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@ -277,6 +277,8 @@
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\begin{frame}
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\frametitle{Aufgabe 2: Abschätzungen von Verteilungen \\(ZGWS)}
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\vspace*{-3mm}
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Im Werk einer Zahnradfabrik werden verschiedene
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Präzisionsmetallteile gefertigt. Während einer Schicht werden
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$5000$ Stück eines Typs A hergestellt. Bei der Qualitätskontrolle
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@ -319,22 +321,38 @@
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\pause\centering
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\rule[1pt]{16cm}{1pt}
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\vspace*{-7mm}
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\vspace*{-10mm}
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\begin{gather*}
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Np(1-p) = 145{,}5 \ge 9 \hspace*{3mm}
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\hspace*{5mm} \rightarrow \hspace*{5mm}
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\tilde{S}_N \sim \mathcal{N}(\mu = \underbrace{Np}_{=150},
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\sigma^2 = \underbrace{Np(1-p)}_{=145{,}5})
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\tilde{S}_N \sim \mathcal{N}(\mu = \overbrace{Np}^{=150},
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\sigma^2 = \overbrace{Np(1-p)}^{=145{,}5})
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\end{gather*}
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\vspace*{-5mm}
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\pause
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\begin{align*}
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P(125 \le \tilde{S}_N \le 180)
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&= P\left(\frac{125 - 150}{\sqrt{145{,}5}} \le \frac{\tilde{S}_N -
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E(\tilde{S}_N)}{\sqrt{V(\tilde{S}_N)}} \le \frac{180 -
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150}{\sqrt{145{,}5}}\right) \\
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&\approx \Phi(2{,}487) - \Phi(-2{,}073) \\
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&= \Phi(2{,}487) - (1 - \Phi(2{,}073)) \approx 0{,}974
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\end{align*}
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\begin{minipage}[t]{0.15\textwidth}
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\centering
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\begin{align*}
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P(125 \le \tilde{S}_N \le 180)
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\end{align*}
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\end{minipage}
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\pause\begin{minipage}[t]{0.3\textwidth}
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\centering
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\vspace*{-10mm}
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\begin{align*}
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&= P\Bigg(\frac{125 - 150}{\sqrt{145{,}5}} \le
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\overbrace{\frac{\tilde{S}_N -
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E(\tilde{S}_N)}{\sqrt{V(\tilde{S}_N)}}}^{\sim
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\mathcal{N}(0,1)} \le \frac{180 -
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150}{\sqrt{145{,}5}}\Bigg)
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\end{align*}
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\vspace*{-9mm}
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\pause
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\begin{align*}
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\hspace{25mm} &\approx \Phi(2{,}487) - \Phi(-2{,}073) \\
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&= \Phi(2{,}487) - (1 - \Phi(2{,}073)) \approx 0{,}974
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\end{align*}
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\end{minipage}
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\end{frame}
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\begin{frame}
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