<description/> <author>LMA</author> <company/> <keywords/> <revisedBy> Marc Artzrouni </revisedBy> </userData> <identityInfo> <revision>5</revision> <documentID>512F3CC3-9CAD-4DEA-833D-FB3F5DBE8C3B</documentID> <versionID>AE60F769-C925-417A-8C64-0F7083D048AD</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="18" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Head24" base-style="Normal"> <blockAttr margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"/> 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<colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="1" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="none" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="4691" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="4691" left="66" top="8.25" width="387" height="41.25" align-x="66" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Normal Distribution and Applications <f size="12"> ( </f> <f size="12"> <inlineAttr underline="false">sample MathCad lecture by Marc.Artzrouni@univ-pau.fr; February 2011)</inlineAttr> </f> </text> </region> <region region-id="3077" left="24" top="87.75" width="485.25" height="24.75" align-x="63" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="22"> <inlineAttr line-through="false">1.Normal distribution:</inlineAttr> </f> </text> </region> <region region-id="3114" left="30" top="128.25" width="477" height="20.25" align-x="41.25" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> The random variable (r.v.) </text> </region> <region region-id="4677" left="12" top="254.25" width="531" height="42.75" align-x="28.5" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">is called a normal </inlineAttr> </f> Example below:<sp count="2"/> <region region-id="3090" left="155.25" top="275.25" width="43.5" height="21.75" align-x="177" align-y="291" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">μ</ml:id> <ml:real>2</ml:real> </ml:globalDefine> </math> <rendering item-idref="2"/> </region> , <region region-id="3091" left="215.25" top="275.25" width="58.5" height="21.75" align-x="237" align-y="291" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">σ</ml:id> <ml:real>0.5</ml:real> </ml:globalDefine> </math> <rendering item-idref="3"/> </region> <sp/> </text> </region> <region region-id="4224" left="24" top="320.25" width="58.5" height="20.25" align-x="45" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> simul : </text> </region> <region region-id="4223" left="108" top="320.25" width="263.25" height="21.75" align-x="142.5" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">si1</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">rnorm</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">μ</ml:id> <ml:id xml:space="preserve">σ</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="1" cols="1"> <ml:real>1.7805145914255278</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="4614" left="0" top="360" width="6008.25" align-x="0" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="38.25"> <area is-collapsed="true" name="" show-name="false" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="2548" bottom-lock-id="2550"> <region region-id="2545" left="72" top="15" width="73.5" height="12.75" align-x="81.75" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">x</ml:id> <ml:range> <ml:sequence> <ml:real>-15</ml:real> <ml:real>-14.9</ml:real> </ml:sequence> <ml:real>15</ml:real> </ml:range> </ml:define> </math> </region> </area> <rendering item-idref="5"/> </region> <region region-id="2542" left="186" top="344.25" width="15.75" height="20.25" align-x="198.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> .<sp count="2"/> </text> </region> <region region-id="4673" left="30" top="372" width="419.25" height="225.75" align-x="30" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="6"/> <rendering item-idref="7"/> </region> <region region-id="4680" left="24" top="602.25" width="428.25" height="40.5" align-x="36" align-y="618" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Calculations done in exercise sessions: <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="3"/>=</inlineAttr> </f> </text> </region> <region region-id="3703" left="12" top="830.25" width="503.25" height="20.25" align-x="18.75" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> which shows that the parameter </text> </region> <region region-id="4683" left="24" top="872.25" width="447" height="20.25" align-x="34.5" align-y="888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Similarly the standard deviation </text> </region> <region region-id="4626" left="18" top="1034.25" width="523.5" height="186" align-x="24.75" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> and therefore<sp count="2"/>. <f family="Arial" charset="0"> <inlineAttr line-through="false">Remarks: </inlineAttr> </f> <f family="Arial" charset="0">1. The<sp count="2"/>density dnorm(x,</f> <f family="Symbol" charset="2" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">m</inlineAttr> </f> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,</inlineAttr> </f> <f family="Symbol" charset="2" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">s</inlineAttr> </f> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) </inlineAttr> </f> <f family="Arial" charset="0"> is centered </f> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">2. The smaller </inlineAttr> </f> <sp/> <f family="Symbol" charset="2">s</f> is, <sp/> 3.<sp count="2"/>dnorm(x,<f family="Symbol" charset="2">m</f>,<f family="Symbol" charset="2">s</f>) does not <f family="Arial" charset="0" size="20"> <sp/> </f> </text> </region> <region region-id="4631" left="6" top="1593.75" width="521.25" height="135" align-x="45" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="22"> <inlineAttr line-through="false">2. Normal distribution as a model for heigths of male and female students collected at first class </inlineAttr> </f> <f family="Arial" charset="0" size="18"> <inlineAttr line-through="false"> <c val="#f00">(make sure path is specificed correctly to<sp count="2"/>EXCEL spreadsheet read below in collapsed area;<sp count="2"/> </c> </inlineAttr> </f> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" line-through="false"> <c val="#f00">male and female heights (random variables) are </c> </inlineAttr> </f> <c val="#f00"> <f size="18">TF, TM)</f> </c> <f family="Arial" charset="0" size="20"> <inlineAttr font-weight="normal" underline="false" line-through="false">i. Histogram</inlineAttr> </f> <c val="#000"> <sp/> </c> </text> </region> <region region-id="4630" left="0" top="1740" width="6008.25" align-x="0" align-y="1740" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="614.25"> <area is-collapsed="true" name="EXCEL spreadsheet (check path, etc)" show-name="true" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="4103" bottom-lock-id="3161"> <region region-id="4609" left="66" top="38.25" width="126.75" height="55.5" align-x="92.25" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="01350083-1122-11DB-9380-000D56C6051A" item-idref="8" disable-calc="false"> <inputs/> <outputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">A</ml:id> </outputs> </component> </region> <region region-id="3446" left="204" top="38.25" width="71.25" height="20.25" align-x="222.75" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> work file </text> </region> <region region-id="3443" left="318" top="69" width="119.25" height="12.75" align-x="331.5" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:id xml:space="preserve">submatrix</ml:id> <ml:sequence> <ml:id xml:space="preserve">A</ml:id> <ml:real>16</ml:real> <ml:real>26</ml:real> <ml:real>5</ml:real> <ml:real>5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> </region> <region region-id="3444" left="582" top="62.25" width="165" height="20.25" align-x="585.75" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> x values: 147.5 etc </text> </region> <region region-id="3222" left="804" top="98.25" width="123.75" height="20.25" align-x="846.75" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Longueurs ici : </text> </region> <region region-id="3227" left="6" top="117" width="122.25" height="12.75" align-x="22.5" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Hf</ml:id> <ml:apply> <ml:id xml:space="preserve">submatrix</ml:id> <ml:sequence> <ml:id xml:space="preserve">A</ml:id> <ml:real>16</ml:real> <ml:real>26</ml:real> <ml:real>7</ml:real> <ml:real>7</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> </region> <region region-id="3228" left="276" top="117" width="125.25" height="12.75" align-x="295.5" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Hm</ml:id> <ml:apply> <ml:id xml:space="preserve">submatrix</ml:id> <ml:sequence> <ml:id xml:space="preserve">A</ml:id> <ml:real>16</ml:real> <ml:real>26</ml:real> <ml:real>6</ml:real> <ml:real>6</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> </region> <region region-id="4194" left="1092" top="135" width="30.75" height="12.75" align-x="1106.25" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">absices</ml:id> </math> </region> <region region-id="3223" left="798" top="141" width="130.5" height="12.75" align-x="813.75" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Xl</ml:id> <ml:apply> <ml:id xml:space="preserve">submatrix</ml:id> <ml:sequence> <ml:id xml:space="preserve">A</ml:id> <ml:real>16</ml:real> <ml:real>28</ml:real> <ml:real>11</ml:real> <ml:real>11</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> </region> <region region-id="3130" left="36" top="140.25" width="195" height="25.5" align-x="70.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math 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<ml:real>167.5</ml:real> <ml:real>172.5</ml:real> <ml:real>177.5</ml:real> <ml:real>182.5</ml:real> <ml:real>187.5</ml:real> <ml:real>192.5</ml:real> <ml:real>197.5</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="11"/> </resultFormat> </math> </region> <region region-id="4280" left="516" top="93" width="51.75" height="12.75" align-x="525.75" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">x</ml:id> <ml:range> <ml:real>140</ml:real> <ml:real>200</ml:real> </ml:range> </ml:define> </math> </region> <region region-id="4249" left="1530" top="81.75" width="501.75" height="205.5" align-x="1569" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="22"> <inlineAttr line-through="false">2. La loi normale</inlineAttr> </f> <f family="Arial" charset="0" size="22"> <inlineAttr font-weight="normal" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial" charset="0" size="22"> <inlineAttr font-weight="normal" font-style="normal" line-through="false">comme modèle de la taille des étudiants</inlineAttr> </f> <f family="Arial" charset="0" size="20"> <inlineAttr font-weight="normal" underline="false" line-through="false">i.</inlineAttr> </f> <f family="Arial" charset="0" size="20"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial" charset="0" size="20"> <inlineAttr font-weight="normal" underline="false" line-through="false">Approximation normale</inlineAttr> </f> Chaque taille de femme et d'homme:<sp count="3"/>v.a.<sp count="2"/>TF ou TM. Ci-dessous les fréquences<sp count="2"/>observées pour les deux sexes (par groupe de taille quinquennal) et les densités normales<sp count="2"/>qui les approximent: les<sp count="3"/> <f family="Arial" charset="0">m</f> fe ,<f family="Arial" charset="0">s</f> fe et<sp count="2"/> <f family="Arial" charset="0">m</f> me ,<f family="Arial" charset="0">s</f> me<sp count="2"/> utilisés sont <c val="#000">des estimateurs</c> <c val="#000"> <sp/> </c> <c val="#000"> des espérances et écarts-types théoriques mais inconnues </c> <c val="#000"> <f family="Arial" charset="0">m</f> </c> <c val="#000"> f </c> <c val="#000">, </c> <f family="Arial" charset="0"> <c val="#000">s</c> </f> <c val="#000"> f </c> <c val="#000"> de TF, et<sp count="2"/> </c> <f family="Arial" charset="0"> <c val="#000">m</c> </f> <c val="#000">m </c> <c val="#000">, </c> <c val="#000"> <f family="Arial" charset="0">s</f> </c> <c val="#000"> m </c> <c val="#000"> <sp/> </c> <c val="#000">de TM.<sp count="2"/>(voir ci-dessous). </c> </text> </region> </area> <rendering item-idref="12"/> </region> <region region-id="4643" left="24" top="1778.25" width="219" height="20.25" align-x="50.25" align-y="1794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> counts and heights, in cm: </text> </region> <region region-id="4265" left="12" top="1817.25" width="721.5" height="51.75" align-x="41.25" align-y="1848" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Df</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="11"> <ml:real>1</ml:real> <ml:real>147.5</ml:real> <ml:real>1</ml:real> <ml:real>152.5</ml:real> <ml:real>2</ml:real> <ml:real>157.5</ml:real> <ml:real>10</ml:real> <ml:real>162.5</ml:real> <ml:real>6</ml:real> <ml:real>167.5</ml:real> <ml:real>2</ml:real> <ml:real>172.5</ml:real> <ml:real>1</ml:real> <ml:real>177.5</ml:real> <ml:real>0</ml:real> <ml:real>182.5</ml:real> <ml:real>0</ml:real> <ml:real>187.5</ml:real> <ml:real>0</ml:real> <ml:real>192.5</ml:real> <ml:real>0</ml:real> <ml:real>197.5</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="matrix" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="13"/> </region> <region region-id="4273" left="12" top="1880.25" width="69" height="25.5" align-x="37.5" align-y="1896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" font="1" subscript="f">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>23</ml:real> </result> </ml:eval> </math> <rendering item-idref="14"/> </region> <region region-id="4645" left="90" top="1880.25" width="177" height="20.25" align-x="120.75" align-y="1896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> = number of females </text> </region> <region region-id="4275" left="24" top="1904.25" width="493.5" height="20.25" align-x="36" align-y="1920" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> In histogram </text> </region> <region region-id="3403" left="12" top="2316" width="532.5" height="225.75" align-x="12" align-y="2316" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="15"/> <rendering item-idref="16"/> </region> <region region-id="4670" left="570" top="2341.5" width="182.25" height="147" align-x="626.25" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#f00">superimposed </c> <c val="#f00">normal density</c> <c val="#f00">explained below -</c> <c val="#f00"> oral comments </c> <c val="#f00"> about roughness </c> <c val="#f00">of fit, small samples, </c> <c val="#f00">etc.<sp count="2"/> </c> </text> </region> <region region-id="4285" left="6" top="2562" width="546" height="225.75" align-x="6" align-y="2562" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="17"/> <rendering item-idref="18"/> </region> <region region-id="4658" left="12" top="2803.5" width="531" height="250.5" align-x="19.5" align-y="2820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="true" lock-width="true"> ii. Normal approximation<sp count="2"/> Male parameters<inlineAttr font-style="normal"> <sp/> </inlineAttr> <sp count="3"/> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#000"> <sp/> </c> </inlineAttr> </f> <c val="#000"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> <sp/> </f> </inlineAttr> </c> <region region-id="3406" left="18" top="3008.25" width="120.75" height="25.5" align-x="51" align-y="3024" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="fe">μ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>164.78260869565219</ml:real> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> <c val="#000">;</c> </f> </inlineAttr> <c val="#000"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> similarly : </f> </inlineAttr> </c> <region region-id="3478" left="174" top="3008.25" width="108" height="25.5" align-x="213.75" align-y="3024" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="me">μ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>177.3</ml:real> </result> </ml:eval> </math> <rendering item-idref="20"/> </region> These estimators are unbiased: </text> </region> <region region-id="3438" left="30" top="3206.25" width="523.5" height="20.25" align-x="35.25" align-y="3222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000">b. To estimate<sp count="2"/> </c> </text> </region> <region region-id="4686" left="12" top="3524.25" width="525.75" height="156" align-x="30.75" align-y="3540" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="true" lock-width="true"> For calculations we need: i. <region region-id="3585" left="24" top="3542.25" width="69" height="25.5" align-x="49.5" align-y="3558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>23</ml:real> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="6"/> </inlineAttr> </f> <region region-id="4320" left="120" top="3542.25" width="75" height="25.5" align-x="151.5" align-y="3558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="m">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>20</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> ii.<sp count="3"/> <region region-id="4321" left="48" top="3572.25" width="129" height="21.75" align-x="114" align-y="3588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">somTF</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>3790</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="8"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="23"/> </region> <sp/> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <region region-id="3584" left="156" top="3572.25" width="132" height="21.75" align-x="225" align-y="3588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">somTM</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>3546</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="8"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="24"/> </region> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">iii. Sums of squares =</inlineAttr> </f> <region region-id="4501" left="186" top="3596.25" width="158.25" height="21.75" align-x="261.75" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">som2TF</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>625478</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <decimal precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="25"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> <sp/> </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18">; </f> </inlineAttr> <region region-id="4328" left="312" top="3596.25" width="161.25" height="21.75" align-x="390.75" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">som2TM</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>629836</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <decimal precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="26"/> </region> Exercise: ladies in lecture calculate<sp count="2"/> <region region-id="4330" left="312" top="3614.25" width="23.25" height="25.5" align-x="322.5" align-y="3630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" subscript="fe" xmlns:ml="http://schemas.mathsoft.com/math30">μ</ml:id> </math> <rendering item-idref="27"/> </region> <sp/> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18">, </f> </inlineAttr> <region region-id="4329" left="348" top="3614.25" width="23.25" height="25.5" align-x="358.5" align-y="3630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" subscript="fe" xmlns:ml="http://schemas.mathsoft.com/math30">σ</ml:id> </math> <rendering item-idref="28"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> ; </f> </inlineAttr> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">gentlemen<sp count="2"/> </inlineAttr> </f> <region region-id="4340" left="108" top="3650.25" width="30" height="25.5" align-x="122.25" align-y="3666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" subscript="me" xmlns:ml="http://schemas.mathsoft.com/math30">μ</ml:id> </math> <rendering item-idref="29"/> </region> <sp/> , <region region-id="4341" left="156" top="3650.25" width="30" height="25.5" align-x="170.25" align-y="3666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" subscript="me" xmlns:ml="http://schemas.mathsoft.com/math30">σ</ml:id> </math> <rendering item-idref="30"/> </region> <sp/> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> <sp count="7"/> </f> </inlineAttr> <inlineAttr underline="false" line-through="false"> <f family="Arial" charset="0" size="14"> <c val="#f00">(done here and now in class)</c> </f> </inlineAttr> </text> </region> <region region-id="4402" left="24" top="4106.25" width="502.5" height="321.75" align-x="24" align-y="4122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="true" lock-width="true"> <f family="Arial" charset="0"> <inlineAttr font-style="normal" line-through="false"> <c val="#800000">Exercise 1:</c> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-style="normal" underline="false" line-through="false"> <c val="#800000"> <sp/> </c> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr underline="false" line-through="false"> <c val="#800000"> We have three sampled values</c> </inlineAttr> </f> <c val="#800000"> X</c> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#800000"> 1 </c> </inlineAttr> </f> <c val="#800000">=0, </c> <c val="#800000"> X</c> <c val="#800000"> 2 </c> <c val="#800000">=0, </c> <c val="#800000"> X</c> <c val="#800000"> 3 </c> <c val="#800000">=1,of a binary rv<sp count="2"/>(e.g. flipped coin). Use<sp count="2"/>eq. (1) and<sp count="2"/>(4) to </c> <c val="#800000">calculate estimated values<sp count="3"/> </c> <f family="Symbol" charset="2"> <c val="#800000">m</c> </f> <c val="#800000">e</c> <c val="#800000"> of </c> <f family="Symbol" charset="2"> <c val="#800000">m</c> </f> <c val="#800000"> <sp count="2"/>and </c> <f family="Symbol" charset="2"> <c val="#800000">s</c> </f> <c val="#800000"> e </c> <c val="#800000"> of </c> <f family="Symbol" charset="2"> <c val="#800000">s. </c> </f> <f family="Symbol" charset="2"> <c val="#800000">m</c> </f> <c val="#800000">e</c> <c val="#800000"> =<sp count="10"/> </c> <c val="#800000"> <sp count="2"/> </c> </text> </region> <region region-id="4550" left="798" top="5660.25" width="132" height="21.75" align-x="852" align-y="5676" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">somL</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>2913.5</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="7"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="31"/> </region> <region region-id="4579" left="672" top="5756.25" width="163.5" height="51" align-x="757.5" align-y="5784" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:div/> <ml:id xml:space="preserve">somL</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="f">n</ml:id> <ml:id xml:space="preserve" subscript="m">n</ml:id> </ml:apply> </ml:parens> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>67.755813953488371</ml:real> </result> </ml:eval> </math> <rendering item-idref="32"/> </region> <region region-id="4574" left="870" top="5816.25" width="111" height="25.5" align-x="937.5" align-y="5832" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:plus/> <ml:id xml:space="preserve" subscript="f">n</ml:id> <ml:id xml:space="preserve" subscript="m">n</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>43</ml:real> </result> </ml:eval> </math> <rendering item-idref="33"/> 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display="false"/> <symbolics derivation-steps-style="vertical-insert" show-comments="false" evaluate-in-place="false"/> <results numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="9" orientation="portrait" print-single-page-width="false" page-width="594.75" page-height="842.25"> <margins left="19.78583" right="19.78583" top="19.78583" bottom="34.01575"/> <header use-full-page-width="false"/> <footer use-full-page-width="false"> <center>{\rtf1\ansi\ansicpg1252\deff0\deflang1036{\fonttbl{\f0\fmodern\fprq9\fcharset0 Arial;}} \viewkind4\uc1\pard\qc\f0\fs18\{n\} \par }</center> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="1" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="none" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="4691" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="4691" left="66" top="8.25" width="387" height="41.25" align-x="66" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Normal Distribution and Applications <f size="12"> ( </f> <f size="12"> <inlineAttr underline="false">sample MathCad lecture by Marc.Artzrouni@univ-pau.fr; February 2011)</inlineAttr> </f> </text> </region> <region region-id="3077" left="24" top="87.75" width="485.25" height="24.75" align-x="63" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="22"> <inlineAttr line-through="false">1.Normal distribution:</inlineAttr> </f> </text> </region> <region region-id="3114" left="30" top="128.25" width="477" height="20.25" align-x="41.25" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> The random variable (r.v.) </text> </region> <region region-id="4677" left="12" top="254.25" width="531" height="42.75" align-x="28.5" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">is called a normal </inlineAttr> </f> Example below:<sp count="2"/> <region region-id="3090" left="155.25" top="275.25" width="43.5" height="21.75" align-x="177" align-y="291" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">μ</ml:id> <ml:real>2</ml:real> </ml:globalDefine> </math> <rendering item-idref="2"/> </region> , <region region-id="3091" left="215.25" top="275.25" width="58.5" height="21.75" align-x="237" align-y="291" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">σ</ml:id> <ml:real>0.5</ml:real> </ml:globalDefine> </math> <rendering item-idref="3"/> </region> <sp/> </text> </region> <region region-id="4224" left="24" top="320.25" width="58.5" height="20.25" align-x="45" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> simul : </text> </region> <region region-id="4223" left="108" top="320.25" width="263.25" height="21.75" align-x="142.5" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">si1</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">rnorm</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">μ</ml:id> <ml:id xml:space="preserve">σ</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="1" cols="1"> <ml:real>1.7805145914255278</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="4614" left="0" top="360" width="6008.25" align-x="0" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="38.25"> <area is-collapsed="true" name="" show-name="false" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="2548" bottom-lock-id="2550"> <region region-id="2545" left="72" top="15" width="73.5" height="12.75" align-x="81.75" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">x</ml:id> <ml:range> <ml:sequence> <ml:real>-15</ml:real> <ml:real>-14.9</ml:real> </ml:sequence> <ml:real>15</ml:real> </ml:range> </ml:define> </math> </region> </area> <rendering item-idref="5"/> </region> <region region-id="2542" left="186" top="344.25" width="15.75" height="20.25" align-x="198.75" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> .<sp count="2"/> </text> </region> <region region-id="4673" left="30" top="372" width="419.25" height="225.75" align-x="30" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="6"/> <rendering item-idref="7"/> </region> <region region-id="4680" left="24" top="602.25" width="428.25" height="40.5" align-x="36" align-y="618" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Calculations done in exercise sessions: <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="3"/>=</inlineAttr> </f> </text> </region> <region region-id="3703" left="12" top="830.25" width="503.25" height="20.25" align-x="18.75" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> which shows that the parameter </text> </region> <region region-id="4683" left="24" top="872.25" width="447" height="20.25" align-x="34.5" align-y="888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Similarly the standard deviation </text> </region> <region region-id="4626" left="18" top="1034.25" width="523.5" height="186" align-x="24.75" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> and therefore<sp count="2"/>. <f family="Arial" charset="0"> <inlineAttr line-through="false">Remarks: </inlineAttr> </f> <f family="Arial" charset="0">1. The<sp count="2"/>density dnorm(x,</f> <f family="Symbol" charset="2" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">m</inlineAttr> </f> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,</inlineAttr> </f> <f family="Symbol" charset="2" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">s</inlineAttr> </f> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) </inlineAttr> </f> <f family="Arial" charset="0"> is centered </f> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">2. The smaller </inlineAttr> </f> <sp/> <f family="Symbol" charset="2">s</f> is, <sp/> 3.<sp count="2"/>dnorm(x,<f family="Symbol" charset="2">m</f>,<f family="Symbol" charset="2">s</f>) does not <f family="Arial" charset="0" size="20"> <sp/> </f> </text> </region> <region region-id="4631" left="6" top="1593.75" width="521.25" height="135" align-x="45" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="22"> <inlineAttr line-through="false">2. Normal distribution as a model for heigths of male and female students collected at first class </inlineAttr> </f> <f family="Arial" charset="0" size="18"> <inlineAttr line-through="false"> <c val="#f00">(make sure path is specificed correctly to<sp count="2"/>EXCEL spreadsheet read below in collapsed area;<sp count="2"/> </c> </inlineAttr> </f> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" line-through="false"> <c val="#f00">male and female heights (random variables) are </c> </inlineAttr> </f> <c val="#f00"> <f size="18">TF, TM)</f> </c> <f family="Arial" charset="0" size="20"> <inlineAttr font-weight="normal" underline="false" line-through="false">i. Histogram</inlineAttr> </f> <c val="#000"> <sp/> </c> </text> </region> <region region-id="4630" left="0" top="1740" width="6008.25" align-x="0" align-y="1740" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="614.25"> <area is-collapsed="true" name="EXCEL spreadsheet (check path, etc)" show-name="true" show-border="true" show-icon="true" show-timestamp="true" allow-expand="false" is-locked="false" timestamp="" top-lock-id="4103" bottom-lock-id="3161"> <region region-id="4609" left="66" top="38.25" width="126.75" height="55.5" align-x="92.25" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="01350083-1122-11DB-9380-000D56C6051A" item-idref="8" disable-calc="false"> <inputs/> <outputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">A</ml:id> </outputs> </component> </region> <region region-id="3446" left="204" top="38.25" width="71.25" height="20.25" align-x="222.75" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> work file </text> </region> <region region-id="3443" left="318" top="69" width="119.25" height="12.75" align-x="331.5" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">X</ml:id> <ml:apply> <ml:id xml:space="preserve">submatrix</ml:id> <ml:sequence> <ml:id xml:space="preserve">A</ml:id> <ml:real>16</ml:real> <ml:real>26</ml:real> <ml:real>5</ml:real> <ml:real>5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> </region> <region region-id="3444" left="582" top="62.25" width="165" height="20.25" align-x="585.75" align-y="78" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> x values: 147.5 etc </text> </region> <region region-id="3222" left="804" top="98.25" width="123.75" height="20.25" align-x="846.75" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Longueurs ici : </text> </region> <region region-id="3227" left="6" top="117" width="122.25" height="12.75" align-x="22.5" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Hf</ml:id> <ml:apply> <ml:id xml:space="preserve">submatrix</ml:id> <ml:sequence> <ml:id xml:space="preserve">A</ml:id> <ml:real>16</ml:real> <ml:real>26</ml:real> <ml:real>7</ml:real> <ml:real>7</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> </region> <region region-id="3228" left="276" top="117" width="125.25" height="12.75" align-x="295.5" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Hm</ml:id> <ml:apply> <ml:id xml:space="preserve">submatrix</ml:id> <ml:sequence> <ml:id xml:space="preserve">A</ml:id> <ml:real>16</ml:real> <ml:real>26</ml:real> <ml:real>6</ml:real> <ml:real>6</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> </region> <region region-id="4194" left="1092" top="135" width="30.75" height="12.75" align-x="1106.25" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">absices</ml:id> </math> </region> <region region-id="3223" left="798" top="141" width="130.5" height="12.75" align-x="813.75" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Xl</ml:id> <ml:apply> <ml:id xml:space="preserve">submatrix</ml:id> <ml:sequence> <ml:id xml:space="preserve">A</ml:id> <ml:real>16</ml:real> <ml:real>28</ml:real> <ml:real>11</ml:real> <ml:real>11</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> </region> <region region-id="3130" left="36" top="140.25" width="195" height="25.5" align-x="70.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="fe">μ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:sequence> <ml:real>27</ml:real> <ml:real>7</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>164.78260869565219</ml:real> </result> </ml:eval> </ml:define> </math> </region> <region region-id="4192" left="282" top="140.25" width="182.25" height="25.5" align-x="323.25" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="me">μ</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">A</ml:id> <ml:sequence> <ml:real>27</ml:real> <ml:real>6</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>177.3</ml:real> </result> </ml:eval> </ml:define> </math> </region> <region region-id="3232" left="558" top="146.25" width="78" height="20.25" align-x="569.25" align-y="162" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> les deux: </text> </region> <region region-id="3224" left="804" top="171" width="129.75" height="12.75" align-x="819" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Hl</ml:id> <ml:apply> <ml:id xml:space="preserve">submatrix</ml:id> <ml:sequence> <ml:id xml:space="preserve">A</ml:id> <ml:real>16</ml:real> 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</ml:eval> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> </region> <region region-id="3888" left="708" top="458.25" width="263.25" height="20.25" align-x="726" align-y="474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> pour densité colorée loi normale </text> </region> </area> <rendering item-idref="10"/> </region> <region region-id="4633" left="0" top="1770" width="6008.25" align-x="0" align-y="1770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="" height="140.25"> <area is-collapsed="true" 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<ml:real>167.5</ml:real> <ml:real>172.5</ml:real> <ml:real>177.5</ml:real> <ml:real>182.5</ml:real> <ml:real>187.5</ml:real> <ml:real>192.5</ml:real> <ml:real>197.5</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat> <table item-idref="11"/> </resultFormat> </math> </region> <region region-id="4280" left="516" top="93" width="51.75" height="12.75" align-x="525.75" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">x</ml:id> <ml:range> <ml:real>140</ml:real> <ml:real>200</ml:real> </ml:range> </ml:define> </math> </region> <region region-id="4249" left="1530" top="81.75" width="501.75" height="205.5" align-x="1569" align-y="102" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial" charset="0" size="22"> <inlineAttr line-through="false">2. La loi normale</inlineAttr> </f> <f family="Arial" charset="0" size="22"> <inlineAttr font-weight="normal" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial" charset="0" size="22"> <inlineAttr font-weight="normal" font-style="normal" line-through="false">comme modèle de la taille des étudiants</inlineAttr> </f> <f family="Arial" charset="0" size="20"> <inlineAttr font-weight="normal" underline="false" line-through="false">i.</inlineAttr> </f> <f family="Arial" charset="0" size="20"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial" charset="0" size="20"> <inlineAttr font-weight="normal" underline="false" line-through="false">Approximation normale</inlineAttr> </f> Chaque taille de femme et d'homme:<sp count="3"/>v.a.<sp count="2"/>TF ou TM. Ci-dessous les fréquences<sp count="2"/>observées pour les deux sexes (par groupe de taille quinquennal) et les densités normales<sp count="2"/>qui les approximent: les<sp count="3"/> <f family="Arial" charset="0">m</f> fe ,<f family="Arial" charset="0">s</f> fe et<sp count="2"/> <f family="Arial" charset="0">m</f> me ,<f family="Arial" charset="0">s</f> me<sp count="2"/> utilisés sont <c val="#000">des estimateurs</c> <c val="#000"> <sp/> </c> <c val="#000"> des espérances et écarts-types théoriques mais inconnues </c> <c val="#000"> <f family="Arial" charset="0">m</f> </c> <c val="#000"> f </c> <c val="#000">, </c> <f family="Arial" charset="0"> <c val="#000">s</c> </f> <c val="#000"> f </c> <c val="#000"> de TF, et<sp count="2"/> </c> <f family="Arial" charset="0"> <c val="#000">m</c> </f> <c val="#000">m </c> <c val="#000">, </c> <c val="#000"> <f family="Arial" charset="0">s</f> </c> <c val="#000"> m </c> <c val="#000"> <sp/> </c> <c val="#000">de TM.<sp count="2"/>(voir ci-dessous). </c> </text> </region> </area> <rendering item-idref="12"/> </region> <region region-id="4643" left="24" top="1778.25" width="219" height="20.25" align-x="50.25" align-y="1794" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> counts and heights, in cm: </text> </region> <region region-id="4265" left="12" top="1817.25" width="721.5" height="51.75" align-x="41.25" align-y="1848" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Df</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="11"> <ml:real>1</ml:real> <ml:real>147.5</ml:real> <ml:real>1</ml:real> <ml:real>152.5</ml:real> <ml:real>2</ml:real> <ml:real>157.5</ml:real> <ml:real>10</ml:real> <ml:real>162.5</ml:real> <ml:real>6</ml:real> <ml:real>167.5</ml:real> <ml:real>2</ml:real> <ml:real>172.5</ml:real> <ml:real>1</ml:real> <ml:real>177.5</ml:real> <ml:real>0</ml:real> <ml:real>182.5</ml:real> <ml:real>0</ml:real> <ml:real>187.5</ml:real> <ml:real>0</ml:real> <ml:real>192.5</ml:real> <ml:real>0</ml:real> <ml:real>197.5</ml:real> </ml:matrix> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="matrix" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="13"/> </region> <region region-id="4273" left="12" top="1880.25" width="69" height="25.5" align-x="37.5" align-y="1896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" font="1" subscript="f">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>23</ml:real> </result> </ml:eval> </math> <rendering item-idref="14"/> </region> <region region-id="4645" left="90" top="1880.25" width="177" height="20.25" align-x="120.75" align-y="1896" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> = number of females </text> </region> <region region-id="4275" left="24" top="1904.25" width="493.5" height="20.25" align-x="36" align-y="1920" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> In histogram </text> </region> <region region-id="3403" left="12" top="2316" width="532.5" height="225.75" align-x="12" align-y="2316" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="15"/> <rendering item-idref="16"/> </region> <region region-id="4670" left="570" top="2341.5" width="182.25" height="147" align-x="626.25" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#f00">superimposed </c> <c val="#f00">normal density</c> <c val="#f00">explained below -</c> <c val="#f00"> oral comments </c> <c val="#f00"> about roughness </c> <c val="#f00">of fit, small samples, </c> <c val="#f00">etc.<sp count="2"/> </c> </text> </region> <region region-id="4285" left="6" top="2562" width="546" height="225.75" align-x="6" align-y="2562" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="17"/> <rendering item-idref="18"/> </region> <region region-id="4658" left="12" top="2803.5" width="531" height="250.5" align-x="19.5" align-y="2820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="true" lock-width="true"> ii. Normal approximation<sp count="2"/> Male parameters<inlineAttr font-style="normal"> <sp/> </inlineAttr> <sp count="3"/> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#000"> <sp/> </c> </inlineAttr> </f> <c val="#000"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> <sp/> </f> </inlineAttr> </c> <region region-id="3406" left="18" top="3008.25" width="120.75" height="25.5" align-x="51" align-y="3024" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="fe">μ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>164.78260869565219</ml:real> </result> </ml:eval> </math> <rendering item-idref="19"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> <c val="#000">;</c> </f> </inlineAttr> <c val="#000"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> similarly : </f> </inlineAttr> </c> <region region-id="3478" left="174" top="3008.25" width="108" height="25.5" align-x="213.75" align-y="3024" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="me">μ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>177.3</ml:real> </result> </ml:eval> </math> <rendering item-idref="20"/> </region> These estimators are unbiased: </text> </region> <region region-id="3438" left="30" top="3206.25" width="523.5" height="20.25" align-x="35.25" align-y="3222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <c val="#000">b. To estimate<sp count="2"/> </c> </text> </region> <region region-id="4686" left="12" top="3524.25" width="525.75" height="156" align-x="30.75" align-y="3540" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="true" lock-width="true"> For calculations we need: i. <region region-id="3585" left="24" top="3542.25" width="69" height="25.5" align-x="49.5" align-y="3558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>23</ml:real> </result> </ml:eval> </math> <rendering item-idref="21"/> </region> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="6"/> </inlineAttr> </f> <region region-id="4320" left="120" top="3542.25" width="75" height="25.5" align-x="151.5" align-y="3558" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="m">n</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>20</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> ii.<sp count="3"/> <region region-id="4321" left="48" top="3572.25" width="129" height="21.75" align-x="114" align-y="3588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">somTF</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>3790</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="8"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="23"/> </region> <sp/> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <region region-id="3584" left="156" top="3572.25" width="132" height="21.75" align-x="225" align-y="3588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">somTM</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>3546</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="8"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="24"/> </region> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">iii. Sums of squares =</inlineAttr> </f> <region region-id="4501" left="186" top="3596.25" width="158.25" height="21.75" align-x="261.75" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">som2TF</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>625478</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <decimal precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="25"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> <sp/> </f> </inlineAttr> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18">; </f> </inlineAttr> <region region-id="4328" left="312" top="3596.25" width="161.25" height="21.75" align-x="390.75" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">som2TM</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>629836</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <decimal precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="26"/> </region> Exercise: ladies in lecture calculate<sp count="2"/> <region region-id="4330" left="312" top="3614.25" width="23.25" height="25.5" align-x="322.5" align-y="3630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" subscript="fe" xmlns:ml="http://schemas.mathsoft.com/math30">μ</ml:id> </math> <rendering item-idref="27"/> </region> <sp/> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18">, </f> </inlineAttr> <region region-id="4329" left="348" top="3614.25" width="23.25" height="25.5" align-x="358.5" align-y="3630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" subscript="fe" xmlns:ml="http://schemas.mathsoft.com/math30">σ</ml:id> </math> <rendering item-idref="28"/> </region> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> ; </f> </inlineAttr> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">gentlemen<sp count="2"/> </inlineAttr> </f> <region region-id="4340" left="108" top="3650.25" width="30" height="25.5" align-x="122.25" align-y="3666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" subscript="me" xmlns:ml="http://schemas.mathsoft.com/math30">μ</ml:id> </math> <rendering item-idref="29"/> </region> <sp/> , <region region-id="4341" left="156" top="3650.25" width="30" height="25.5" align-x="170.25" align-y="3666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" subscript="me" xmlns:ml="http://schemas.mathsoft.com/math30">σ</ml:id> </math> <rendering item-idref="30"/> </region> <sp/> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial" charset="0" size="18"> <sp count="7"/> </f> </inlineAttr> <inlineAttr underline="false" line-through="false"> <f family="Arial" charset="0" size="14"> <c val="#f00">(done here and now in class)</c> </f> </inlineAttr> </text> </region> <region region-id="4402" left="24" top="4106.25" width="502.5" height="321.75" align-x="24" align-y="4122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="true" lock-width="true"> <f family="Arial" charset="0"> <inlineAttr font-style="normal" line-through="false"> <c val="#800000">Exercise 1:</c> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr font-style="normal" underline="false" line-through="false"> <c val="#800000"> <sp/> </c> </inlineAttr> </f> <f family="Arial" charset="0"> <inlineAttr underline="false" line-through="false"> <c val="#800000"> We have three sampled values</c> </inlineAttr> </f> <c val="#800000"> X</c> <f family="Arial" charset="0" size="18"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#800000"> 1 </c> </inlineAttr> </f> <c val="#800000">=0, </c> <c val="#800000"> X</c> <c val="#800000"> 2 </c> <c val="#800000">=0, </c> <c val="#800000"> X</c> <c val="#800000"> 3 </c> <c val="#800000">=1,of a binary rv<sp count="2"/>(e.g. flipped coin). Use<sp count="2"/>eq. (1) and<sp count="2"/>(4) to </c> <c val="#800000">calculate estimated values<sp count="3"/> </c> <f family="Symbol" charset="2"> <c val="#800000">m</c> </f> <c val="#800000">e</c> <c val="#800000"> of </c> <f family="Symbol" charset="2"> <c val="#800000">m</c> </f> <c val="#800000"> <sp count="2"/>and </c> <f family="Symbol" charset="2"> <c val="#800000">s</c> </f> <c val="#800000"> e </c> <c val="#800000"> of </c> <f family="Symbol" charset="2"> <c val="#800000">s. </c> </f> <f family="Symbol" charset="2"> <c val="#800000">m</c> </f> <c val="#800000">e</c> <c val="#800000"> =<sp count="10"/> </c> <c val="#800000"> <sp count="2"/> </c> </text> </region> <region region-id="4550" left="798" top="5660.25" width="132" height="21.75" align-x="852" align-y="5676" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" 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