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Schwankungsintervall - Versionsgeschichte
2024-03-28T16:41:41Z
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https://wikis.hu-berlin.de/mmstat/w/index.php?title=Schwankungsintervall&diff=1262&oldid=prev
Jacobdan am 30. Mai 2018 um 15:26 Uhr
2018-05-30T15:26:40Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Nächstältere Version</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Version vom 30. Mai 2018, 16:26 Uhr</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{| </del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins style="font-weight: bold; text-decoration: none;">iframe k</ins>="<ins style="font-weight: bold; text-decoration: none;">wiwi</ins>" <ins style="font-weight: bold; text-decoration: none;">p</ins>="<ins style="font-weight: bold; text-decoration: none;">examples</ins>/<ins style="font-weight: bold; text-decoration: none;">stat_Schwankungsintervall_Schwankungsintervall_N_R00480004800000000000000_plot</ins>.<ins style="font-weight: bold; text-decoration: none;">html</ins>" /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|</del><<del style="font-weight: bold; text-decoration: none;">R output</del>="<del style="font-weight: bold; text-decoration: none;">display</del>"<del style="font-weight: bold; text-decoration: none;">></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">curve(from=-4, to=4, dnorm(x, mean=0, sd=1), xaxt="n", ylab="f(z)", xlab="z", main="Zentrales Schwankungsintervall einer normalverteilten Zufallsvariablen", col="red", ylim=c(0.0,0.4), lty=1, lwd=4, font.lab=2, "xaxs"="i" ,"yaxs"="i", bty="l")</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">abline(v=-2, col="orange", lwd=2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">abline(v=2, col="orange", lwd=2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">#abline(h=0.0)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">text(0, 0.15, expression(paste("1-" , alpha)), col = "black", cex=2.5)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">text(-2.3, 0.01, expression(paste(alpha, "/2")), col </del>= "<del style="font-weight: bold; text-decoration: none;">black", cex=1.2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">text(2.3, 0.01, expression(paste(alpha, "</del>/<del style="font-weight: bold; text-decoration: none;">2")), col = "black", cex=1</del>.<del style="font-weight: bold; text-decoration: none;">2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">axis( side=1, at=c(-2, 2, 4), labels=c(expression(paste(mu, </del>"<del style="font-weight: bold; text-decoration: none;">-z", sigma)), expression(paste(mu, "+z", sigma)), "z"), tick=FALSE)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><</del>/<del style="font-weight: bold; text-decoration: none;">R</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wegen</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Wegen</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l117">Zeile 117:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Mit einer [[Wahrscheinlichkeit]] von 99% liegt die [[Zufallsvariable]] <math>X\,</math> im Intervall <math>\left[74,2;125,8\right]</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Mit einer [[Wahrscheinlichkeit]] von 99% liegt die [[Zufallsvariable]] <math>X\,</math> im Intervall <math>\left[74,2;125,8\right]</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{|</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins style="font-weight: bold; text-decoration: none;">iframe k</ins>="<ins style="font-weight: bold; text-decoration: none;">wiwi</ins>" <ins style="font-weight: bold; text-decoration: none;">p</ins>="<ins style="font-weight: bold; text-decoration: none;">examples/stat_Schwankungsintervall_Schwankungsintervall_N_100_10_R00480004800000000000000_plot</ins>.<ins style="font-weight: bold; text-decoration: none;">html</ins>" /></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|</del><<del style="font-weight: bold; text-decoration: none;">R output</del>="<del style="font-weight: bold; text-decoration: none;">display</del>"<del style="font-weight: bold; text-decoration: none;">></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">pdf(rpdf,width=7,height=7)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">curve(from=60, to=140, dnorm(x, mean=100, sd=10), xaxt</del>="<del style="font-weight: bold; text-decoration: none;">n", ylab="f(x)", xlab="x", col="blue", ylim=c(0.0,0.04), lty=1, lwd=4, font</del>.<del style="font-weight: bold; text-decoration: none;">lab=2, "xaxs</del>"<del style="font-weight: bold; text-decoration: none;">="i" ,"yaxs"="i", bty="l")</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">axis( side=1, at=c(60, 74.2, 125.8), labels=c("60", "74,2", "125,8"), tick=FALSE)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">par(new=TRUE)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">xx <-c(74.5:125.5, 125.5:74.5)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">yy <-c(c(dnorm(c(74.5:125.5), mean=100, sd=10)),c(rep(0,52)))</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">polygon(xx, yy, col="grey", border=NA)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">par(new=TRUE)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">curve(from=60, to=140, dnorm(x, mean=100, sd=10), xaxt="n", ylab="f(x)", xlab="x", col="blue", ylim=c(0.0,0.04), lty=1, lwd=4, font.lab=2, "xaxs"="i" ,"yaxs"="i", bty="l")</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">abline(v=74.2, col="black", lwd=2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">abline(v=125.8, col="black", lwd=2)</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><</del>/<del style="font-weight: bold; text-decoration: none;">R</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">|}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
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Jacobdan
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=={{Vorlage:Überschrift}}==<br />
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===Schwankungsintervall, Sicherheits- und Überschreitungswahrscheinlichkeit===<br />
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Ein ''Schwankungsintervall'' für die [[Zufallsvariable]] <math>X\,</math> ist ein Bereich mit festen Grenzen <math>x_{u}\,</math> und <math>x_{o}(x_{u}\leq x_{o})</math>, in dem die [[Zufallsvariable]] <math>X</math> [[Realisation]]en <math>x</math> mit einer vorgegebenen ''Sicherheitswahrscheinlichkeit'' <math>1 - \alpha</math> annimmt, d.h. <math>(1 - \alpha)\cdot 100%</math> aller [[Realisation]]en von <math>X\,</math> liegen in diesem Intervall und <math>\alpha </math> aller [[Realisation]]en von <math>X\,</math> außerhalb des Intervalls. <br />
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<math>\alpha</math> wird als ''Überschreitungswahrscheinlichkeit'' bezeichnet.<br />
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===Zentrales Schwankungsintervall===<br />
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Konstruiert man das Intervall um den bekannten [[Erwartungswert]] <math>\mu</math> der [[Zufallsvariable]]n <math>X\,</math> derart, dass den<br />
beiden Bereichen außerhalb der Grenzen des Intervalls jeweils die gleiche [[Wahrscheinlichkeit]] <math>\alpha/2</math> zugeordnet ist, dann heißt<br />
<br />
<math>[x_u \leq X \leq x_o] = [\mu - k \leq X \leq \mu + k]</math><br />
<br />
ein ''zentrales Schwankungsintervall'' mit der [[Sicherheitswahrscheinlichkeit]]<br />
<br />
<math>P(x_u \leq X \leq x_o) = 1 - \alpha</math>.<br />
<br />
Um die Bedeutung der [[Standardabweichung (stochastisch)|Standardabweichung]] als Streuungs[[parameter]] hervorzuheben, misst man die Abweichung <math>k</math> von <math>\mu</math> oftmals in Vielfachen von <math>\sigma</math>, so dass das zentrale Schwankungsintervall die Form<br />
<br />
<math>[\mu - c\cdot \sigma \leq X \leq \mu + c\cdot \sigma]</math><br />
<br />
hat.<br />
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Ist die [[Zufallsvariable]] <math>X\; N(\mu, \sigma)</math>-verteilt, so folgt für <math>x= \mu + c\cdot \sigma</math><br />
<br />
<math>z= \frac{x - \mu}{\sigma} = \frac{\mu + c\cdot \sigma - \mu}{\sigma}= c</math><br />
<br />
und <math>P(Z \leq z) = \Phi(z) = 1 - \frac{\alpha}{2}</math>.<br />
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Der Wert <math>z_{1-\frac{\alpha}{2}}</math> kann für die [[Wahrscheinlichkeit]] <math>1 -\frac{\alpha}{2}</math> aus der Tabelle der [[Verteilungsfunktion (stochastisch, eindimensional)|Verteilungsfunktion]] der [[Standardnormalverteilung]] entnommen werden.<br />
<br />
Somit ist das zentrale Schwankungsintervall für eine [[Normalverteilung|normalverteilte]] [[Zufallsvariable]] mit<br />
<br />
<math>[\mu - z_{1-\frac{\alpha}{2}}\cdot \sigma \leq X \leq \mu + z_{1-\frac{\alpha}{2}}\cdot \sigma]</math><br />
<br />
und die [[Wahrscheinlichkeit]] dieses Intervalls mit<br />
<br />
<math>P(\mu - z_{1-\frac{\alpha}{2}}\cdot \sigma \leq X \leq \mu + z_{1-\frac{\alpha}{2}}\cdot \sigma) = 1 - \alpha</math><br />
<br />
gegeben.<br />
<br />
{| <br />
|<R output="display"><br />
pdf(rpdf,width=7,height=7)<br />
<br />
curve(from=-4, to=4, dnorm(x, mean=0, sd=1), xaxt="n", ylab="f(z)", xlab="z", main="Zentrales Schwankungsintervall einer normalverteilten Zufallsvariablen", col="red", ylim=c(0.0,0.4), lty=1, lwd=4, font.lab=2, "xaxs"="i" ,"yaxs"="i", bty="l")<br />
abline(v=-2, col="orange", lwd=2)<br />
abline(v=2, col="orange", lwd=2)<br />
#abline(h=0.0)<br />
text(0, 0.15, expression(paste("1-" , alpha)), col = "black", cex=2.5)<br />
text(-2.3, 0.01, expression(paste(alpha, "/2")), col = "black", cex=1.2)<br />
text(2.3, 0.01, expression(paste(alpha, "/2")), col = "black", cex=1.2)<br />
axis( side=1, at=c(-2, 2, 4), labels=c(expression(paste(mu, "-z", sigma)), expression(paste(mu, "+z", sigma)), "z"), tick=FALSE)<br />
<br />
</R><br />
|}<br />
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Wegen<br />
<br />
<math>P(-z \leq Z \leq z) = P(Z \leq z) - P(Z \leq -z) = P(Z \leq z) - [1 - P(Z \leq z)] = 2P(Z \leq z)-1</math>,<br />
<br />
folgt<br />
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<math>P(\mu - z_{1-\frac{\alpha}{2}}\cdot \sigma \leq X \leq \mu + z_{1-\frac{\alpha}{2}}\cdot \sigma) = 2 \Phi(z) - 1</math>.<br />
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Für vorgegebenes <math>z</math> lässt sich die [[Sicherheitswahrscheinlichkeit]] für das zentrale Schwankungsintervall ermitteln, z.B.<br />
<br />
<math>P(\mu -z_{1-\frac{\alpha}{2}}\cdot \sigma \leq X \leq \mu +z_{1-\frac{\alpha}{2}}\cdot \sigma )=<br />
\begin{cases}<br />
0,6827\quad \mbox{, wenn}\quad z_{1-\frac{\alpha}{2}}=1\\<br />
0,9545\quad \mbox{, wenn}\quad z_{1-\frac{\alpha}{2}}=2\\<br />
0,9973\quad \mbox{, wenn}\quad z_{1-\frac{\alpha}{2}}=3<br />
\end{cases}</math><br />
<br />
Umgekehrt findet man für eine vorgegebene [[Sicherheitswahrscheinlichkeit]] <math>1-\alpha</math> den zugehörigen <math>z</math>-Wert, z.B. für <math>P(\mu -z_{1-\frac{\alpha}{2}}\cdot \sigma \leq X \leq\mu+z_{1-\frac{\alpha}{2}}\cdot \sigma ) = 0,95</math> den Wert <math>z = 1,96</math>.<br />
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=={{Vorlage:Beispiele}}==<br />
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===Schwankungsintervall===<br />
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Gegeben sei eine [[Zufallsvariable]] <math>X\,</math>, die <math>N(100;10)</math>-verteilt ist.<br />
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Gesucht ist ein symmetrischer Bereich um den [[Mittelwert]], so dass 99% der [[Realisation]]en von <math>X</math> in diesem Bereich liegen.<br />
<br />
{|<br />
|<math>0,99\,</math><br />
|<math>=P(x_u\leq X\leq x_o)</math><br />
|-<br />
|<br />
|<math>=P(\frac{x_u-100}{10}\leq Z\leq \frac{x_o-100}{10})</math><br />
|-<br />
|<br />
|<math>=P(-z\leq Z\leq z)</math><br />
|-<br />
|<br />
|<math>=2\cdot \Phi(z)-1</math><br />
|-<br />
|<math>\Phi(z)=\frac{1,99}{2}=0,995</math><br />
|}<br />
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Für die [[Wahrscheinlichkeit]] von 0,995 findet man aus der Tabelle der [[Verteilungsfunktion (stochastisch, eindimensional)|Verteilungsfunktion]] der [[Standardnormalverteilung]] den Wert <math>z = 2,58</math>. <br />
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Damit sind<br />
<br />
{|<br />
|<math>x_o = \mu + z \cdot \sigma = 100 + 2,58 \cdot 10 = 125,8</math><br />
|-<br />
|<math>x_u = \mu - z \cdot \sigma = 100 - 2,58 \cdot 10 = 74,2</math><br />
|}<br />
<br />
und somit <math>P(74,2 \leq X \leq 125,8) = 0,99</math>.<br />
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Mit einer [[Wahrscheinlichkeit]] von 99% liegt die [[Zufallsvariable]] <math>X\,</math> im Intervall <math>\left[74,2;125,8\right]</math>.<br />
<br />
{|<br />
|<R output="display"><br />
pdf(rpdf,width=7,height=7)<br />
<br />
curve(from=60, to=140, dnorm(x, mean=100, sd=10), xaxt="n", ylab="f(x)", xlab="x", col="blue", ylim=c(0.0,0.04), lty=1, lwd=4, font.lab=2, "xaxs"="i" ,"yaxs"="i", bty="l")<br />
axis( side=1, at=c(60, 74.2, 125.8), labels=c("60", "74,2", "125,8"), tick=FALSE)<br />
<br />
par(new=TRUE)<br />
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xx <-c(74.5:125.5, 125.5:74.5)<br />
yy <-c(c(dnorm(c(74.5:125.5), mean=100, sd=10)),c(rep(0,52)))<br />
<br />
polygon(xx, yy, col="grey", border=NA)<br />
<br />
par(new=TRUE)<br />
<br />
curve(from=60, to=140, dnorm(x, mean=100, sd=10), xaxt="n", ylab="f(x)", xlab="x", col="blue", ylim=c(0.0,0.04), lty=1, lwd=4, font.lab=2, "xaxs"="i" ,"yaxs"="i", bty="l")<br />
<br />
<br />
abline(v=74.2, col="black", lwd=2)<br />
abline(v=125.8, col="black", lwd=2)<br />
<br />
</R><br />
|}</div>
Jacobdan