$(A_1):$ $\mbox{(1): } V(\hat{\beta}_0)= \frac{\sigma^2}{n}(1+ \frac{\overline{a}^2}{s_{aa}}), \qquad \mbox{(2): } V(\hat{\beta}_1)= \frac{\sigma^2}{n} \frac{1}{s_{aa}},$
 $(A_2):$ [スチューデント化]. 上の(H$_1$)に動機付けられて, 次を得る： \begin{align} & T_{\beta_0} := \frac{\sqrt{n}(\hat{\beta}_0-{\beta}_0)} {\sqrt{ {\hat{\sigma}^2(1+ \overline{a}^2/ s_{aa})}}} \sim t_{n-2}, \qquad T_{\beta_1} := \frac{\sqrt{n}(\hat{\beta}_1-{\beta}_1)} {\sqrt{ {\hat{\sigma}^2/ s_{aa}}}} \sim t_{n-2} \tag{15.27} \end{align} ここに,$t_{n-2}$は自由度($n-2$)のスチューデント分布の確率密度関数とする.
$\square \quad$

 $(B_1):$ 帰無仮説 $H_{N} = { \{ \beta_0 \}} (\subseteq \Theta_0={\mathbb R})$において, 棄却域は以下のようになる. \begin{align} {\widehat R}_{{H_N}}^{\alpha; X} & = {\widehat{E}_0}^{-1}( {\widehat R}_{{H_N}}^{\alpha; {\Theta_0}}) = \bigcap_{\omega \in \Omega \mbox{ such that } \pi_0(\omega) \in {H_N}} \{ x (\in X) : d^x_{\Theta_0} ({}{\widehat{E}_0}(x), \pi_0(\omega ) ) \ge \eta^\alpha_{\omega } \} \nonumber \\ & = \Big\{ x \in X \;:\; \frac{ |\hat{\beta}_0 (x) -{\beta}_0| }{ {\sqrt{ {\frac{\hat{\sigma}^2(x)}{n}(1+ \overline{a}^2/ s_{aa})}}} } \ge t_{n-2}(\alpha/2) \Big\} \tag{15.34} \end{align}
 $(B_2):$ 帰無仮説 $H_{N} = { \{ \beta_0 \}} (\subseteq \Theta_0={\mathbb R})$において, 棄却域は以下のようになる. \begin{align} {\widehat R}_{{H_N}}^{\alpha; X} & = {\widehat{E}_1}^{-1}( {\widehat R}_{{H_N}}^{\alpha; {\Theta_1}}) = \bigcap_{\omega \in \Omega \mbox{ such that } \pi_1(\omega) \in {H_N}} \{ x (\in X) : d^x_{\Theta_1} ({}{\widehat{E}_1}(x), \pi_1(\omega ) ) \ge \eta^\alpha_{\omega } \} \nonumber \\ & = \Big\{ x \in X \;:\; \frac{ |\hat{\beta}_1 (x) -{\beta}_1| }{ {\sqrt{ {\frac{\hat{\sigma}^2(x)}{n}(1/ s_{aa})}}} } \ge t_{n-2}(\alpha/2) \Big\} \tag{15.35} \end{align}