Confidence Level and Confidence Interval
2013-12-17 02:55
This applet demonstrates confidence levels and confidence intervals in statistics. Drag the red dot to adjust the confidence level and observe the interval changes. It combines binomial and normal distributions to show sample proportions, means, standard deviations, and probabilities, aiding the understanding of interval estimation.
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.4" version="4.4.2.0" id="3b2190e8-1a8c-4b54-96f8-7317f9b0f12e" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="862" height="630" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.6276102088167054" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="942,301,700,550" />
<view id="4" toolbar="0 59 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="300" window="100,100,600,400" />
<view id="2" visible="false" inframe="true" stylebar="true" location="3,3" size="200" window="1005,316,250,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="3" size="851" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="true" location="1" size="361" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="true" stylebar="true" location="1" size="150" window="271,102,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 59 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="851" height="493"/>
<coordSystem xZero="176.44675300971235" yZero="335.1173900603429" scale="328.48862794892443" yscale="45.54872486438877"/>
<evSettings axes="true" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="0" grid="20"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="0.1"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true" positiveAxis="true"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="dhConfidence">
<value val="0.8"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-0.4" y="4.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Sequence">
<input a0="E + (0, (k * dhConfidence))" a1="k" a2="0" a3="2"/>
<output a0="confidencePoints"/>
</command>
<element type="list" label="confidencePoints">
<show object="true" label="false"/>
<objColor r="255" g="102" b="153" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Point">
<input a0="confidencePoints"/>
<output a0="ConfidencePoint"/>
</command>
<element type="point" label="ConfidencePoint">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<javascript onUpdate="resetCount();"/>
<coords x="-0.4" y="4.8" z="1.0"/>
<pointSize val="7"/>
<pointStyle val="2"/>
</element>
<element type="numeric" label="p">
<value val="0.54"/>
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="100.0" x="40.0" y="425.0" fixed="false" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
<javascript onUpdate="resetCount();"/>
</element>
<expression label="q" exp="1 - p" />
<element type="numeric" label="q">
<value val="0.45999999999999996"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<element type="numeric" label="n">
<value val="16.0"/>
<show object="true" label="true"/>
<objColor r="0" g="153" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="50" absoluteScreenLocation="true" width="100.0" x="275.0" y="425.0" fixed="false" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<javascript onUpdate="resetCount();"/>
</element>
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-3.0" y="-2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="BinomialDist">
<input a0="n" a1="p" a2="false"/>
<output a0="barsBino"/>
</command>
<element type="numeric" label="barsBino">
<value val="1.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<selectionAllowed val="false"/>
<tags>
</tags>
</element>
<expression label="σ" exp="sqrt(((n * p) * q))" />
<element type="numeric" label="σ">
<value val="1.993589727100338"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Sequence">
<input a0="k" a1="k" a2="0" a3="n"/>
<output a0="X"/>
</command>
<element type="list" label="X">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="((Binomial[n, k] * p^(k)) * q^(n - k))" a1="k" a2="0" a3="n"/>
<output a0="f_X"/>
</command>
<element type="list" label="f_X">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<expression label="Xbar" exp="X / n" />
<element type="list" label="Xbar">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<expression label="μ" exp="(n * p)" />
<element type="numeric" label="μ">
<value val="8.64"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Normal">
<input a0="μ" a1="σ" a2="x"/>
<output a0="f"/>
</command>
<element type="function" label="f">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<command name="Segment">
<input a0="(μ, 0)" a1="(μ, f(μ))"/>
<output a0="segMu"/>
</command>
<element type="segment" label="segMu">
<show object="true" label="false"/>
<condition showObject="showSD"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.20011252815878594" y="0.0" z="1.7289722432919106"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="(μ + σ, 0)" a1="(μ + σ, f(μ))"/>
<output a0="segSigmaPlus"/>
</command>
<element type="segment" label="segSigmaPlus">
<show object="true" label="false"/>
<condition showObject="showSD"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.20011252815878594" y="0.0" z="2.1279145236933434"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="(μ - σ, 0)" a1="(μ - σ, f(μ))"/>
<output a0="segSigmaMinus"/>
</command>
<element type="segment" label="segSigmaMinus">
<show object="true" label="false"/>
<condition showObject="showSD"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.20011252815878594" y="0.0" z="1.330029962890478"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="Mu" exp="(μ, f(μ))" type="point" />
<element type="point" label="Mu">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="8.64" y="0.20011252815878594" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="Sigma" exp="(μ + σ, f(μ))" type="point" />
<element type="point" label="Sigma">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="10.633589727100338" y="0.20011252815878594" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="B" exp="(Mu + Sigma) / 2" type="point" />
<element type="point" label="B">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.636794863550168" y="0.20011252815878594" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="B" a1="Mu"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false"/>
<condition showObject="showSD"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.9967948635501678" y="0.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="B"/>
</element>
<command name="Vector">
<input a0="B" a1="Sigma"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false"/>
<condition showObject="showSD"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.9967948635501696" y="0.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="B"/>
</element>
<element type="boolean" label="showSD">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="31" y="438"/>
<labelMode val="0"/>
<caption val="顯示標準差 σ"/>
</element>
<command name="Sequence">
<input a0="(k - 0.5, 0)" a1="k" a2="0" a3="n + 1"/>
<output a0="xvalues"/>
</command>
<element type="list" label="xvalues">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Point">
<input a0="xvalues"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="false"/>
<condition showObject="showP1"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="4.5" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="xvalues"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="false"/>
<condition showObject="showP1"/>
<objColor r="255" g="204" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="6.5" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Min">
<input a0="x(C)" a1="x(D)"/>
<output a0="xmin"/>
</command>
<element type="numeric" label="xmin">
<value val="4.5"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Max">
<input a0="x(C)" a1="x(D)"/>
<output a0="xmax"/>
</command>
<element type="numeric" label="xmax">
<value val="6.5"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Integral">
<input a0="f" a1="xmin" a2="xmax"/>
<output a0="prob2"/>
</command>
<element type="numeric" label="prob2">
<value val="0.12262009369716786"/>
<show object="true" label="false"/>
<condition showObject="showP2"/>
<objColor r="255" g="0" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<selectionAllowed val="false"/>
</element>
<command name="Sequence">
<input a0="Polygon[(k - 0.5, 0), (k + 0.5, 0), (k + 0.5, Element[f_X, k + 1]), (k - 0.5, Element[f_X, k + 1])]" a1="k" a2="xmin + 0.5" a3="xmax - 0.5"/>
<output a0="barsSelected"/>
</command>
<element type="list" label="barsSelected">
<show object="true" label="false"/>
<condition showObject="showP1"/>
<objColor r="0" g="51" b="204" alpha="0.5" fillType="1" hatchAngle="45" hatchDistance="5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sum">
<input a0="barsSelected"/>
<output a0="prob1"/>
</command>
<element type="numeric" label="prob1">
<value val="0.12336664322778089"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
</element>
<expression label="txtP1" exp=""P(" + (LaTeX[xmin + 0.5]) + "≤X≤" + (LaTeX[xmax - 0.5]) + ")=" + (LaTeX[prob1]) + """ />
<element type="text" label="txtP1">
<show object="true" label="false"/>
<condition showObject="showP1"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<decimals val="4"/>
<startPoint x="9.081421434587556" y="-0.1784686224215169" z="1.0"/>
</element>
<element type="boolean" label="showP2">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="277" y="438"/>
<labelMode val="0"/>
<caption val="常態分布機率"/>
</element>
<expression label="ab" exp="(barsBino * showSD)" />
<element type="numeric" label="ab">
<value val="0.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<auxiliary val="true"/>
</element>
<expression label="dx" exp="0.5 / n" />
<element type="numeric" label="dx">
<value val="0.03125"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="Xnorm" exp="X / n" />
<element type="list" label="Xnorm">
<objColor r="0" g="100" b="0" alpha="0.1"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sequence">
<input a0="Polygon[(Element[Xnorm, k] - dx, 0), (Element[Xnorm, k] + dx, 0), (Element[Xnorm, k] + dx, (n * Element[f_X, k])), (Element[Xnorm, k] - dx, (n * Element[f_X, k]))]" a1="k" a2="1" a3="n + 1"/>
<output a0="barsNorm"/>
</command>
<element type="list" label="barsNorm">
<show object="true" label="false" ev="3"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<command name="Sum">
<input a0="barsNorm"/>
<output a0="probOne"/>
</command>
<element type="numeric" label="probOne">
<value val="1.0000000000000002"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="txtSD" exp=""σ=\sqrt{npq}=" + (LaTeX[σ]) + """ />
<element type="text" label="txtSD">
<show object="true" label="false"/>
<condition showObject="showSD"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint exp="B"/>
<labelOffset x="-5" y="-29"/>
</element>
<element type="boolean" label="showP1">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="167" y="438"/>
<labelMode val="0"/>
<caption val="計算機率"/>
</element>
<expression label="txtMu" exp=""μ=np=" + (LaTeX[μ]) + """ />
<element type="text" label="txtMu">
<show object="true" label="false"/>
<condition showObject="showSD"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint exp="Mu2"/>
<labelOffset x="-9" y="2"/>
</element>
<expression label="Mu2" exp="(x(Mu), (-0.06))" type="point" />
<element type="point" label="Mu2">
<show object="false" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="8.64" y="-0.06" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="6"/>
</element>
<expression label="txtSND" exp=""f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}""/>
<element type="text" label="txtSND">
<show object="false" label="false" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="12" style="0"/>
<startPoint x="1.224000000000007" y="5.767922583397844" z="1.0"/>
</element>
<command name="Textfield">
<input a0="p"/>
<output a0="inputP"/>
</command>
<element type="textfield" label="inputP">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="1.0"/>
<layer val="0"/>
<labelOffset x="160" y="414"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<linkedGeo exp="p"/>
<length val="3"/>
</element>
<command name="Textfield">
<input a0="n"/>
<output a0="inputN"/>
</command>
<element type="textfield" label="inputN">
<show object="true" label="false"/>
<objColor r="0" g="153" b="0" alpha="1.0"/>
<layer val="0"/>
<labelOffset x="398" y="413"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<linkedGeo exp="n"/>
<length val="3"/>
</element>
<expression label="txtP2" exp=""常態機率: " + (LaTeX[prob2]) + """ />
<element type="text" label="txtP2">
<show object="true" label="false"/>
<condition showObject="showP2"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<decimals val="4"/>
<startPoint x="11.119912329691807" y="-0.32524740671810914" z="1.0"/>
</element>
<expression label="txtError" exp=""誤差: " + (LaTeX[abs(prob1 - prob2)]) + """ />
<element type="text" label="txtError">
<show object="true" label="false"/>
<condition showObject="showP2"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="true" sizeM="1.4" size="5" style="0"/>
<decimals val="4"/>
<startPoint x="12.072720524375661" y="-0.41865926211690246" z="1.0"/>
</element>
<expression label="txtFuncNormal" exp=""f(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}{\left(\frac{x-\mu}{\sigma}\right)}^2}""/>
<element type="text" label="txtFuncNormal">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="12" style="0"/>
<startPoint x="5.088695145313391" y="0.5921181820810604" z="1.0"/>
</element>
<command name="Normal">
<input a0="p" a1="σ / n" a2="x"/>
<output a0="g"/>
</command>
<element type="function" label="g">
<show object="true" label="false" ev="2"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
</element>
<expression label="text2" exp=""\frac{σ}{n}=\frac{\sqrt{npq}}{n}=\sqrt{\frac{pq}{n}}=" + (LaTeX[σ / n]) + """ />
<element type="text" label="text2">
<show object="true" label="false" ev="3"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2.0" size="12" style="0"/>
<decimals val="4"/>
<startPoint x="-0.058621114915892225" y="-2.406555997888542" z="1.0"/>
</element>
<expression label="Mu3" exp="(p, (-0.6))" type="point" />
<element type="point" label="Mu3">
<show object="true" label="false" ev="2"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.54" y="-0.6" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="6"/>
</element>
<expression label="txtP" exp=""p=" + (LaTeX[p]) + """ />
<element type="text" label="txtP">
<show object="true" label="false" ev="2"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint exp="Mu3"/>
<labelOffset x="-7" y="5"/>
</element>
<expression label="Mu4" exp="(p, g(p))" type="point" />
<element type="point" label="Mu4">
<show object="false" label="true" ev="3"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.54" y="3.201800450540575" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="Sigma2" exp="(p + σ / n, y(Mu4))" type="point" />
<element type="point" label="Sigma2">
<show object="false" label="true" ev="3"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.6645993579437711" y="3.201800450540575" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="Mid2" exp="(Mu4 + Sigma2) / 2" type="point" />
<element type="point" label="Mid2">
<show object="false" label="true" ev="3"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.6022996789718855" y="3.201800450540575" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="Mid2" a1="Mu4"/>
<output a0="w"/>
</command>
<element type="vector" label="w">
<show object="true" label="false" ev="3"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.062299678971885486" y="0.0" z="0.0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="Mid2"/>
</element>
<command name="Vector">
<input a0="Mid2" a1="Sigma2"/>
<output a0="b"/>
</command>
<element type="vector" label="b">
<show object="true" label="false" ev="3"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.0622996789718856" y="0.0" z="0.0"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
<coordStyle style="cartesian"/>
<startPoint exp="Mid2"/>
</element>
<command name="Segment">
<input a0="(p, 0)" a1="Mu4"/>
<output a0="segMu2"/>
</command>
<element type="segment" label="segMu2">
<show object="true" label="false" ev="2"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-3.201800450540575" y="0.0" z="1.7289722432919106"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="Sigma2" a1="(x(Sigma2), 0)"/>
<output a0="segSigma2"/>
</command>
<element type="segment" label="segSigma2">
<show object="true" label="false" ev="3"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="3.201800450540575" y="0.0" z="-2.1279145236933434"/>
<lineStyle thickness="2" type="10" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text4" exp=""\frac{σ}{n}=" + (LaTeX[σ / n]) + """ />
<element type="text" label="text4">
<show object="true" label="false" ev="3"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<decimals val="4"/>
<startPoint exp="Mid2"/>
<labelOffset x="-10" y="-42"/>
</element>
<element type="button" label="btnResample">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<bgColor r="255" g="192" b="203" alpha="255"/>
<layer val="0"/>
<labelOffset x="515" y="397"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<javascript val="resample();"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<file name="/geogebra/main/nav_play.png"/>
<caption val="抽樣"/>
</element>
<element type="numeric" label="pHat">
<value val="0.375"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<expression label="sigmaHat" exp="sqrt((pHat * (1 - pHat) / n))" />
<element type="numeric" label="sigmaHat">
<value val="0.12103072956898178"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="PHatPoint" exp="(pHat, 0)" type="point" />
<element type="point" label="PHatPoint">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.375" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="zIndex" exp="1 + y(ConfidencePoint - E) / dhConfidence" />
<element type="numeric" label="zIndex">
<value val="1.9999999999999998"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="CIRight" exp="PHatPoint + ((zIndex * sigmaHat), 0)" type="point" />
<element type="point" label="CIRight">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.6170614591379635" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="CILeft" exp="PHatPoint - ((zIndex * sigmaHat), 0)" type="point" />
<element type="point" label="CILeft">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.13293854086203646" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="numeric" label="dh">
<value val="0.2"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="ratioDW">
<value val="0.1"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<command name="PolyLine">
<input a0="CIRight + {(((-ratioDW) * dh), dh), (0, dh), (0, (-dh)), (((-ratioDW) * dh), (-dh))}"/>
<output a0="bracketRight"/>
</command>
<element type="polyline" label="bracketRight">
<lineStyle thickness="3" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="PolyLine">
<input a0="CILeft + {((ratioDW * dh), dh), (0, dh), (0, (-dh)), ((ratioDW * dh), (-dh))}"/>
<output a0="bracketLeft"/>
</command>
<element type="polyline" label="bracketLeft">
<lineStyle thickness="3" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="Polygon">
<input a0="{CILeft + (0, dh), CILeft + (0, (-dh)), CIRight + (0, (-dh)), CIRight + (0, dh)}"/>
<output a0="CI" a1="a" a2="c" a3="d" a4="e"/>
</command>
<element type="polygon" label="CI">
<lineStyle thickness="0" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.5" fillType="1" hatchAngle="45" hatchDistance="5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="segment" label="a">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.4" y="0.0" z="-0.05317541634481459"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="0.484122918275927" z="0.09682458365518541"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-0.4" y="0.0" z="0.24682458365518542"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.0" y="-0.484122918275927" z="0.09682458365518541"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="PHatMarker" exp="PHatPoint + (0, (1.5 * dh))" type="point" />
<element type="point" label="PHatMarker">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.375" y="0.30000000000000004" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="7"/>
</element>
<expression label ="txtConfidence" exp="{"68%", "95%", "99.7%"}"/>
<element type="list" label="txtConfidence">
<show object="false" label="false"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<pointSize val="3"/>
<pointStyle val="-1"/>
</element>
<expression label="txtConf68" exp=""68% ( 1 個標準差 )""/>
<element type="text" label="txtConf68">
<show object="true" label="false"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<startPoint x="-0.34467776085666596" y="3.8925975322226813" z="1.0"/>
</element>
<expression label="txtConf95" exp=""95% ( 2 個標準差 )""/>
<element type="text" label="txtConf95">
<show object="true" label="false"/>
<objColor r="0" g="0" b="204" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<startPoint x="-0.3455479475812467" y="4.686242934261053" z="1.0"/>
</element>
<expression label="txtConf99" exp=""99.7% ( 3 個標準差 )""/>
<element type="text" label="txtConf99">
<show object="true" label="false"/>
<objColor r="102" g="0" b="153" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<startPoint x="-0.34256943019080227" y="5.502501689368673" z="1.0"/>
</element>
<expression label="txtConfTitle" exp=""信心水準""/>
<element type="text" label="txtConfTitle">
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="2.0" size="12" style="1"/>
<startPoint x="-0.4630680606189508" y="6.358274472731557" z="1.0"/>
</element>
<element type="numeric" label="guessTotal">
<value val="200.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="5"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="guessRight">
<value val="187.0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="5"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<expression label="guessRatio" exp="guessRight / guessTotal" />
<element type="numeric" label="guessRatio">
<value val="0.935"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="text1" exp=""\frac{" + (LaTeX[guessRight]) + "}{" + (LaTeX[guessTotal]) + "}=" + (LaTeX[guessRatio]) + """ />
<element type="text" label="text1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="204" alpha="0.0"/>
<bgColor r="255" g="255" b="102" alpha="255"/>
<layer val="5"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-0.3910644478286668" y="1.9355585193336091" z="1.0"/>
</element>
<element type="boolean" label="guess">
<value val="true"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
</element>
<element type="image" label="picHappy">
<file name="a72dad5cc6782068803b087dc13f5b7b/emoticonHappy.png"/>
<inBackground val="false"/>
<absoluteScreenLocation x="576" y="145"/> <show object="true" label="false"/>
<condition showObject="guess"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="5"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="image" label="picSad">
<file name="7e2793dec711d10734919e385fe40402/emoticonSad.png"/>
<inBackground val="false"/>
<absoluteScreenLocation x="576" y="145"/> <show object="true" label="false"/>
<condition showObject="¬guess"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<layer val="5"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="button" label="btnReset">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="102" g="204" b="0" alpha="255"/>
<layer val="5"/>
<labelOffset x="763" y="398"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<ggbscript val="guessRight=0 guessTotal=0"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<file name="/geogebra/main/nav_pause.png"/>
<caption val="重設"/>
</element>
<expression label="text3" exp=""實際猜中比例""/>
<element type="text" label="text3">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<font serif="false" sizeM="2.0" size="12" style="0"/>
<startPoint x="-0.4557530910276895" y="2.344869136715513" z="1.0"/>
</element>
<element type="button" label="btn100times">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="51" b="102" alpha="255"/>
<layer val="5"/>
<labelOffset x="655" y="397"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<javascript val="resampleByTimes(100);"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<file name="/geogebra/gui/images/nav_fastforward.png"/>
<caption val="抽 100 次"/>
</element>
<expression label="txtPopulationP" exp=""母體比例""/>
<element type="text" label="txtPopulationP">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint x="-0.4278311336929239" y="-2.9213645061995726" z="1.0"/>
</element>
<expression label="txtSampleSize" exp=""樣本大小""/>
<element type="text" label="txtSampleSize">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint x="0.28452229376388255" y="-2.9652735290307177" z="1.0"/>
</element>
<expression label="txtPHat" exp=""\widehat{p}=" + (LaTeX[pHat]) + """ />
<element type="text" label="txtPHat">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<decimals val="4"/>
<startPoint exp="PHatMarker"/>
<labelOffset x="-7" y="-27"/>
</element>
<expression label="text5" exp=""信賴區間""/>
<element type="text" label="text5">
<show object="true" label="false"/>
<objColor r="220" g="20" b="60" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<font serif="false" sizeM="2.0" size="12" style="1"/>
<startPoint x="1.1095128016305256" y="2.347718233537859" z="1.0"/>
</element>
<expression label="txtCI1" exp=""[" + (pHat - (zIndex * sigmaHat)) + ", " + (pHat + (zIndex * sigmaHat)) + "]"" />
<element type="text" label="txtCI1">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<decimals val="4"/>
<startPoint x="1.210116588034873" y="1.5935261304380295" z="1.0"/>
</element>
<expression label="txtCI2" exp=""= " + pHat + " ± " + ((zIndex * sigmaHat)) + """ />
<element type="text" label="txtCI2">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="5"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<decimals val="4"/>
<startPoint x="1.2107966854342023" y="0.8144563835590781" z="1.0"/>
</element>
<element type="button" label="btn50times">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="1.0"/>
<bgColor r="255" g="153" b="204" alpha="255"/>
<layer val="5"/>
<labelOffset x="571" y="397"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<javascript val="resampleByTimes(50);"/>
<font serif="false" sizeM="1.4" size="5" style="0"/>
<file name="/geogebra/gui/images/nav_fastforward.png"/>
<caption val="抽 50 次"/>
</element>
</construction>
</geogebra>