Låt $x_1, \ldots, x_6$ vara observationerna av Global Terrorism Index.

Låt $y_1, \ldots, y_6$ vara observationerna av Global Peace Index.

Vårt test är

$H_0$ : $a = 1$ .

$H_{A}$ : $a \neq 1$ .

\begin{align*} \sum_{i=1}^{6} x_i &= 22.596. \tag{1} \\ \sum_{i=1}^{6} y_i &= 12.793. \tag{2} \\ \sum_{i=1}^{6} x_i^2 &= 115.30435. \tag{3} \\ \sum_{i=1}^{6} x_i y_i &= 57.272799. \tag{4} \\ \sum_{i=1}^{6} y_i^2 &= 30.465455. \tag{5} \end{align*}

Från (1)-(5) beräknar vi

\begin{align*} S_{xx} &= \sum_{i=1}^{6} x_i^2 - \left( \sum_{i=1}^{6} x_i \right)^2 / 6 = 115.30435 - 22.596^2 / 6 \\ &= 30.207814. \tag{6} \\ S_{xy} &= \sum_{i=1}^{6} x_i y_i - \left( \sum_{i=1}^{6} x_i \right) \left( \sum_{i=1}^{6} y_i \right) / 6 = 57.272799 - 22.596 \cdot 12.793 / 6 \\ &= 9.094361. \tag{7} \\ S_{yy} &= \sum_{i=1}^{6} y_i^2 - \left( \sum_{i=1}^{6} y_i \right)^2 / 6 = 30.465455 - 12.793^2 / 6 \\ &= 3.188646833 \dots \tag{8} \end{align*}

Under $H_0$ gäller

\frac{a^* - 1}{S_a} \in t(6 - 2).

Enligt (6)-(8) blir vår testvariabel

\begin{aligned} u &= \frac{a^* - 1}{s_a} = \frac{\bar{y} - b^* \bar{x} - 1}{s \sqrt{\frac{1}{6} + \frac{\bar{x}^2}{S_{xx}}}} = \frac{\bar{y} - b^* \bar{x} - 1}{\sqrt{\frac{Q_0}{6 - 2}} \sqrt{\frac{1}{6} + \frac{\bar{x}^2}{S_{xx}}}} = \frac{\bar{y} - \frac{S_{xy}}{S_{xx}} \bar{x} - 1}{\sqrt{\frac{S_{yy} - S_{xy}^2 / S_{xx}}{4}} \sqrt{\frac{1}{6} + \frac{\bar{x}^2}{S_{xx}}}} \\ &= \frac{2(12.793/6 - \frac{9.094361}{30.207814} \cdot 22.596/6 - 1)}{\sqrt{3.188646833 \ldots - 9.094361^2 / 30.207814} \sqrt{\frac{1}{6} + \frac{(22.596/6)^2}{30.207814}}} \approx -0.006. \end{aligned}

Eftersom $|u| \approx |-0.006| < 2.776 \approx t_{0.05/2}(6 - 2)$ förkastas inte $H_0$ .

SVAR: $H_0$ förkastas inte.