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zJ$h%;+~0yl<;9n#uKc<;cz0RaZ;q9_IUk)m^Q~@SX!g@Ax;`$%Ws$ur=%Z*EQ}A+OIqB8~z1eZYz9k zvs_lm^~u+^E#f;`b>OA`^~X>BBhF{-OW)db?p9XK&RH*_Q*D#F*DP_<4qk8B79r=f z@rL3r!KC#|jJ1o`?`#W@%FH;rt*AEd59?E&dZ(QSE^JVb<6Wmv@i0Cm>)+Ky&+gYr z{cBwG(9C(+j-m<`d_AZ@iNIzm*!y>t+zCZ<g2#x71KhR!B-3WSvaZQ+IvgBh6_D1e5+>SfPc>mLr3Irj0nWt8j1 zT_IPO>irGAzH7Dj>a{LO8*Uuso=_nuv|0Q>&H*LERadzu+*q4rCi%hUwEF$~mg@VT zFdWEdFq*r_yM&udz{4?Abahhrn$Fyf>Sjllt?2BzAsD?#=bGd@{=KUdqg)gwsoSmY z*wxW3G123;Hh;%b+FHM^!s^gUjey77ELIO}0=sd|=~Q`AJK W|KVkp \mu_0$. + Der Student-$t$-Test ist dann gegeben durch + \[ + \phi_c^r = \mathbbm{1}_{\sqrt{n}(\overline{X_n} - \mu_0) \geq c \hat{S}_n} + \] + mit $ c= t_{(n-1), (1-\alpha)}$. + Wir berechnen also zunächst + \[ + \overline{X_n} = \frac{1}{n}\sum_{i = 1}^{n} X_i = 103.64, + \] + \[ + \hat{S}_n = \sqrt{\frac{1}{n-1} \sum_{i = 1}^{n} (X_i - \overline{X}_n)^2} \approx 5.22 + \] + und + \[ + c = t_{(n-1, 1-\alpha)} = t_{9, 0.95} = 1.833. + \] + Daraus folgt + \[ + \phi_c^r = \mathbbm{1}_{\sqrt{10}(103.64-100) \geq 1.833 \cdot 5.22} = \mathbbm{1}_{11.51 \geq 9.57} = 1, + \] + wir lehnen also ab. + \item Wir wollen als Partition in richtige und falsche Parameter $\mathcal{R}_\mu = \{\mu\}$ und $\mathcal{F}_\mu = \R\setminus\{\mu\}$. + Dann erhalten wir als assoziierte Familie von Partitionen und Null- und Alternativhypothesen + $\mathscr H_\mu^0 = \{\mu\}$ und $\mathscr H_\mu^1 = \R \setminus \{\mu\}$. + Da der beidseitige Student-$t$-Test ein $\alpha$-Test der Nullhypothese $H_0\colon \mathscr H_\mu^0$ gegen die + Alternative $H_1 \colon \mathscr H_\mu^1$ für jedes $\mu \in \R$ ist, muss die assoziierte Bereichsschätzfunktion + für $(\{\mathcal R_\mu, \mathcal F_\mu\})_{\mu \in \R}$ ein $(1-\alpha)$-Konfidenzbereich sein. + Die assoziierte Bereichsschätzfunktion zum beidseitigen Student-$t$-Test + $\phi^b_{t_{(n-1), (1 - \alpha/2)}, \mu}(X_1, \dots X_n)$ ist gegeben durch + \begin{align*} + B(X_1, \dots, X_n) &= \{\mu \in \R: \phi^b_{t_{(n-1), (1 - \alpha/2)}, \mu}(X_1, \dots X_n) = 0\}\\ + &= \{\mu \in \R: \sqrt{n}|\overline{X_n} - \mu| \leq t_{(n-1), (1-\alpha/2)} S\}\\ + &= \{\mu \in \R: |\overline{X_n} - \mu| \leq \frac{S}{\sqrt{n}} t_{(n-1), (1-\alpha/2)}\}\\ + &= \left[\overline{X_n} - \frac{S}{\sqrt{n}} t_{(n-1), (1-\alpha/2)}, \overline{X_n} + \frac{S}{\sqrt{n}} t_{(n-1), (1-\alpha/2)}\right] + \end{align*} + Das war zu zeigen. + \item Mithilfe unserer numerischen Resultate aus der (a) sowie der Aussage von Teilaufgabe (b) folgern wir, dass + \begin{align*} + [103.64 - \frac{5.22}{\sqrt{10}}t_{9, 0.975}, 103.64 + \frac{5.22}{\sqrt{10}}t_{9, 0.975}] &\subset [99.90, 107.38] + \end{align*} + ein 95\%-Konfidenzintervall ist. + \end{enumerate} +\end{aufgabe} + \end{document}