}}{\chi^{2}_{1-\frac{\alpha}{2}}(n)} \end{equation}$

母平均が未知の場合

統計量

T = \frac{n S^{2}}{σ_{0}^{2}} \sim χ^{2} (n - 1)

$\begin{equation} T = \frac{n{S}^{2}}{\sigma_{0}^{2}} \sim \chi^{2}(n-1) \end{equation}$

S^{2} = \frac{1}{n} \sum_{i = 1}^{n} (X_{i} - \bar{X})^{2}

$\begin{equation} {S}^{2}= \frac{1}{n}\sum_{i=1}^{n}(X_i - \bar{X})^2 \end{equation}$

検定

$H_{0}:~~\sigma^{2} = {\sigma_{0}}^{2}~~~~H_{1}:~\sigma^{2} \ne {\sigma_{0}}^{2}$

T < χ_{1 - \frac{α}{2}}^{2} (n - 1), T > χ_{\frac{α}{2}}^{2} (n - 1)

$\begin{equation} T < \chi^{2}_{1-\frac{\alpha}{2}}(n-1),~~T > \chi^{2}_{\frac{\alpha}{2}}(n-1) \end{equation}$

$H_{0}:~~\sigma^{2} = {\sigma_{0}}^{2}~~~~H_{1}:~\sigma^{2} > {\sigma_{0}}^{2}$

T > χ_{α}^{2} (n - 1)

$\begin{equation} T > \chi^{2}_{\alpha}(n-1) \end{equation}$

$H_{0}:~~\sigma^{2} = {\sigma_{0}}^{2}~~~~H_{1}:~\sigma^{2} < {\sigma_{0}}^{2}$

T < χ_{1 - α}^{2} (n - 1)

$\begin{equation} T < \chi^{2}_{1-\alpha}(n-1) \end{equation}$

区間推定

\frac{n S^{2}}{χ_{\frac{α}{2}}^{2} (n - 1)} \leq {σ_{0}}^{2} \leq \frac{n S^{2}}{χ_{1 - \frac{α}{2}}^{2} (n - 1)}

$\begin{equation} \frac{n{S}^{2}}{\chi^{2}_{\frac{\alpha}{2}}(n-1)} \le {\sigma_0}^2 \le \frac{n{S}^{2}}{\chi^{2}_{1-\frac{\alpha}{2}}(n-1)} \end{equation}$

第4講 統計的検定

正規母集団の母分散の検定と推定

母平均が既知の場合

統計量

検定

区間推定

母平均が未知の場合

統計量

検定

区間推定

第4講統計的検定