204 lines
6.3 KiB
XML
204 lines
6.3 KiB
XML
<?xml version="1.0" encoding="UTF-8"?>
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<!-- When creating a new activity, make a copy of this file with appropriate name -->
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<worksheet xml:id="quiz-05" margin="0.5in">
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<title>Quiz 5</title>
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<!-- Optional introduction -->
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<introduction>
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<p>
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The following work should be completed individually.
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Use of notes or textbooks is not allowed.
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You may use a scientific calculator, not a graphing calculator or phone app.
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</p>
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<p>
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Show all work unless instructed otherwise.
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</p>
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</introduction>
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<page>
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<!-- Exercises start here. -->
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<exercise>
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<statement>
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<p>
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The joint and marginal distributions of <m>X</m> and <m>Y</m> are given below.
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Find <m>\Cov(X, Y)</m> and <m>\rho_{X, Y}</m>.
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</p>
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<table>
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<title>Joint Distribution Table</title>
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<tabular halign="center">
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<row header="yes" bottom="minor">
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<cell right="minor"></cell>
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<cell><m>Y = 1</m></cell>
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<cell><m>Y = 2</m></cell>
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<cell><m>Y = 3</m></cell>
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</row>
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<row>
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<cell right="minor"><m>X = 0</m></cell>
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<cell><m>0.1</m></cell>
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<cell><m>0.15</m></cell>
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<cell><m>0.1</m></cell>
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</row>
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<row>
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<cell right="minor"><m>X = 1</m></cell>
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<cell><m>0.15</m></cell>
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<cell><m>0.2</m></cell>
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<cell><m>0.3</m></cell>
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</row>
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</tabular>
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</table>
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<table>
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<title>Distribution Table for <m>X</m></title>
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<tabular halign="center">
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<row header="yes" bottom="minor">
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<cell right="minor"><m>x</m></cell>
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<cell><m>0</m></cell>
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<cell><m>1</m></cell>
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</row>
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<row>
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<cell right="minor"><m>\Pr(X = x)</m></cell>
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<cell><m>0.35</m></cell>
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<cell><m>0.65</m></cell>
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</row>
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</tabular>
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</table>
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<table>
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<title>Distribution Table for <m>Y</m></title>
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<tabular halign="center">
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<row header="yes" bottom="minor">
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<cell right="minor"><m>y</m></cell>
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<cell><m>1</m></cell>
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<cell><m>2</m></cell>
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<cell><m>2</m></cell>
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</row>
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<row>
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<cell right="minor"><m>\Pr(Y = y)</m></cell>
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<cell><m>0.25</m></cell>
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<cell><m>0.35</m></cell>
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<cell><m>0.4</m></cell>
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</row>
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</tabular>
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</table>
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</statement>
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<solution>
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<p>
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<m>\Cov(X, Y) = \E(XY) - \E(X)\E(Y)</m>. We have:
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<md>
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<mrow> \E(X) \amp = 0(0.35) + 1(0.65) = 0.65 </mrow>
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<mrow> \E(Y) \amp = 1(0.25) + 2(0.35) + 3(0.4) = 2.15 </mrow>
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<mrow> \E(XY) \amp = (0)(1)(0.1) + (0)(2)(0.15) + (0)(3)(0.1) + </mrow>
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<mrow> \amp \quad\, (1)(1)(0.15) + (1)(2)(0.2) + (1)(3)(0.3) = 1.45 \\ </mrow>
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<mrow> \Cov(X, Y) \amp = 1.45 - (0.65)(2.15) = \boxed{0.0525} </mrow>
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</md>
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Since <m>X</m> is an indicator random variable, we have <m>\Var(X) = (0.65)(0.35) = 0.2275</m>.
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For <m>\Var(Y)</m>:
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<md>
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<mrow> \E(Y^2) \amp = 1^2(0.25) + 2^2(0.35) + 3^2(0.4) = 5.25 </mrow>
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<mrow> \Var(Y) \amp = \E(Y^2) - (\E(Y))^2 = 5.25 - (2.15)^2 = 0.6275 </mrow>
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</md>
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So the correlation is:
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<md>
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<mrow> \rho_{X, Y} \amp = \frac{\Cov(X, Y)}{\sqrt{\Var(X)\Var(Y)}} = \frac{0.0525}{\sqrt{(0.2275)(0.6275)}} \approx \boxed{0.139}. </mrow>
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</md>
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</p>
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</solution>
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</exercise>
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</page>
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<page>
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<exercise>
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<statement>
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<p>
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A sample of 30 measurements are taken and a best fit line is calculated, resulting in the data below (the final row of the table shows the sums for each column).
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Find the RSS, SST, and coefficient of determination.
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</p>
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<table>
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<title>Sample Data</title>
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<tabular>
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<row header="yes" bottom="minor">
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<cell><m>x_i</m></cell>
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<cell><m>y_i</m></cell>
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<cell>best fit predicted <m>y_i</m></cell>
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<cell>res<m>^2</m></cell>
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<cell><m>(y - \textrm{avg }y)^2</m></cell>
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</row>
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<row>
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<cell><m>1.94</m></cell>
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<cell><m>3.31</m></cell>
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<cell><m>10.53</m></cell>
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<cell><m> 52.13</m></cell>
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<cell><m>886.55</m></cell>
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</row>
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<row>
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<cell><m>2.63</m></cell>
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<cell> <m>10.38</m></cell>
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<cell><m> 17.37</m> </cell>
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<cell><m>48.76</m></cell>
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<cell><m> 515.34</m></cell>
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</row>
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<row>
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<cell><m>\vdots</m></cell>
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<cell><m>\vdots</m></cell>
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<cell><m>\vdots</m></cell>
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<cell><m>\vdots</m></cell>
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<cell><m>\vdots</m></cell>
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</row>
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<row>
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<cell><m>3.94</m></cell>
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<cell><m>19.47</m></cell>
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<cell><m> 30.31</m> </cell>
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<cell><m>117.35</m></cell>
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<cell><m> 185.24 </m></cell>
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</row>
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<row bottom="minor">
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<cell><m>6.35</m></cell>
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<cell><m> 53.42</m></cell>
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<cell><m>54.22</m></cell>
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<cell><m>0.64</m></cell>
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<cell><m>413.51</m></cell>
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</row>
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<row>
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<cell>sum:</cell>
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<cell><m>992.55</m></cell>
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<cell><m>992.55</m></cell>
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<cell><m> 953.03</m></cell>
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<cell><m>38397.35</m></cell>
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</row>
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</tabular>
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</table>
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</statement>
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<solution>
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<p>
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The RSS is the sum of res<m>^2</m>, <m>953.03</m>, and the SST is the sum of <m>(y - \textrm{avg})^2</m>, <m>38397.35</m>.
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Then the coefficient of determination is:
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<md>
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<mrow> r^2 = 1 - \frac{\text{RSS}}{\text{SST}} = 1 - \frac{953.03}{38397.35} \approx \boxed{0.975} </mrow>
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</md>
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</p>
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</solution>
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</exercise>
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</page>
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</worksheet> |