第4講統計的検定

正規母集団の母平均の差の検定と推定

母分散が既知の場合

統計量

T = \frac{{\bar{X}}_{1} - {\bar{X}}_{2}}{\sqrt{\frac{{σ_{1}}^{2}}{n_{1}} + \frac{{σ_{2}}^{2}}{n_{2}}}} \sim N (0, 1)

$\begin{equation} T = \frac{\bar{X}_1 - \bar{X}_2}{\ds \sqrt{\frac{{\sigma_1}^2}{n_1} + \frac{{\sigma_2}^2}{n_2}}} \sim N(0, 1) \end{equation}$

検定

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu \ne \mu_0$

| T | > z_{\frac{α}{2}}

$\begin{equation} |T| > z_{\frac{\alpha}{2}} \end{equation}$

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu > \mu_0$

T > z_{α}

$\begin{equation} T > z_{\alpha} \end{equation}$

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu < \mu_0$

T < - z_{α}

$\begin{equation} T < -z_{\alpha} \end{equation}$

区間推定

{\bar{X}}_{1} - {\bar{X}}_{2} - z_{\frac{α}{2}} \sqrt{(\frac{{σ_{1}}^{2}}{n_{1}} + \frac{{σ_{2}}^{2}}{n_{2}})} \leq μ_{1} - μ_{2} \leq {\bar{X}}_{1} - {\bar{X}}_{2} + z_{\frac{α}{2}} \sqrt{(\frac{{σ_{1}}^{2}}{n_{1}} + \frac{{σ_{2}}^{2}}{n_{2}})}

$\begin{equation} \bar{X}_1 - \bar{X}_2 - z_{\frac{\alpha}{2}} \sqrt{\left(\frac{{\sigma_1}^2}{n_1} + \frac{{\sigma_2}^2}{n_2}\right)} \le \mu_1 - \mu_2 \le \bar{X}_1 - \bar{X}_2 + z_{\frac{\alpha}{2}} \sqrt{\left(\frac{{\sigma_1}^2}{n_1} + \frac{{\sigma_2}^2}{n_2}\right)} \end{equation}$

母分散は未知であるが等しい場合

${\sigma_{1}}^{2} = {\sigma_{2}}^{2} = \sigma^{2}$ とする．

統計量

T = \frac{{\bar{X}}_{1} - {\bar{X}}_{2}}{\sqrt{(\frac{1}{n_{1}} + \frac{1}{n_{2}}) U^{2}}} \sim t (n_{1} + n_{2} - 2)

$\begin{equation} T = \frac{\bar{X}_1 - \bar{X}_2}{\ds \sqrt{\left(\frac{1}{n_1} + \frac{1}{n_2}\right)U^2}} \sim t(n_1 + n_2 - 2) \end{equation}$

U^{2} = \frac{(n_{1} - 1) {U_{1}}^{2} + (n_{2} - 1) {U_{2}}^{2}}{n_{1} + n_{2} - 2}

$\begin{equation} {U}^{2}= \frac{(n_1 - 1){U_1}^2 + (n_2 - 1){U_2}^2}{n_1 + n_2 - 2} \end{equation}$

{U_{1}}^{2} = \frac{1}{n_{1} - 1} \sum_{i = 1}^{n_{1}} (X_{1 i} - {\bar{X}}_{1})^{2}

$\begin{equation} {U_1}^{2}= \frac{1}{n_1 - 1}\sum_{i=1}^{n_1}(X_{1i} - \bar{X}_1)^2 \end{equation}$

{U_{2}}^{2} = \frac{1}{n_{2} - 1} \sum_{i = 1}^{n_{2}} (X_{2 i} - {\bar{X}}_{2})^{2}

$\begin{equation} {U_2}^{2}= \frac{1}{n_2 - 1}\sum_{i=1}^{n_2}(X_{2i} - \bar{X}_2)^2 \end{equation}$

検定

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu \ne \mu_0$

| T | > t_{\frac{α}{2}} (n_{1} + n_{2} - 2)

$\begin{equation} |T| > t_{\frac{\alpha}{2}}(n_1 + n_2 - 2) \end{equation}$

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu > \mu_0$

T > t_{α} (n_{1} + n_{2} - 2)

$\begin{equation} T > t_{\alpha}(n_1 + n_2 - 2) \end{equation}$

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu < \mu_0$

T < - t_{α} (n_{1} + n_{2} - 2)

$\begin{equation} T < -t_{\alpha}(n_1 + n_2 - 2) \end{equation}$

区間推定

{\bar{X}}_{1} - {\bar{X}}_{2} - t_{\frac{α}{2}} (n_{1} + n_{2} - 2) \sqrt{(\frac{1}{n_{1}} + \frac{1}{n_{2}}) U^{2}} \leq μ_{1} - μ_{2} \leq {\bar{X}}_{1} - {\bar{X}}_{2} + t_{\frac{α}{2}} (n_{1} + n_{2} - 2) \sqrt{(\frac{1}{n_{1}} + \frac{1}{n_{2}}) U^{2}}

$\begin{equation} \bar{X}_1 - \bar{X}_2 - t_{\frac{\alpha}{2}}(n_1+n_2-2) \sqrt{\left(\frac{1}{n_1} + \frac{1}{n_2}\right)U^2} \le \mu_1 - \mu_2 \le \bar{X}_1 - \bar{X}_2 + t_{\frac{\alpha}{2}}(n_1+n_2-2) \sqrt{\left(\frac{1}{n_1} + \frac{1}{n_2}\right)U^2} \end{equation}$

母分散が未知であり等しいとは限らない場合

統計量

T = \frac{{\bar{X}}_{1} - {\bar{X}}_{2}}{\sqrt{\frac{{U_{1}}^{2}}{n_{1}} + \frac{{U_{2}}^{2}}{n_{2}}}} \sim t (υ^{*})

$\begin{equation} T = \frac{\bar{X}_1 - \bar{X}_2}{\ds \sqrt{\frac{{U_1}^{2}}{n_1} + \frac{{U_2}^{2}}{n_2}}} \sim t(\upsilon^{*}) \end{equation}$

自由度

υ^{*}

$\upsilon^{*}$ は，ウェルチの近似法により

υ = \frac{{(\frac{{U_{1}}^{2}}{n_{1}} + \frac{{U_{2}}^{2}}{n_{2}})}^{2}}{\frac{({U_{1}}^{2} / n_{1})^{2}}{n_{1} - 1} + \frac{({U_{2}}^{2} / n_{2})^{2}}{n_{2} - 1}}

$\begin{equation} \upsilon = \frac{\ds\left(\frac{{U_{1}}^{2}}{n_1} + \frac{{U_{2}}^{2}}{n_2}\right)^{2}}{\ds\frac{({U_{1}}^{2}/n_1)^{2}}{n_1-1} + \frac{({U_{2}}^{2}/n_2)^{2}}{n_2-1}} \end{equation}$

に最も近い整数．

{U_{1}}^{2} = \frac{1}{n_{1} - 1} \sum_{i = 1}^{n_{1}} (X_{1 i} - {\bar{X}}_{1})^{2}

$\begin{equation} {U_1}^{2}= \frac{1}{n_1 - 1}\sum_{i=1}^{n_1}(X_{1i} - \bar{X}_1)^2 \end{equation}$

{U_{2}}^{2} = \frac{1}{n_{2} - 1} \sum_{i = 1}^{n_{2}} (X_{2 i} - {\bar{X}}_{2})^{2}

$\begin{equation} {U_2}^{2}= \frac{1}{n_2 - 1}\sum_{i=1}^{n_2}(X_{2i} - \bar{X}_2)^2 \end{equation}$

検定

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu \ne \mu_0$

| T | > t_{\frac{α}{2}} (υ^{*})

$\begin{equation} |T| > t_{\frac{\alpha}{2}}(\upsilon^{*}) \end{equation}$

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu > \mu_0$

T > t_{α} (υ^{*})

$\begin{equation} T > t_{\alpha}(\upsilon^{*}) \end{equation}$

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu < \mu_0$

T < - t_{α} (υ^{*})

$\begin{equation} T < -t_{\alpha}(\upsilon^{*}) \end{equation}$

区間推定

{\bar{X}}_{1} - {\bar{X}}_{2} - t_{\frac{α}{2}} (υ^{*}) \sqrt{\frac{{U_{1}}^{2}}{n_{1}} + \frac{{U_{2}}^{2}}{n_{2}}} \leq μ_{1} - μ_{2} \leq {\bar{X}}_{1} - {\bar{X}}_{2} + t_{\frac{α}{2}} (υ^{*}) \sqrt{\frac{{U_{1}}^{2}}{n_{1}} + \frac{{U_{2}}^{2}}{n_{2}}}

$\begin{equation} \bar{X}_1 - \bar{X}_2 - t_{\frac{\alpha}{2}}(\upsilon^{*}) \sqrt{\frac{{U_1}^2}{n_1} + \frac{{U_2}^2}{n_2}} \le \mu_1 - \mu_2 \le \bar{X}_1 - \bar{X}_2 + t_{\frac{\alpha}{2}}(\upsilon^{*}) \sqrt{\frac{{U_1}^2}{n_1} + \frac{{U_2}^2}{n_2}} \end{equation}$

2つの分布からの標本の大きさが等しく，対応がある場合

$n_1 = n_2 = n$ とする．

統計量

T = \frac{{\bar{X}}_{1} - {\bar{X}}_{2}}{\sqrt{\frac{U^{2}}{n}}} \sim t (n - 1)

$\begin{equation} T = \frac{\bar{X}_1 - \bar{X}_2}{\ds \sqrt{\frac{{U}^{2}}{n}}} \sim t(n - 1) \end{equation}$

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

$\begin{equation} {U}^{2}= \frac{1}{n - 1}\sum_{i=1}^{n}\{(X_{1i} - X_{2i}) - (\bar{X}_1 - \bar{X}_2)\}^2 \end{equation}$

対比較の検定と呼ぶことがある．

{σ_{1}}^{2} = {σ_{2}}^{2}

${\sigma_1}^2 = {\sigma_2}^2$ の検定は不要．

検定

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu \ne \mu_0$

| T | > t_{\frac{α}{2}} (n - 1)

$\begin{equation} |T| > t_{\frac{\alpha}{2}}(n-1) \end{equation}$

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu > \mu_0$

T > t_{α} (n - 1)

$\begin{equation} T > t_{\alpha}(n-1) \end{equation}$

$H_{0}:~\mu_{1} = \mu_{2}~~~~H_{1}:~\mu < \mu_0$

T < - t_{α} (n - 1)

$\begin{equation} T < -t_{\alpha}(n-1) \end{equation}$

区間推定

{\bar{X}}_{1} - {\bar{X}}_{2} - t_{\frac{α}{2}} (n - 1) \sqrt{\frac{U^{2}}{n}} \leq μ_{1} - μ_{2} \leq {\bar{X}}_{1} - {\bar{X}}_{2} + t_{\frac{α}{2}} (n - 1) \sqrt{\frac{U^{2}}{n}}

$\begin{equation} \bar{X}_1 - \bar{X}_2 - t_{\frac{\alpha}{2}}(n-1) \sqrt{\frac{{U}^2}{n}} \le \mu_1 - \mu_2 \le \bar{X}_1 - \bar{X}_2 + t_{\frac{\alpha}{2}}(n-1) \sqrt{\frac{{U}^2}{n}} \end{equation}$

第4講 統計的検定

正規母集団の母平均の差の検定と推定

母分散が既知の場合

統計量

検定

区間推定

母分散は未知であるが等しい場合

統計量

検定

区間推定

母分散が未知であり等しいとは限らない場合

統計量

検定

区間推定

2つの分布からの標本の大きさが等しく，対応がある場合

統計量

検定

区間推定

第4講統計的検定