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<h2>HL Paper 1</h2><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right)"> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right) = {x^3} + 3{x^2} + 8x - 24"><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>24</mn></math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}"> <mi>x</mi> <mo>∈<!-- ∈ --></mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the remainder when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is divided by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 2} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the remainder when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is divided by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 3} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> has only one real zero.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the transformation that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 8{x^3} + 12{x^2} + 16x - 24"><mi>y</mi><mo>=</mo><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mo>−</mo><mn>24</mn></math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> follows a Poisson distribution with a mean of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6{\text{P}}\left( {X = 3} \right) = 3{\text{P}}\left( {X = 2} \right) - 2{\text{P}}\left( {X = 1} \right) + 3{\text{P}}\left( {X = 0} \right)">
<mn>6</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( 2 \right) = 8 - 12 + 16 - 24">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mo>−</mo>
<mn>12</mn>
<mo>+</mo>
<mn>16</mn>
<mo>−</mo>
<mn>24</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for a valid attempt at remainder theorem or polynomial division.</p>
<p>= −12 <em><strong>A1</strong></em></p>
<p>remainder = −12</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( 3 \right) = 27 - 27 + 24 - 24">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>27</mn>
<mo>−</mo>
<mn>27</mn>
<mo>+</mo>
<mn>24</mn>
<mo>−</mo>
<mn>24</mn>
</math></span> = 0 <em><strong>A1</strong></em> </p>
<p>remainder = 0</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 3">
<mi>x</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> (is a zero) <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: bold;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">A1</span></p>
<p><strong>Note:</strong> Can be seen anywhere.</p>
<p><strong>EITHER</strong></p>
<p>factorise to get <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 3} \right)\left( {{x^2} + 8} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>8</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 8 \ne 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>8</mn>
<mo>≠</mo>
<mn>0</mn>
</math></span> (for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>) (or equivalent statement) <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>if correct two complex roots are given.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p'\left( x \right) = 3{x^2} - 6x + 8">
<msup>
<mi>p</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mi>x</mi>
<mo>+</mo>
<mn>8</mn>
</math></span> <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: bold;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">A1</span></p>
<p>attempting to show <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p'\left( x \right) \ne 0">
<msup>
<mi>p</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>≠</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><em>eg</em> discriminant = 36 – 96 < 0, completing the square</p>
<p>no turning points<em><strong> R1</strong></em></p>
<p><strong>THEN</strong></p>
<p>only one real zero (as the curve is continuous) <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>new graph is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( {2x} \right)"> <mi>y</mi> <mo>=</mo> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p>stretch parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis (with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> invariant), scale factor 0.5 <em><strong>A</strong><strong>1</strong></em></p>
<p><strong>Note:</strong> Accept “horizontal” instead of “parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis”.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{6{\lambda ^3}{e^{ - \lambda }}}}{6} = \frac{{3{\lambda ^2}{e^{ - \lambda }}}}{2} - 2\lambda {e^{ - \lambda }} + 3{e^{ - \lambda }}">
<mfrac>
<mrow>
<mn>6</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mn>6</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mn>2</mn>
<mi>λ</mi>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Allow factorials in the denominator for <em><strong>A1</strong></em>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\lambda ^3} - 3{\lambda ^2} + 4\lambda - 6 = 0">
<mn>2</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>λ</mi>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A</strong><strong>1</strong></em></p>
<p><strong>Note:</strong> Accept any correct cubic equation without factorials and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{e^{ - \lambda }}}">
<mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</msup>
</mrow>
</mrow>
</math></span>.</p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4\left( {2{\lambda ^3} - 3{\lambda ^2} + 4\lambda - 6} \right) = 8{\lambda ^3} - 12{\lambda ^2} + 16\lambda - 24 = 0">
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>λ</mi>
<mo>−</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>12</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>16</mn>
<mi>λ</mi>
<mo>−</mo>
<mn>24</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\lambda = 3">
<mn>2</mn>
<mi>λ</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {2\lambda - 3} \right)\left( {{\lambda ^2} + 2} \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>λ</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> = 1.5 <em><strong>A</strong><strong>1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the time, in days, from December <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mtext>st</mtext></math> and the percentage of Christmas trees in stock at a shop on the beginning of that day.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The following table shows the natural logarithm of both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> on these days to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> decimal places.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the data in the second table to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> for the regression line, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>m</mi><mo>(</mo><mi>ln</mi><mo> </mo><mi>d</mi><mo>)</mo><mo>+</mo><mi>b</mi></math>.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming that the model found in part (a) remains valid, estimate the percentage of trees in stock when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>25</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>695</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>695383</mn><mo>…</mo></mrow></mfenced><mo>;</mo><mo> </mo><mi>b</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>63</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>4</mn><mo>.</mo><mn>62974</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>695</mn><mfenced><mrow><mi>ln</mi><mo> </mo><mn>25</mn></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>63</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>39288</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>9</mn><mo>%</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Those candidates who did this question were often successful. There were a number, however, who found an equation of a line through two of the points instead of using their technology to find the equation of the regression line. A common problem was to introduce rounding errors at various stages throughout the problem. Some candidates failed to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> from that of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi></math>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Those candidates who did this question were often successful. There were a number, however, who found an equation of a line through two of the points instead of using their technology to find the equation of the regression line. A common problem was to introduce rounding errors at various stages throughout the problem. Some candidates failed to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> from that of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi></math>.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The weights of apples from Tony’s farm follow a normal distribution with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>158</mn><mtext> g</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mtext> g</mtext></math>. The apples are sold in bags that contain six apples.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the mean weight of a bag of apples.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the standard deviation of the weights of these bags of apples.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a bag selected at random weighs more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>158</mn><mo>×</mo><mn>6</mn><mo>=</mo><mn>948</mn><mo> </mo><mfenced><mtext>g</mtext></mfenced></math> <em><strong> (M1)</strong></em><em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>variance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><msup><mn>13</mn><mn>2</mn></msup></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>SD</mtext><mo>=</mo><mn>31</mn><mo>.</mo><mn>8</mn><mo> </mo><mfenced><mtext>g</mtext></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>13</mn><msqrt><mn>6</mn></msqrt><mo>,</mo><mo> </mo><mn>31</mn><mo>.</mo><mn>8433</mn><mo>…</mo></mrow></mfenced></math> <em><strong> </strong></em><em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>948</mn><mo>,</mo><mo> </mo><mn>31</mn><mo>.</mo><mn>8433</mn><msup><mo>…</mo><mn>2</mn></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>></mo><mn>1000</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>0512</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0512350</mn><mo>…</mo></mrow></mfenced></math> <em><strong> (M1)A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0510</mn><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0510014</mn><mo>…</mo></mrow></mfenced></math> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> sf value <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>31</mn><mo>.</mo><mn>8</mn></math> is used.<br>Award <em><strong>(M1)A1FT</strong></em> if the answer is correct for their SD, even if no working is shown. e.g. If the SD is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>78</mn></math> then accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>252</mn></math>.</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A manufacturer of chocolates produces them in individual packets, claiming to have an average of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>85</mn></math> chocolates per packet.</p>
<p>Talha bought <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> of these packets in order to check the manufacturer’s claim.</p>
<p>Given that the number of individual chocolates is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, Talha found that, from his <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> packets, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Σ</mtext><mi>x</mi><mo>=</mo><mn>2506</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Σ</mtext><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>209</mn><mo> </mo><mn>738</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an unbiased estimate for the mean number <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>μ</mi><mo>)</mo></math> of chocolates per packet.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the formula <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>s</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mfrac><mrow><mtext>Σ</mtext><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mfenced><mrow><mtext>Σ</mtext><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mi>n</mi></mfrac></mstyle></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> to determine an unbiased estimate for the variance of the number of chocolates per packet.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mo>%</mo></math> confidence interval for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math>. You may assume that all conditions for a confidence interval have been met.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Suggest, with justification, a valid conclusion that Talha could make.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mtext>Σ</mtext><mi>x</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mfrac><mn>2506</mn><mn>30</mn></mfrac><mo>=</mo><mn>83</mn><mo>.</mo><mn>5</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>83</mn><mo>.</mo><mn>5333</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msubsup><mi>s</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mfrac><mrow><mtext>Σ</mtext><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mfenced><mrow><mtext>Σ</mtext><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mi>n</mi></mfrac></mstyle></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mrow><mn>209738</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mn>2506</mn><mn>2</mn></msup><mn>30</mn></mfrac></mstyle></mrow><mn>29</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>13</mn><mo>.</mo><mn>9</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>13</mn><mo>.</mo><mn>9126</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>82</mn><mo>.</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>84</mn><mo>.</mo><mn>9</mn></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>82</mn><mo>.</mo><mn>1405</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>84</mn><mo>.</mo><mn>9261</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A2</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>85</mn></math> is outside the confidence interval and therefore Talha would suggest that the manufacturer’s claim is incorrect <em><strong>R1</strong></em><br><br><strong>Note:</strong> The conclusion must refer back to the original claim.</p>
<p> Allow use of a two sided <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>-test giving a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value rounding to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>04</mn><mo><</mo><mn>0</mn><mo>.</mo><mn>05</mn></math> and therefore Talha would suggest that the manufacturer’s claims in incorrect.</p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A factory, producing plastic gifts for a fast food restaurant’s Jolly meals, claims that just <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>%</mo></math> of the toys produced are faulty.</p>
<p>A restaurant manager wants to test this claim. A box of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn></math> toys is delivered to the restaurant. The manager checks all the toys in this box and four toys are found to be faulty.</p>
</div>
<div class="specification">
<p>The restaurant manager performs a one-tailed hypothesis test, at the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> significance level, to determine whether the factory’s claim is reasonable. It is known that faults in the toys occur independently.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Identify the type of sampling used by the restaurant manager.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the null and alternative hypotheses.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value for the test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the conclusion of the test. Give a reason for your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Convenience <em><strong> A1</strong></em> </p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mi mathvariant="normal">:</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>%</mo></math> of the toys produced are faulty <em><strong> A1</strong></em> <br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub><mi mathvariant="normal">:</mi></math> More than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>%</mo></math> are faulty <em><strong> A1</strong></em> </p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>200</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>01</mn></mrow></mfenced></math> <em><strong> (M1)</strong></em> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≥</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>142</mn></math> <em><strong> A1</strong></em> </p>
<p><strong><br>Note:</strong> Any attempt using Normal approximation to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value is awarded <em><strong>M0A0</strong></em>.</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn><mo>%</mo><mo>></mo><mn>10</mn><mo>%</mo></math> <em><strong> R1</strong></em> </p>
<p>so there is insufficient evidence to reject <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math>. <em><strong> A1</strong></em> </p>
<p><strong><br>Note:</strong> Do not award <em><strong>R0A1</strong></em>. Accept “fail to reject <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math>” or “accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math>”.</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Saloni wants to find a model for the temperature of a bottle of water after she removes it from the fridge. She uses a temperature probe to record the temperature of the water, every 5 minutes.</p>
<p><img src="data:image/png;base64,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"></p>
<p>After graphing the data, Saloni believes a suitable model will be</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T = 28 - a{b^t}">
<mi>T</mi>
<mo>=</mo>
<mn>28</mn>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mrow>
<msup>
<mi>b</mi>
<mi>t</mi>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{,}}\,\,b \in {\mathbb{R}^ + }">
<mi>a</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="28 - T">
<mn>28</mn>
<mo>−</mo>
<mi>T</mi>
</math></span> can be modeled by an exponential function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the least squares exponential regression curve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="28 - T">
<mn>28</mn>
<mo>−</mo>
<mi>T</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coefficient of determination, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{R^2}">
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Interpret what the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{R^2}">
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> tells you about the model.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence predict the temperature of the water after 3 minutes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Rearranging the model gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="28 - T = a{b^t}">
<mn>28</mn>
<mo>−</mo>
<mi>T</mi>
<mo>=</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>b</mi>
<mi>t</mi>
</msup>
</mrow>
</math></span><em><strong> A1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="28 - T">
<mn>28</mn>
<mo>−</mo>
<mi>T</mi>
</math></span> can be modeled by an exponential function. <em><strong> AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAc8AAAAtCAYAAADIg+AWAAARN0lEQVR4Ae2dzYsdRRfGOy9uxURdibhwdCEKisa40GwCGjHgKhA/IBtBiYobIYGouIgmRowr4xcK2YToymwifoAuom5UGEFwoZEgWY4mxj9gXn7l+9z3TN3q29V9q/vemXsKZrq7uqqec5469V23etPq6upq1cH9+eef1Ycffli9//771U033VR99tlnjals2rSpWlpaqp588slq//79jeE9gDPgDDgDzoAzMI8MbOraeM6jMi6TM+AMOAPOgDMwBANXxCCMDt05A86AM+AMOAPOwP8ZiCdpxxpPgsaB/h/d7xaRATpUs7QJx19s/ilzbgOLbQPzkP9x3f+f2MOfnQFnwBlwBpwBZ2AyA954TubH3zoDzoAz4Aw4A2MMeOM5Rol7OAPOgDPgDDgDkxnwxrOqKn528/TTT1dXX3119frrr09mbJ2//eabb8L60d133119+umn61ybyeJ/9NFHFfq6q2cAu190t+gcuP4dywA/VbHu3/1C1mft/dGjR/ldaO3fzp07RxEIR/h5d1u3bl3dt2/f6srKylSinj17NvDCdd7du+++my1rk01I1zNnzqwuLS2FdLmeOnVKr6a65uCL+9g2S+RFDr4UxIb27NmzChcpR3nYsmVL4Ai7W15eTgVb41cKv67srgFLPOTgozfpK//RMVX2u+iPSKVkGJqDgwcPjjHahYOu+sf4fepvFcWusYFU+etL/xz8kvqzi3KNa8okwCkgamgIL4LIqHlqPGkQkXeSo4JHB+kzKWzTO1Xg4qMpfJv3cF46XSp5/ppck00QX7qrwZSRlpC5DX6TLl3e5+CTLg2mGsaU3nDCeyoWNbI8N9leSXxbPnO5yMEnXdsB5Z54loeu+iNnSRmG4kBlQGUCPbpy0EX/Ovy+9Jc9Yc90DOP871v/XPxS+ndqPK0xWIIgrYtgUrr0lQzEgCa5uMGfFHZW7+DV8lxKDo0+m9LLKbhUlvBtHQ0D/tO6HHw13tNipeLn4FMmZGt1eUUHyI4EaEQJa8tTn/jI16V85ugfyy2bFSe876o/cUvJMCQHskk7u9CVgy76p/CH0J8yr7rFdp6msYE2+k/CL6l/6zVPjtV75JFHkssk11xzzeiYPubR+W2O1hB/+umnsKaIH/cPPvhgeM+VNUfCsebIH2tV1vGOIwCJy1od8Zsc4X744YfqwIEDIR5xU+7HH3+sNm/evOZVV1klo9bZ0ANcdGR9Ue9feOGFEZ78UnEIhCz33HNPCL99+/ZRenpHXoABbzZd3lveCBO/v/XWW0O6Jf598cUX1QMPPLAmqW3btlW///77Gr+N+kA+TDpyEh7OnTtX3XvvvSMKbr/99nD/xx9/jPy63jThd023a7y///47RL3tttvCtW/9U3LGMqTC9OFHfUZ5f+utt6pTp05VyuehOKjD70PXOE3qvL/++qt66qmn4lehLuizDAA4CX9MoGk94h5jmxZePcK4d6E0Scv2PLnHj943PVP1vOkNKw3u7QiGXrmmukiXuPTechxpWfxUnLowXWRVT0+6gEf6yE9PDKcemXqj6qHbOKnRMLzZMKQFDxrZnTt3LuBoFCNuhYM/sliXkte+132OTcR5TVzwcuIKp+6akwZ6wrOmi7hvyvs6vNg/B9/GIXycV3VcEzbOF5sW9yXwSUdlCbshTa51a7NWhrb4xI2XBKbRvwsHKRnw65sD6al8tXagd9ZPupWyAWGk8PvWnzqI8kedJr2srpLN+ilcCf2b8Evr33rkOW1jTfxXX321YpSqHtmOHTuq++67LyTNPSNGuWPHjlX79u0bhX3iiSdCDz5n9Kk0prm2kbUOhxGYemK7d+8Owf75559whYfYXXXVVbHX2DM9W3pxhw4dCu9uvPHGMCNw+vTpNWE///zzMLJnZJJzeP+ayAUetmzZUiCV5iSwJXq833//fTgN6ciRI2HWIZ7FaE5p+BDxzEdfEmADcPTbb79VKysrYaZg165dxWcHmPHANo8fP56lSh/618nQNwfUY/R30J8yyWyRZpUmkVGKgyb8PvXfs2dP9cEHH4S63epKXjRxUEL/HPyS+s+k8bTENt3TkB4+fHg09cpXWXBqfJriz9v7VGPZRcaff/45RLv22mtH3LzzzjvVpUuXgj+NyZkzZ8JXbwjDNDrTOX05GklNk1kMOg6zcHRWtm7dWsWdiVnIAuaVV14ZoC9fvjwmwp133jnm17cHdqiO15dfflkMjs7K+fPnQ0fN2vqQ+tfJECvZFwfg0HCK3++++y5AD8lBCr9P/d97770w6LnjjjtG9RF4dB5YOsP1qX8Ofmn9577xpLE8evRo6M3Ro9OfRqoxIYvyfMMNNwRVxYeudnT50EMPhVHG8vJyGJE988wzvdHDeifrntYxwrnrrrus16D3Fy9eHBRvEhidGToY33777SiY1oO1Ljh6MdBNqrMzDTSN1okTJ6q33347jD6YHWLUgRtK/0kypHQrzYHFsJ2HITmQDDG+/O21lP50VlUH6QrO2bNngz/1dZ82kINv9db9NPpP1XhqJHPhwgXJUvzKtz9fe+210bCfAsnQW46NMCqg8rPXr776Kjwy8lJlZd9zrzCxfx/P4iwegbC5AMf0Bt9IpeGJHT1Y4qP//fffHypjjSjxp/dF5YGDE97hMFo6IRqVBk/zr0RHZO/evaHnKXxkYYqQafYhHJuy0Fn8gs/U2bPPPjsE/AhD+KkywfIDswPYMOHeeOONMDqmk1PKTcKnrOhgDOHToGspYRoZ0IkllpMnT46SYcnAur71z5GhTw7iukj1kuW3Tw5y8PvU3+Z1nR32qX8OflH9w8qu+ddmc4A2HhBHG1WUFBtZ8OePzRza0MGzfluYCqM42gjD4jP3bADhHQvL2gQDFn7ajCNsXfW7O+LG8ikMm3PsBiX8u8oqPsDDgSl9JKMNIz20iYF4hNNmJXFAWvIjvuKhX90GGdIUFjIQTvGCcGbzkp7rrsTPcciIDsKLNwbkpJEKk4MPtrgQfs5mmBRe7JeDD7fYpvQnDs/W7mTLvOOPcsAmhyZXCh88axM8xzaRkiUH33Iv/biSL3Jd9Sd+KRn65AB7I33pDydxGejKQY7+Ofh96q985io7ozxYG+tT/xz8kvqPfQyblvlfW7Xt+Hze07u5+eabw4iHOf4ujpHSo48+Olrg75LGeo3DyJTRoUaLdXrM2iYcf7Zlctb8Y5ezlsHxF9sGU/k/1bRtXWU7lP+bb75Zff3112FxvismU6BMaU6a1u2a9jzHY1qTKcShpzXnmROXzRlwBpyBXAbWdePJz0hYz5vWsdGFkSu7M7VOMW2a8xqfNVV6UWwpZzduifXOedXV5XIGnAFnoC8GktO2fYF5us6AM+AMOAPOwHpkIF7OvCKlRBwoFcb9FoeB1Hz/kNo7/mKvN2FrbgOLbQPzkP9xnbeup21jZfzZGXAGnAFnwBkYggFvPIdg2TGcAWfAGXAGNhQD3nhuqOx0ZZwBZ8AZcAaGYGDhGk9+G8rPUviE17Q7aznVhnRITydqDJFpjuEMrAcG+P1w04Hg60EPl9EZSDHQqfGkoaDh0bcoUw2RjtFjoZc/wk7bWEkBGi2lm7pOwiEu7tdffx19f5FCjg788ftH62xjC5ZtLDlHloPr+ZLH448/bqMt3H2OTVhS4Bw+Szvk4Le7OoYulT5h+LYpNmmPekyFbeuXg0/Z4HuzhC3l6soEOg7h9BMoWx45fGRIB59xvRR/w7ZPeVL4k+qivmTJscG+sEl3YfDtkUbc5xwDxbFjHB3HUUs47omno6jw51gmfbeTMDqqzh5XFiJ3+Ac+xyzhwJTM4PLOHglmk5cMkpt3HI/GUVJc8bfH2PHMO/x0jJq+xyldSYPjp5DBHkNlcdf7vfidpEeTTdi48KRj7Kx/3X0OPnE5nkzp2vyx6QobG8nNr5L44JJebprInhMW/mWjxMF24aJEecvBt+XQ8l3qPkeG2AbF9VAcxPhxvTgNFzn6k35OGegih+OPH1E65pNLks0ACirx1GipIOFvHcalMNa/7b2tKISlNDCeOozUR6aJb8Nzjx8O46cCivVInVlJOJuO5NkI1xI2IR7gEv7UCZH/pGsOPhWk+Ce88tCmq4azbWVaCh/7wz5jm7Uypu5z8ON48ItNlnA5+G11aitXjgxxmpIpt5MUx7fPXfDjetGm1/Y+Bz+nDLTFVXjHH2sqwye+xE+45pC0JsL/Rm/Eo2LAYTQUXDs6xZDxs73jOJ0uzyogOXFpdDViVXjkqRt54h+HV7z4WqpjEKc7D88lbEJ6YBNU7G3yrS0+4Uk/duQl+dTWlcIXbhvdidMWnzjYrjoTwu16zcFvq1NbWXJkUJrUP9RF5HfbjpLSiK9t8BWXuoV4qhfl3+XaFp/wqTLQBZs4jj/eeHZa84znyw8ePFjxFW99WonvyLGmyLF3+ljzww8/PPU5tDFul+f4w8Mcy/fKK6+Eo/k4ZP75558fHfnHJ63i8JMwh/y02SQ55uFdbBPIxDokB9Hz7b2hHeswH3/8cTiGkXVOrc2x2WujOdaTh/wcHPzxoWM+b8ZarvYGzGK9j7VX6pxdu3aFT/Bdf/31M8veVBmYmTAOXJyBqRtPCgiNzPHjx9cIx4eX+UDyysoKTXZ15MiRiq+M1+2+U2UWX4cogFSmVDb8ld5AsoaUBXlI2QTfUn355ZfH7ARKtImrT3p++eWXUfKHDx8ONsmHemlo4k1io4Dr9ObFF1+s+G5izseQS6nIGdOUHzbPqbwfOHCg8Ys9pfCVDmc1g0+dRMd4+/bttXWO4vRxTZWBPnA8zdkxMFXjScVz/vz5il2ntqDSQNLL1xflUY/RBgev66PPscoYfOpv//79cdBBn9HPXT4DdTZBL5ydyZqJoFLD0VmKP5qcj9Y+5GOPPTb6Cg8V7bZt26pPPvmkfUJzGgP+aTiG+gh5HQ0q76dPn64L0qs/DeehQ4cCBh+RH9LVlYEhZXCs/hno3HhiICdOnBg1kGy/j0eJTJXF7tKlS7HX3D4zFY2esWNqqm4EHYddpOdJNsE72zli1IeTX988XXfddQHiwoULY1CbN28e81uvHpRJRp00HrN2Fy9enKkItkM/lCCTysBQMjjOMAx0ajxpKI8dO1adPHlyJKUdPdxyyy1h/eOll14a/ZaNqTFGHs8999woTomby5cvh2SYFsxxbdYl+dYlFYA9BEF6pLB27NiR8l4IvyabiElom29x/LpnddjiRpLGZOfOncFuZSt0gLDbvXv31iXX2r8O3ybUl+7SZxZryky903mW/pQTRsBDfi+WWQzbgdf97t27Lf293bctA30JojyIy0BfeHG6C4Mf777K2VXFTw0IF//Z3X1sD2e3m8IQp8SuMyuvfnoABjt5m7ak81MB5GjjkNnqyy5G+/tVpQW+1V/+G+EKv03OcqQ855rixOYbfDa5HHzSZCct+SB8nu1uS3Zh6rd3hMm1yVL4yIJMkg9ZeW6y2xx8OOy6m7gE/+SztYFcbpuw9T6HA+2wFb/IUGrHaQ6+1V8ycE2VAemVe83BzykDuXhxOMcfrwOT3/NkKm0jOqZUOPVEmwlK6chIZmlpqVpeXh7t1C2V9jykQ49+ljbh+IvNP2XAbWCxbWAe83+hGk8KIceV8cdmphLrQjScrI2yvsLGqY3o5tFwh+TZ9Z9txU1eex7MNg+c/3H+O615Dllxlcbit6c0muz81ZpIVwzWeUiHDUR2/bdreh7PGXAGnAFnYH0wkBx5rg/RXUpnwBlwBpwBZ2AYBuKlqyti2DhA/N6fnQFnwBlwBpyBRWdg4aZtFz3DXX9nwBlwBpyB6RnwxnN6Dj0FZ8AZcAacgQVjwBvPBctwV9cZcAacAWdgegb+C5NpDxRK7xFgAAAAAElFTkSuQmCC"><em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="28 - T = 22.7{\left( {0.925} \right)^t}">
<mn>28</mn>
<mo>−</mo>
<mi>T</mi>
<mo>=</mo>
<mn>22.7</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.925</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>t</mi>
</msup>
</mrow>
</math></span><em><strong> M1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{R^2} = 0.974">
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0.974</mn>
</math></span><strong><em><span style="font-family: 'Verdana',sans-serif;"> A1</span></em></strong></span></p>
<p style="text-align: start;"><strong><em><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">[1 mark]</span></em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Since the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{R^2}">
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> is close to +1, the model is a good fit for the data. <em><strong> R1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T = 28 - \left( {22.69 \ldots } \right){\left( {0.9250 \ldots } \right)^3} = 10.0">
<mi>T</mi>
<mo>=</mo>
<mn>28</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>22.69</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.9250</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>10.0</mn>
</math></span> minutes <em><strong> M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A newspaper vendor in Singapore is trying to predict how many copies of <em>The Straits Times</em> they will sell. The vendor forms a model to predict the number of copies sold each weekday. According to this model, they expect the same number of copies will be sold each day.</p>
<p>To test the model, they record the number of copies sold each weekday during a particular week. This data is shown in the table.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAArMAAABHCAYAAAAUe889AAAakElEQVR4Ae2agbEkt82EncdfTsClBFyKwIrAisCKwIrAysCKwIrAisCKwMrgMrgM7q9v5Xb14Tg75LzhPM66WbU3OyQIgN0AiH3S7z5lBIEgEASCQBAIAkEgCASBmyLwu5v6HbeDQBAIAkEgCASBIBAEgsCnNLMJgiAQBIJAEAgCQSAIBIHbIvBZM/v73//fp3yCQWIgMZAYSAwkBhIDiYHEwMox4J33F82sL+Z7EHhFBEjOjHUQCB/hYh0E1vIkubEOH+FibS7SzK7DTzy5CIEUpYuA7jQTPjqBukAsXFwA8oCJ8DEA1mTRcDEZ4AH1LS7SzA4AGNHXQKCVCK9xsnueInysw1u4WIcLPAkf6/ARLtbmIs3sOvzEk4sQSFG6COhOM+GjE6gLxMLFBSAPmAgfA2BNFg0XkwEeUN/iIs3sAIARfQ0EWonwGie75ynCxzq8hYt1uMCT8LEOH+FibS7SzK7DTzy5CIEUpYuA7jQTPjqBukAsXFwA8oCJ8DEA1mTRcDEZ4AH1LS7SzA4AGNHXQKCVCK9xsnueInysw1u4WIcLPAkf6/ARLtbmIs3sOvzEk4sQSFG6COhOM+GjE6gLxMLFBSAPmAgfA2BNFg0XkwEeUN/iIs3sAIARfQ0EWonwGie75ynCxzq8hYt1uMCT8LEOH+FibS7SzK7DTzy5CIEUpYuA7jQTPjqBukAsXFwA8oCJ8DEA1mTRcDEZ4AH1LS4ON7PfffeXx69GlPrn22///OnXX/894FZEg8C1CLQS4UwPiH/lRCsXPnz48HT9TF+++eZPn3744W9nqjxd1ww+fvzx7//FWFzUZ4ub0w/XULgyJ71c/PLLvx748vSh+e+//6tPf1LM//TTPz6b7335+PHjw97PP/+zd8vpctgGH3y5avTysedPbz5QK7766g976qavr8B3PeRZXFS9e+9wgu3Wh1qyNfa4fI943vJ1dL7FxenNLEZIBopXRhBYEYFWIpzpp5pZ8oBLpA7mWMOP2Q3Vyo2TcJnNB3bAe5WmfmVORrhoxbdi++uv/yh6H0+aWHQfvRdWaG7e4/If4eMzwHdetvJhrwHaUXva8gp818PM4qLaqe+zOHmPeK5nO/re4uLNzaxf1hQq/jKLoWe/GI4eIPuCwBkItBLhDL3SoWaWv07VSx0ZckP/ZSPN7DX/X+DW5S3Orny+SjNLrefjg3jnTiDH/K+2W7nge599X6G5eY/Lf1at2sqHWY3TM25bayvwXf2axUW1U99ncfIe8VzPdvS9xcWpzSyO6T8nYUy/wr3JZZ4PlzkBy4fEYs6LnwpiLZZHD599QUAIEGszh5pZigWx7Q0rucCcComvUbSUH8j4D0XJM6d8oSli3oeaZOVYbZzcBjKsywf0su5D/9nYc9PXz/g+mw98bJ2tdWG25lSL8LPygm4aNda21vc4QT9NoOvQf46n/sFRHS0/qsyRd3zoHYpFySu2eVdTqzXePbY4n5+59b8lcHZhIg4U7+gCF8ceed052CWuXUddRxc6ZKO1Lh95yhYxwsCWc4sedLAubCQrHFjn0ztG+OjViVwrH5gHV9b8XDoT6638qHPiBT34D9d7WIMldsRF5fsZ1qpRig3hIA70/tbnLC72/BInz+SQAT/FM7Fa94HPe8fzszOMrLW4OL2ZxSEBpgtQACtQ9VQBU+IAvoZ0EJAZQeBMBFqJcKZ+LlFsUDyIcY9hvjPHGjJqJMkBYl7v+s+y2it5v1j4zh4NdJBrutBrXpFfyOuCRY539DC0Ln088dVt+NpZ32fzgZ+ty7tewsjVOTBxTOEBXeKFJ/gIc/Gm2rfHiXiVPD7ABzYYWldcMCcbsvkQPOmfES7URMgP/FIsETf6zjp6dUZhyJPBOhiDlYbHJZzoDtEeeEEndqSDPXpnj/OEDP54LNeYwIZ8Vg6LZ96Rxya6GS7Pu7hiT8u+cACn3jHCR69O5OrZtXcLV3HDufBJPLCvzoEL+jkvn5a9it0e31XesUY/6/LxYfA/fYjiQXNvec7iYs8nOAHPZ0O8KbbA3fetEs/PzjCy1uJiSjNLYNWAd0cJMNYVfApMFRolPQSqcPj+fA8Cb0GglQhv0Vf3qnAQ13wU18iRG5rHD2QV7ypE0keeaK9yRA0BMsijgxzZ0rF1ackGOSgb8tv9YL8udO05+zmbD/xt4VAvYeTqHPscD2TUwPJdHLXq1FFO0A8m7GfAj1/KNFxquh4CJ/4zyoXjg0/CiidrDP/OO3J+Hua8MUbez+/r5AHDL+rHxH+4IL8YzpHWecpfcbMV2+LV94qXFteSQ7/O5rnFOvuFieT3nqN87OnTOn6AYR1buKpG1Pxgf52DA4/PPax7+K5+8u5YixvljOLJ62VLx8jcLC72fIATbNePx1KLN59bJZ73ztq73uJiSjNL4GPMA4kiBLgkuEhRM8sBtIcLVYHp672HjFwQ2EOglQh7e0bW1RTq4iW21bSqAKk5Zd6/ux3lAZeFZPwi9TkVb/T54GIh7zRY550Pa2Ahn5Bhjg9D+qtO6TrrOZsP/OSMjgNz9RKuc5xbtao+hRmXp2qXsK4XasVPcsIPP8QJTYBsaR9r2EOOD9+JjRljlAtvTPFLZ8c33olL/PfmhnmdsT6R52zI+KhcoVNxKjmf83um2hB2yGuNy17z6MPfev8oH/BFg0YMPXx0Lu2TPE8G/mpN+/ee+Ddj4Cs+16Fz+LzPVR6Qq3Otc6JjC+sevrHzDGt88DNhj7w8c8ziYs9Hx39LtiXjc6vE85b/o/MtLk5vZhVUGFNhA0jeAZcCzZN3T2zNEdgkA+sEb0YQOBsBYmvmUBOkS0wXJbGtv9rookPWv7tfyOMrOSUZvmv4XE8zK334QL6RX+SgNw7MYRO/WKsNg2yf+ZzNB776RSffwRLb4Kjhc+IRbPcGOsBUjS049nAiGWok++FIPGGfQR3FT61xFo+DPd9G1ke54JycmXPUWOFM+Kyn/MB/5rcGax6TyDkvvINVtedzxC52ewb7kOfs0sle5nx4vjGPjOIKHfDFu+8DG94VSx5rrnvr+ygfW3rqvPyu85yjYu9zlQf21zkwdAzcRgvrHr57sKaugTf+8MTWmWMWF3s+Ov5bsi0Zn1slnrf8H51vcXF6MwtoXhRUrEkQgowPwY6MB7zLsVYTavSwkQ8CWwi0EmFL9sh8vbiIbYorH11muhiRVbNCY+BDxZk5yZM/Gj7HPDlTdTCnos732kSQizXXdBG05GX7zOdsPvDVcZDvYIZtx0x1CGyFacVM+7ee1EBw1X7XX32hBiLvA77wi9jQQA6dyFZ5yZzxHOVCsUusVpx4x1ewB1cNnUPv9am41h9DWHdeeAcj9PjwOb7XuHbZ1ne3wX7ywAfnAR94VY77ucS332vsUeyN+oPtUT7c32ff5VOVaeHmc5wRnzymHTf0wYtjUG3w7nv2+O7FWnL4i4/OTcuH0blZXOz54fhvybZkfI7vK8Tzlv+j8y0u3tzMorR+SBQVIhU7l2Gd91qUAVtyFMeMIDADAWJs5lBRpUhrENteTFTAkWWocdU7lwV+qkGQPJeJRp2TDuUe7+igkDGw7znHPOv1klURZE26ZHPGEzuzx9bl7ZiALfjgj7gDQ/bqHTyQ0WWtJlO8sI5OYb7Hida1H95VHxULYCOuWfNG4mzcjnDBefHL/cUvmgnm+fiosc0aOKJHwxveFi/g+6yZhQfseryrGcVPPpxV+YVdxT32tF88Io9/7NG673cfFRvoZF4YwPXoOMJHjw180tlcXhg8mwMH4ernVo7UZnYPa2w94xsuerCWHs7mseRnecv3WVzs+dTipO5pyfjcKvFc/T763uLi9GaWICd4fah44YDW+U7Q+VCxYe3sX1VuJ9//txFoJcKZiKh4q7ijm8Lil5maE88Vj3+KsV+0kufy0GjNYYfz8VGjxRwDW1wavi6bFDsNXR61WdD62c/ZfODv1uXtmCAjPJw7zQk35xE+wFlrPIW3cHrGiTcD7KU+yl5tWokJfPQYkI2znvgwOtSQt/Z54+PrnI014ca5PQYrLsJEvIBpjc86B7folQ3ktR9fWj74uscGvnJOdAl/9sOH9ONjbcqxo32u27F49h3dM8ZWPoAhaz7qnOOCbOUGnL2hR9ce1nt892KNHJjh89ljFhd7flb8W/ItmTrnvL1XPLd8PzLX4uJwM3vEgb09CkSAzggCsxBoJcIsW3fUy8WiS+oK/8NHH8qtJqFvZ79UuOjHqleSpuLonRY+elH+TY4fDGBG43b2CBe/IfqWeD6LkxYXSzWzFGucBKyMIDALgVYizLJ1R738peXo5XvkvOFjHzX9P4YzLmm3Hi4cjbd/54chuUROHRnhYww1/hJf/yo8pmFbOlz89r/NvCWet9EdW2lxsUQzS/OKc3xmBeIYVJF+ZQRaifDK5+09Gxcv2PBX2SP/SbTXTpULHxWRz9/1I/9oQ/S5tudv4eI5PiOrutdosI6O8NGHnP4iS674/67St7tP6n+dizPiuQ/pfakWF0s0s/uuRyIInIdAKxHO0x5NowiEj1HE5smHi3nYHtEcPo6gNmdPuJiD6xGtLS7SzB5BMntujUArEW59oJs7Hz7WITBcrMMFnoSPdfgIF2tzkWZ2HX7iyUUIpChdBHSnmfDRCdQFYuHiApAHTISPAbAmi4aLyQAPqG9xkWZ2AMCIvgYCrUR4jZPd8xThYx3ewsU6XOBJ+FiHj3CxNhdpZtfhJ55chECK0kVAd5oJH51AXSAWLi4AecBE+BgAa7JouJgM8ID6FhdpZgcAjOhrINBKhNc42T1PET7W4S1crMMFnoSPdfgIF2tz8UUzC2H5BIPEQGIgMZAYSAwkBhIDiYFVY8Db6y+aWV/M9yDwigiQmBnrIBA+wsU6CKzlSXJjHT7CxdpcpJldh594chECKUoXAd1pJnx0AnWBWLi4AOQBE+FjAKzJouFiMsAD6ltcpJkdADCir4FAKxFe42T3PEX4WIe3cLEOF3gSPtbhI1yszUWa2XX4iScXIZCidBHQnWbCRydQF4iFiwtAHjARPgbAmiwaLiYDPKC+xUWa2QEAI/oaCLQS4TVOds9ThI91eAsX63CBJ+FjHT7CxdpcpJldh594chECKUoXAd1pJnx0AnWBWLi4AOQBE+FjAKzJouFiMsAD6ltcpJkdADCir4FAKxFe42T3PEX4WIe3cLEOF3gSPtbhI1yszUWa2XX4iScXIZCidBHQnWbCRydQF4iFiwtAHjARPgbAmiwaLiYDPKC+xUWa2QEAI/oaCLQS4TVOds9ThI91eAsX63CBJ+FjHT7CxdpcpJldh594chECKUoXAd1pJnx0AnWBWLi4AOQBE+FjAKzJouFiMsAD6ltcHG5mv/vuL49fjd9886fPXPj1138/5r/66g+fzc94+fjx48MWB+P7KgMMwAW/+Pz00z/e1TW4wA/8Wn30+KrY++GHvx06DljcdSi/FFt61jzU+cBoa00y7/28Mx/UnW+//fN/c/3HH//+GZy8iyN/Ho3dz5RPeLkzF8ABrsKZOtEayHz99R8//fzzP1vLS83dmY96D9bccKBZgxPOu+o9tToXxDMxv1XvwZha1ao9I1w5b+/1vcXFm5tZlHpR0GX7v9zMqtkCm4rPe5Df0yC+h18tmz2+Ct9WUrZ01rlWIlSZVd85cy3233//10+ti+KXX/71iL+t4rbKGe/MB5eDmiZ4IX6dC9bqD20ubbhZcdyZC3An1sGbD9/hxwe5whn5+L3lMit9vysf4E8uKDc+fPjwaFY9N8BZPJETdW0lHvBlZS7AV3Hdqvf8QU3rrXuTPEEHQzny3n+Ee8Z/i4tTmlkCUSPN7KdHEQXs1ZNTnK30TDP7nI3ayCJN/rXm1Wi1ittzK9eutgrTtR4cs6YfC7oE0ELOE8MMNVWuHZ68XvraCt/vygXYgbtfwOKn/nDQHZVmdl7EgS2x5HWJxlbNrSxTm/iQK6uPO+QG+D6r9+RIbWbJGecJHqhR9YfgSvy0uHhzM6vmQ42bCoUKOgAALsYlwxyg+5zeeQIia+hAHx/AZQ5dujxIAOakR74gU8mBQMmiy4ue1rAtHVuFDpv65YI+5P1cOqtssb41dPFJtgYZZxAWyKDb/VLBYF5nQI49Xhx0JsfE5Sse+OtnlE7fX8+EL/Xsjgvye9ghs+crZxMmFa/q09Y753mVwUUNf3WADWvENLysPO7KB/GN755rWw2U8GcPubXquCsX1CZ89/oIxrU+M7cluyInd+fDazR12+8E1uBH9/mK+LtPd+Bir96Dt3Pi5/Pv6sN8bqXvLS7e3MwSnADEh6KuQsG7hpocD2TAwiHN6Z05/3BRo8vnSAoG9nzev8sf5GpjJjkVPsjVnJ7113w9i+T01AWls2recZAOnpxbMv5UoJHg9dySUyOuZlbz/vQGRnrUjB7BA92tpomzwINsuA98F7/IVWwkK+yQkZ49X9krrNg3Mtj7KgPsHGPORewKm73itgIOd+WDPMR3rxWqf6otFV9yyOXr+nu/35UL6iW+11ygnigXhO0eR5Jb4XlXPsBO9wyc8Kk8kAv+xwnet/ImXPQhsFfvW/nQ0gwXNZdacu8118qLU5pZDo1ygleFAtA01MQ4OGpeNad3QPSmGL2sMWRHur2ZVaJQ1NDBPuRV5NjDd4YuITXF7EVetiX3ELZ/ZB85NVuaY7/mWuc1NY+v+CMfmeCC411nEx74yDkZmsM+w5tZFQGf06UpW/jXg4fsCNOHsSf/iHP8l69VXDjtYee+okPv8gVb6MCW5qqtvXf2vspwPHUmxTXvcOk/bCSz0vOufBDrxKfnqC5w1QLHWbHrc6t9vysX4Eicez6ozlN7fKheqWb62mrf78wHWOouqTVI9x35wiCXkCGftu6Q9+bmDlzs1Xvw3bs3yY/K13tjX+23uDilmcWQGgwVEEDTABiMe1FRkGtO7wKagGYPHxUdNWrS7TKeALpQ0KU90uVP6UGOeSWW/K5P+Sifte5nZ651XsnyVDHFpvvtMvjGuhpS1tSIMs93na0GXvVHurCrPeioH+GhQqN1Lmsw2mry8Q0ZyfNE3nHqxc593cLJ+XXMer/j3ysMeIJrH2ADbhrgXuNDa6s878wH+aR8Iwf4VE6EM/mwV2Mk+17PO3NBfVIdIuZVczwfwFV1Be5WH3fmA5zhgLjnHHCj+073kHOje4ceYsVxBy726j33K3fzs4EO5+WZ7Huttbg4rZlVcKoZUWPEYVvNnYqOGh4VHgHtjaqAdRvodRklCfPe7GgPh2992IdN1mR7iyD5KJ8lp8tMSdg6r2R5ch754n67jHDsaWbr5Vn9kS7s9uCBH8JF2OAvep41tJwfeXHLHl3evdi5r1s4Ob+OWe93/HqFAQ4ei3CmuGo9XXal878KH2BK7m3VEdY8n1fiQL68EhfUnNYPOdUVauHq4658UIuo5Yp37g3iH04Yaly1zpzqV+rU8ajcinlphJOt+oQM2N81L05rZgHCmxhA09A8TwbFRE2LAlfNjoBWYJPMyDPUiEm3y6hpUtKwj+aKd76zR3oeyuwfbCIj27b02Vd8RY6klC7NMa9Gb6+ZRanOL5uOCXocM87JEEbYZwgPbAtHn1OhkC1s9ODxUF7+wabbKctfvII98uJKOO1h576iVO8tnDT3hfGdCfx6heFYbp1nr7ht7bty/lX4eIY1uae8vRLbUVuvwgX1htqhmuw4wAXnvOul7WdZ9bvqveMvTuQz/DCnobtJ95bmV3neITee1SBwBPOtexMu9Ae5VTDf8qPFxanNrIoEhgBNA/CYa30UzGrUBLQ3quhlqFGTbpepurk4ahNYZbDJkH+y/Zhs/IM+9FY9vKuZZltPM8u5W3rkk365tmQUcMKjJaMfDvgDXsgIR2Fd98m2Gum67jocnme+SmcvdtVXcdPyZY8v99G/o+vuA+5bf3Wq5wL/Hrm678r3u/MBF+QMOKvmVPyI1aPxWnXNfL87FzRExDx1RPWu4qV6pTpa11d6vysf6gVU/8GUHPF37kDniTW/t1biAV/uwAX4PfvRDN7eqwhjaph6MZ+70w+LU5tZQAAoSAc0DQo8IDPPBxkCl+8CUO8q+OyRvIqSmjfpdhklBnu4VLRHPsgv6XRCscm8bGtP64lN14UvOoPksY++Oq91Pd1n5N0nZDiD44ZeMNAQHszrDOgBS/zUwEfmHRM/Q8u262MdP9y2dOvJmvvKnopnD3YtX90XbMhO1S9f9p74dvcBf3vxxRmJBeJj5XFXPsgnfG/VgIo3F8yqF4P7elcuOINqB7nh9c/PR85wRn2O1hDXOfP7nfngx4L/8afeb+DmfNR7ayauR3SvzoXiX7HtOaBapTW/E5wDrfN81hQfwe/MPfhXx+FmtirK+/UIeDN7vfX7Wmwlwn1Pc3/Pw8c6HIaLdbjAk/CxDh/hYm0u0syuw8+wJ2lmhyF7bEhROobbrF3hYxay43rDxThmM3eEj5nojukOF2N4zZRucZFmdibik3WnmT0GcCsRjmnKrjMQCB9noHiOjnBxDo5naQkfZyH5dj3h4u0YnqWhxUWa2bPQjZ7bINBKhNs4/4KOho91SA0X63CBJ+FjHT7CxdpcpJldh594chECKUoXAd1pJnx0AnWBWLi4AOQBE+FjAKzJouFiMsAD6ltcpJkdADCir4FAKxFe42T3PEX4WIe3cLEOF3gSPtbhI1yszUWa2XX4iScXIZCidBHQnWbCRydQF4iFiwtAHjARPgbAmiwaLiYDPKC+xUWa2QEAI/oaCLQS4TVOds9ThI91eAsX63CBJ+FjHT7CxdpcpJldh594chECKUoXAd1pJnx0AnWBWLi4AOQBE+FjAKzJouFiMsAD6ltcfNHMIpRPMEgMJAYSA4mBxEBiIDGQGFg1Brz//ayZ9YV8DwJBIAgEgSAQBIJAEAgCqyOQZnZ1huJfEAgCQSAIBIEgEASCwCYCaWY3oclCEAgCQSAIBIEgEASCwOoIpJldnaH4FwSCQBAIAkEgCASBILCJQJrZTWiyEASCQBAIAkEgCASBILA6Av8PPr6i1KriPqUAAAAASUVORK5CYII="></p>
<p>A goodness of fit test at the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> significance level is used on this data to determine whether the vendor’s model is suitable. The critical value for the test is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>49</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an estimate for how many copies the vendor expects to sell each day.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the null and alternative hypotheses for this test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the degrees of freedom for this test.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the conclusion to the test. Give a reason for your answer.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfrac><mrow><mn>74</mn><mo>+</mo><mn>97</mn><mo>+</mo><mn>91</mn><mo>+</mo><mn>86</mn><mo>+</mo><mn>112</mn></mrow><mn>5</mn></mfrac></mfenced><mo>=</mo><mn>92</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mtext>:</mtext></math> The data satisfies the model <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub><mtext>:</mtext></math> The data does not satisfy the model <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Do not accept “<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mtext>:</mtext></math> The same number of copies will be sold each day” but accept a similar statement if the word ‘expect’ or ‘expected’ is included. Similarly for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub></math>. </p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><msup><mi>χ</mi><mn>2</mn></msup><mpadded lspace="-1px"><mi>calc</mi></mpadded></msub><mo>=</mo><mn>8</mn><mo>.</mo><mn>54</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>8</mn><mo>.</mo><mn>54347</mn><mo>…</mo></mrow></mfenced></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0736</mn><mo> </mo><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>0735802</mn><mo>…</mo><mo>)</mo><mo> </mo></math> <em><strong>A2</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>54</mn><mo><</mo><mn>9</mn><mo>.</mo><mn>49</mn></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0736</mn><mo>></mo><mn>0</mn><mo>.</mo><mn>05</mn></math> <strong><em>R1</em></strong></p>
<p>therefore there is insufficient evidence to reject <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math> <em><strong>A1</strong></em></p>
<p>(i.e. the data satisfies the model)</p>
<p><br><strong>Note:</strong> Do not award <em><strong>R0A1</strong></em>. Accept “accept” or “do not reject” in place of “insufficient evidence to reject”. <br> Award the <em><strong>R1</strong></em> for comparing their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>05</mn></math> or their <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>χ</mi><mn>2</mn></msup></math> value with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>49</mn></math> and then <em><strong>FT</strong></em> their final conclusion.</p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>Two unbiased tetrahedral (four-sided) dice with faces labelled 1, 2, 3, 4 are thrown and the scores recorded. Let the random variable <em>T</em> be the maximum of these two scores.</p>
<p>The probability distribution of <em>T</em> is given in the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>a</em> and the value of <em>b</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the expected value of <em>T</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{3}{{16}}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{5}{{16}}">
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)A1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for consideration of the possible outcomes when rolling the two dice.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( T \right) = \frac{{1 + 6 + 15 + 28}}{{16}} = \frac{{25}}{8}\left( { = 3.125} \right)">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>6</mn>
<mo>+</mo>
<mn>15</mn>
<mo>+</mo>
<mn>28</mn>
</mrow>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>25</mn>
</mrow>
<mn>8</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>3.125</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Allow follow through from part (a) even if probabilities do not add up to 1.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The cars for a fairground ride hold four people. They arrive at the platform for loading and unloading every <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> seconds.</p>
<p>During the hour from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> am the arrival of people at the ride in any interval of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> minutes can be modelled by a Poisson distribution with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mi>t</mi><mo> </mo><mfenced><mrow><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mn>60</mn></mrow></mfenced></math>.</p>
<p>When the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> am car leaves there is no one in the queue to get on the ride.</p>
<p>Shunsuke arrives at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>01</mn></math> am.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> people arrive at the ride before Shunsuke.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability there will be space for him on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>01</mn></math> car.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the number of people who arrive between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>00</mn></math> am and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>01</mn></math> am</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>Po</mtext><mfenced><mn>9</mn></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>></mo><mn>7</mn></mrow></mfenced><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≥</mo><mn>8</mn></mrow></mfenced></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>676</mn><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>67610</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Mean number of people arriving each <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> seconds is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn></math> <strong>(M1)</strong></p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub></math> be the number who arrive in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> seconds and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>2</mn></msub></math> the number who arrive in the second <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> seconds.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>(Shunsuke will be able to get on the ride)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>≤</mo><mn>4</mn></mrow></mfenced><mo>×</mo><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>2</mn></msub><mo>≤</mo><mn>3</mn></mrow></mfenced><mo>+</mo><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>5</mn></mrow></mfenced><mo>×</mo><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>2</mn></msub><mo>≤</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>6</mn></mrow></mfenced><mo>×</mo><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>2</mn></msub><mo>≤</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>7</mn></mrow></mfenced><mo>×</mo><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow></mfenced></math> <strong>M1</strong><strong>M1</strong></p>
<p> </p>
<p><strong>Note: M1</strong> for first term, <strong>M1</strong> for any of the other terms.</p>
<p> </p>
<p>null <strong>(A1)(A1)</strong></p>
<p> </p>
<p><strong>Note: (A1)</strong> for one correct value, <strong>(A1)(A1)</strong> for four correct values.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>221</mn><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>220531</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[6 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Chloe and Selena play a game where each have four cards showing capital letters A, B, C and D.<br>Chloe lays her cards face up on the table in order A, B, C, D as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_14.39.35.png" alt="N17/5/MATHL/HP1/ENG/TZ0/10"></p>
<p>Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Chloe’s card directly above.<br>Chloe wins if <strong>no</strong> matches occur; otherwise Selena wins.</p>
</div>
<div class="specification">
<p>Chloe and Selena repeat their game so that they play a total of 50 times.<br>Suppose the discrete random variable <em>X </em>represents the number of times Chloe wins.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability that Chloe wins the game is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{8}"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the mean of <em>X</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the variance of <em>X</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p>number of possible “deals” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4! = 24"> <mo>=</mo> <mn>4</mn> <mo>!</mo> <mo>=</mo> <mn>24</mn> </math></span> <strong><em>A1</em></strong></p>
<p>consider ways of achieving “no matches” (Chloe winning):</p>
<p>Selena could deal B, C, D (<em>ie</em>, 3 possibilities)</p>
<p>as her first card <strong><em>R1</em></strong></p>
<p>for each of these matches, there are only 3 possible combinations for the remaining 3 cards <strong><em>R1</em></strong></p>
<p>so no. ways achieving no matches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3 \times 3 = 9"> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo>=</mo> <mn>9</mn> </math></span> <strong><em>M1A1</em></strong></p>
<p>so probability Chloe wins <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{9}{{23}} = \frac{3}{8}"> <mo>=</mo> <mfrac> <mn>9</mn> <mrow> <mn>23</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </math></span> <strong><em>A1AG</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>number of possible “deals” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4! = 24"> <mo>=</mo> <mn>4</mn> <mo>!</mo> <mo>=</mo> <mn>24</mn> </math></span> <strong><em>A1</em></strong></p>
<p>consider ways of achieving a match (Selena winning)</p>
<p>Selena card A can match with Chloe card A<em>, </em>giving 6 possibilities for this happening <strong><em>R1</em></strong></p>
<p>if Selena deals B as her first card, there are only 3 possible combinations for the remaining 3 cards. Similarly for dealing C and dealing D <strong><em>R1</em></strong></p>
<p>so no. ways achieving one match is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 6 + 3 + 3 + 3 = 15"> <mo>=</mo> <mn>6</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>15</mn> </math></span> <strong><em>M1A1</em></strong></p>
<p>so probability Chloe wins <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 - \frac{{15}}{{24}} = \frac{3}{8}"> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <mn>15</mn> </mrow> <mrow> <mn>24</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </math></span> <strong><em>A1AG</em></strong></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>systematic attempt to find number of outcomes where Chloe wins (no matches)</p>
<p>(using tree diag. or otherwise) <strong><em>M1</em></strong></p>
<p>9 found <strong><em>A1</em></strong></p>
<p>each has probability <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{4} \times \frac{1}{3} \times \frac{1}{2} \times 1"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>×</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>×</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>1</mn> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{24}}"> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>24</mn> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p>their 9 multiplied by their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{24}}"> <mfrac> <mn>1</mn> <mrow> <mn>24</mn> </mrow> </mfrac> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{3}{8}"> <mo>=</mo> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \sim {\text{B}}\left( {50,{\text{ }}\frac{3}{8}} \right)"> <mi>X</mi> <mo>∼</mo> <mrow> <mtext>B</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>50</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = np = 50 \times \frac{3}{8} = \frac{{150}}{8}{\text{ }}\left( { = \frac{{75}}{4}} \right){\text{ }}( = 18.75)"> <mi>μ</mi> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo>=</mo> <mn>50</mn> <mo>×</mo> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>150</mn> </mrow> <mn>8</mn> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>75</mn> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>18.75</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\sigma ^2} = np(1 - p) = 50 \times \frac{3}{8} \times \frac{5}{8} = \frac{{750}}{{64}}{\text{ }}\left( { = \frac{{375}}{{32}}} \right){\text{ }}( = 11.7)"> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mi>n</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>50</mn> <mo>×</mo> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> <mo>×</mo> <mfrac> <mn>5</mn> <mn>8</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>750</mn> </mrow> <mrow> <mn>64</mn> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>375</mn> </mrow> <mrow> <mn>32</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>11.7</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A company produces bags of sugar with a labelled weight of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mtext>kg</mtext></math>. The weights of the bags are normally distributed with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mtext>kg</mtext></math> and a standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mtext>g</mtext></math>. In an inspection, if the weight of a randomly chosen bag is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>950</mn><mo> </mo><mtext>g</mtext></math> then the company fails the inspection.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the company fails the inspection.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A statistician in the company suggests it would be fairer if the company passes the inspection when the mean weight of five randomly chosen bags is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>950</mn><mo> </mo><mtext>g</mtext></math>.</p>
<p>Find the probability of passing the inspection if the statistician’s suggestion is followed.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the weight of sugar in the bag</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo><</mo><mn>950</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>308537</mn><mo>…</mo><mo>≈</mo><mn>0</mn><mo>.</mo><mn>309</mn></math> <em><strong>(M1)A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>X</mi><mo>¯</mo></mover></math> be the mean weight of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> bags of sugar</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mover><mi>X</mi><mo>¯</mo></mover></mfenced><mo>=</mo><mn>1000</mn></math> <em><strong>(A1)</strong></em></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mover><mi>X</mi><mo>¯</mo></mover></mfenced><mo>=</mo><mfrac><msup><mi>σ</mi><mn>2</mn></msup><mi>n</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mover><mi>X</mi><mo>¯</mo></mover></mfenced><mo>=</mo><mfrac><msup><mn>100</mn><mn>2</mn></msup><mn>5</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>2000</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>X</mi><mo>¯</mo></mover><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>1000</mn><mo>,</mo><mo> </mo><mn>2000</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mover><mi>X</mi><mo>¯</mo></mover><mo>></mo><mn>950</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>868223</mn><mo>…</mo><mo>≈</mo><mn>0</mn><mo>.</mo><mn>868</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>86</mn><mo>.</mo><mn>8</mn><mo>%</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> be the total weight of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> bags of sugar</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>T</mi></mfenced><mo>=</mo><mn>5000</mn></math> <em><strong>(A1)</strong></em></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><msub><mi>X</mi><mn>2</mn></msub></mrow></mfenced><mo>=</mo><mtext>Var</mtext><mfenced><msub><mi>X</mi><mn>1</mn></msub></mfenced><mo>+</mo><mtext>Var</mtext><mfenced><msub><mi>X</mi><mn>2</mn></msub></mfenced></math> for independent random variables <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mi>T</mi></mfenced><mo>=</mo><mn>5</mn><mo>×</mo><msup><mn>100</mn><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>50</mn><mo> </mo><mn>000</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>5000</mn><mo>,</mo><mo> </mo><mn>50</mn><mo> </mo><mn>000</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>T</mi><mo>></mo><mn>4750</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>868223</mn><mo>…</mo><mo>≈</mo><mn>0</mn><mo>.</mo><mn>868</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>86</mn><mo>.</mo><mn>8</mn><mo>%</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) was straightforward, and a good number of candidates showed their knowledge in achieving a correct answer. Candidates are advised to not use calculator notation, as examiners cannot be familiar with all variations of GDC syntax; instead, correct mathematical notation and/or a written commentary will ensure the method is communicated to the examiner. Rounding errors once again caused problems for some. Good answers to part (b) were much less common and this was a challenging question for many. A few understood how to use the central limit theorem to find the sampling distribution of the sample mean and a few used the mean and variance of the sum of independent random variables.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The discrete random variable <em>X</em> has the following probability distribution, where<em> p</em> is a constant.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>p</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>μ</em>, the expected value of <em>X</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find P(<em>X</em> > <em>μ</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>equating sum of probabilities to 1 (<em>p</em> + 0.5 − <em>p</em> + 0.25 + 0.125 + <em>p</em><sup>3</sup> = 1) <em><strong>M1</strong></em></p>
<p><em>p</em><sup>3</sup> = 0.125 = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{8}">
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
</math></span></p>
<p><em>p</em>= 0.5 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>μ</em> = 0 × 0.5 + 1 × 0 + 2 × 0.25 + 3 × 0.125 + 4 × 0.125 <em><strong> M1</strong></em></p>
<p>= 1.375 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \frac{{11}}{8}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>11</mn>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>P(<em>X</em> > <em>μ</em>) = P(<em>X</em> = 2) + P(<em>X</em> = 3) + P(<em>X</em> = 4) <em><strong>(M1)</strong></em></p>
<p>= 0.5 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award follow through <em><strong>A</strong></em> marks in (b)(i) from an incorrect value of <em>p</em>.</p>
<p><strong>Note:</strong> Award <em><strong>M</strong> </em>marks in both (b)(i) and (b)(ii) provided no negative probabilities, and provided a numerical value for <em>μ</em> has been found.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The sex of cuttlefish is difficult to determine visually, so it is often found by weighing the cuttlefish.</p>
<p>The weights of adult male cuttlefish are known to be normally distributed with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mtext>kg</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<p>The weights of adult female cuttlefish are known to be normally distributed with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo> </mo><mtext>kg</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<p>A zoologist uses the null hypothesis that in the absence of information a cuttlefish is male.</p>
<p>If the weight is found to be above <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>kg</mtext></math> the cuttlefish is classified as female.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>%</mo></math> of adult cuttlefish are male.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of making a Type I error when weighing a male cuttlefish.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of making a Type II error when weighing a female cuttlefish.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of making an error using the zoologist’s method.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>(Type I error) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>P</mtext></math>(stating female when male) </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>P</mtext><mfenced><mrow><msub><mi>W</mi><mrow><mi>M</mi><mi>a</mi><mi>l</mi><mi>e</mi></mrow></msub><mo>></mo><mn>11</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>00135</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00134996</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>(Type II error) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>P</mtext></math>(stating male when female) </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>P</mtext><mfenced><mrow><msub><mi>W</mi><mrow><mi>F</mi><mi>e</mi><mi>m</mi><mi>a</mi><mi>l</mi><mi>e</mi></mrow></msub><mo><</mo><mn>11</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>309</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>308537</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use the total probability <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>(error) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>9</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>00134996</mn><mo>…</mo><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>308537</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0321</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0320687</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This was a straightforward problem on Type I and Type II errors which some candidates answered successfully in a couple of lines but many candidates were unable to do the correct calculations.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider two events, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( A \right) = {\text{P}}\left( {A' \cap B} \right) = 0.4">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩<!-- ∩ --></mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.4</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cap B} \right) = 0.1">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∩<!-- ∩ --></mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By drawing a Venn diagram, or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cup B} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are not independent.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><img src="data:image/png;base64,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"> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for a Venn diagram with at least one probability in the correct region.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cap B'} \right) = 0.3">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∩</mo>
<msup>
<mi>B</mi>
<mo>′</mo>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.3</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cup B} \right) = 0.3 + 0.4 + 0.1 = 0.8">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.3</mn>
<mo>+</mo>
<mn>0.4</mn>
<mo>+</mo>
<mn>0.1</mn>
<mo>=</mo>
<mn>0.8</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( B \right) = 0.5">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>B</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.5</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cup B} \right) = 0.5 + 0.4 - 0.1 = 0.8">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.5</mn>
<mo>+</mo>
<mn>0.4</mn>
<mo>−</mo>
<mn>0.1</mn>
<mo>=</mo>
<mn>0.8</mn>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( A \right){\text{P}}\left( B \right) = 0.4 \times 0.5">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>B</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.4</mn>
<mo>×</mo>
<mn>0.5</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 0.2 <em><strong>A1</strong></em></p>
<p>statement that their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( A \right){\text{P}}\left( B \right) \ne {\text{P}}\left( {A \cap B} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>B</mi>
<mo>)</mo>
</mrow>
<mo>≠</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for correct reasoning from their value.</p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> not independent <strong><em>AG</em></strong></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\left. A \right|B} \right) = \frac{{{\text{P}}\left( {A \cap B} \right)}}{{{\text{P}}\left( B \right)}} = \frac{{0.1}}{{0.5}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mi>A</mi>
<mo>|</mo>
</mrow>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>B</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.1</mn>
</mrow>
<mrow>
<mn>0.5</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 0.2 <em><strong>A1</strong></em></p>
<p>statement that their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\left. A \right|B} \right) \ne {\text{P}}\left( A \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mi>A</mi>
<mo>|</mo>
</mrow>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>≠</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
</math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for correct reasoning from their value.</p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> not independent <strong><em>AG</em></strong></p>
<p><strong>Note:</strong> Accept equivalent argument using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\left. B \right|A} \right) = 0.25">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mi>B</mi>
<mo>|</mo>
</mrow>
<mi>A</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.25</mn>
</math></span>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>It is believed that the power <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> of a signal at a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> km from an antenna is inversely proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>d</mi><mi>n</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<p>The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is recorded at distances of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><mi mathvariant="normal">m</mi></math> and the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>d</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>P</mi></math> are plotted on the graph below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The values of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>d</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>P</mi></math> are shown in the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why this graph indicates that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is inversely proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>d</mi><mi>n</mi></msup></math>.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the least squares regression line of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>P</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>d</mi></math>.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your answer to part (b) to write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> to the nearest integer.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p>a straight line with a negative gradient <strong>A1A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mo> </mo><mi>P</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>.</mo><mn>040</mn><mo>…</mo><mo> </mo><mi>log</mi><mo> </mo><mi>d</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>12632</mn><mo>…</mo><mo>≈</mo><mo>-</mo><mn>2</mn><mo>.</mo><mn>04</mn><mo> </mo><mi>log</mi><mo> </mo><mi>d</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>126</mn></math> <strong>A1A1</strong></p>
<p> </p>
<p><strong>Note: A1</strong> for each correct term.</p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>126</mn><mo>…</mo></mrow></msup><msup><mi>d</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mn>0</mn><mo>.</mo><mn>748</mn><msup><mi>d</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of fish that can be caught in one hour from a particular lake can be modelled by a Poisson distribution.</p>
<p>The owner of the lake, Emily, states in her advertising that the average number of fish caught in an hour is three.</p>
<p>Tom, a keen fisherman, is not convinced and thinks it is less than three. He decides to set up the following test. Tom will fish for one hour and if he catches fewer than two fish he will reject Emily’s claim.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State a suitable null and alternative hypotheses for Tom’s test.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of a Type I error.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The average number of fish caught in an hour is actually 2.5.</p>
<p>Find the probability of a Type II error.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{H}}_0}\,{\text{:}}\,m = 3">
<mrow>
<msub>
<mrow>
<mtext>H</mtext>
</mrow>
<mn>0</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>m</mi>
<mo>=</mo>
<mn>3</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{H}}_1}\,{\text{:}}\,m < 3">
<mrow>
<msub>
<mrow>
<mtext>H</mtext>
</mrow>
<mn>1</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>m</mi>
<mo><</mo>
<mn>3</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept equivalent statements in words.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>(let </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span><em> be the number of fish caught)</em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X \leqslant 1\left| {m = 3} \right.} \right) = 0.199">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>⩽</mo>
<mn>1</mn>
<mrow>
<mo>|</mo>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>3</mn>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.199</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X \geqslant 2\left| {m = 2.5} \right.} \right)\,\,\,\,\left( { = 1 - {\text{P}}\left( {X \leqslant 1\left| {m = 2.5} \right.} \right)} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>⩾</mo>
<mn>2</mn>
<mrow>
<mo>|</mo>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>2.5</mn>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>⩽</mo>
<mn>1</mn>
<mrow>
<mo>|</mo>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mn>2.5</mn>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> = 2.5 to evaluate a probability, award <em><strong>A1</strong></em> for also having <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> ≥ 2 .</p>
<p>= 0.713 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="indent2" style="margin-top:12.0pt;">The number of cars passing a certain point in a road was recorded during 80 equal time intervals and summarized in the table below.</p>
<p class="indent2" style="margin-top:12.0pt;"><img style="display: block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p class="indent2" style="margin-top:12.0pt;">Carry out a <span class="mjpage"><math alttext="{\chi ^2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> </math></span> goodness of fit test at the 5% significance level to decide if the above data can be modelled by a Poisson distribution.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>H<sub>0</sub> : The data can be modeled by a Poisson distribution.</p>
<p>H<sub>1</sub> : The data cannot be modeled by a Poisson distribution.</p>
<p><span class="mjpage"><math alttext="\sum {f = 80{\text{,}}\,\,} \frac{{\sum {fx} }}{{\sum f }} = \frac{{0 \times 4 + 1 \times 18 + 2 \times 19 + \ldots + 5 \times 8}}{{80}} = \frac{{200}}{{80}} = 2.5" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>∑</mo> <mrow> <mi>f</mi> <mo>=</mo> <mn>80</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </mrow> <mfrac> <mrow> <mo>∑</mo> <mrow> <mi>f</mi> <mi>x</mi> </mrow> </mrow> <mrow> <mo>∑</mo> <mi>f</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>0</mn> <mo>×</mo> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mo>×</mo> <mn>18</mn> <mo>+</mo> <mn>2</mn> <mo>×</mo> <mn>19</mn> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mn>5</mn> <mo>×</mo> <mn>8</mn> </mrow> <mrow> <mn>80</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>200</mn> </mrow> <mrow> <mn>80</mn> </mrow> </mfrac> <mo>=</mo> <mn>2.5</mn> </math></span> <em><strong>A1</strong></em></p>
<p>Theoretical frequencies are</p>
<p><span class="mjpage"><math alttext="f\left( 0 \right) = 8.0{{\text{e}}^{ - 2.5}} = 6.5668" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>8.0</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>2.5</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>6.5668</mn> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math alttext="f\left( 1 \right) = \frac{{2.5}}{1} \times 6.5668 = 16.4170" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2.5</mn> </mrow> <mn>1</mn> </mfrac> <mo>×</mo> <mn>6.5668</mn> <mo>=</mo> <mn>16.4170</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math alttext="f\left( 2 \right) = \frac{{2.5}}{2} \times 16.4170 = 20.5212" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2.5</mn> </mrow> <mn>2</mn> </mfrac> <mo>×</mo> <mn>16.4170</mn> <mo>=</mo> <mn>20.5212</mn> </math></span></p>
<p><span class="mjpage"><math alttext="f\left( 3 \right) = \frac{{2.5}}{3} \times 20.5212 = 17.1010" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2.5</mn> </mrow> <mn>3</mn> </mfrac> <mo>×</mo> <mn>20.5212</mn> <mo>=</mo> <mn>17.1010</mn> </math></span></p>
<p><span class="mjpage"><math alttext="f\left( 4 \right) = \frac{{2.5}}{4} \times 17.1010 = 10.6882" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2.5</mn> </mrow> <mn>4</mn> </mfrac> <mo>×</mo> <mn>17.1010</mn> <mo>=</mo> <mn>10.6882</mn> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong><em> </em>Award A1 for <span class="mjpage"><math alttext="f\left( 2 \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math alttext="f\left( 4 \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math alttext="f\left( 4 \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </math></span>.</p>
<p><span class="mjpage"><math alttext="f" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> </math></span>(5 or more) <span class="mjpage"><math alttext=" = 80 - \left( {6.5668 + 16.4170 + 20.5212 + 17.1010 + 10.6882} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mn>80</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>6.5668</mn> <mo>+</mo> <mn>16.4170</mn> <mo>+</mo> <mn>20.5212</mn> <mo>+</mo> <mn>17.1010</mn> <mo>+</mo> <mn>10.6882</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> <span class="mjpage"><math alttext=" = 8.7058" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mn>8.7058</mn> </math></span></p>
<p><img src="data:image/png;base64,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"></p>
<p><span class="mjpage"><math alttext="{\chi ^2} = \frac{{{{\left( {4 - 6.5668} \right)}^2}}}{{6.5668}} + \frac{{{{\left( {18 - 16.4170} \right)}^2}}}{{16.4170}} + \frac{{{{\left( {19 - 20.5212} \right)}^2}}}{{20.5212}} + \frac{{{{\left( {20 - 17.1010} \right)}^2}}}{{17.1010}} + \frac{{{{\left( {11 - 10.6882} \right)}^2}}}{{10.6882}} + \frac{{{{\left( {8 - 8.7058} \right)}^2}}}{{8.7058}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>−</mo> <mn>6.5668</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>6.5668</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>18</mn> <mo>−</mo> <mn>16.4170</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>16.4170</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>19</mn> <mo>−</mo> <mn>20.5212</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>20.5212</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>20</mn> <mo>−</mo> <mn>17.1010</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>17.1010</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>11</mn> <mo>−</mo> <mn>10.6882</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>10.6882</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mo>−</mo> <mn>8.7058</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>8.7058</mn> </mrow> </mfrac> </math></span></p>
<p> <span class="mjpage"><math alttext=" = 1.83" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mn>1.83</mn> </math></span> (accept 1.82) <em><strong>(M1)(A1)</strong></em></p>
<p> <span class="mjpage"><math alttext="v = 4" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>v</mi> <mo>=</mo> <mn>4</mn> </math></span> (six frequencies and two restrictions) <em><strong>(A1)</strong></em></p>
<p> <span class="mjpage"><math alttext="{\chi ^2}\left( 4 \right) = 9.488" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>9.488</mn> </math></span> at the 5% level. <em><strong>A1</strong></em></p>
<p> Since 1.83 < 9.488 we accept H<sub>0</sub> and conclude that the distribution can be modeled by a Poisson distribution. <em><strong>R1 N0</strong></em></p>
<p><em><strong>[11 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A zoologist believes that the number of eggs laid in the Spring by female birds of a certain breed follows a Poisson law. She observes 100 birds during this period and she produces the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The zoologist wishes to determine whether or not a Poisson law provides a suitable model.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Calculate the mean number of eggs laid by these birds.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Write down appropriate hypotheses.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Carry out a test at the 1% significance level, and state your conclusion.</p>
<div class="marks">[14]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Mean <span class="mjpage"><math alttext=" = \frac{{1 \times 19 + 2 \times 34 + \ldots + 5 \times 4}}{{100}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>×</mo> <mn>19</mn> <mo>+</mo> <mn>2</mn> <mo>×</mo> <mn>34</mn> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mn>5</mn> <mo>×</mo> <mn>4</mn> </mrow> <mrow> <mn>100</mn> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math alttext=" = 2.16" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mn>2.16</mn> </math></span> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>H<sub>0</sub> : Poisson law provides a suitable model <em><strong>A1</strong></em></p>
<p>H<sub>1</sub> : Poisson law does not provide a suitable model <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The expected frequencies are</p>
<p><img src="data:image/png;base64,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"> <em><strong>A1A1A1A1A1A1</strong></em></p>
<p><em> <strong>Note:</strong></em> Accept expected frequencies rounded to a minimum of three significant figures.</p>
<p><span class="mjpage"><math alttext="{\chi ^2} = \frac{{{{\left( {10 - 11.533} \right)}^2}}}{{11.533}} + \ldots + \frac{{{{\left( {4 - 6.824} \right)}^2}}}{{6.824}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>10</mn> <mo>−</mo> <mn>11.533</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>11.533</mn> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>−</mo> <mn>6.824</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>6.824</mn> </mrow> </mfrac> </math></span> <em><strong>(M1)(A2)</strong></em></p>
<p><span class="mjpage"><math alttext=" = 5.35" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mn>5.35</mn> </math></span> (accept 5.33 and 5.34) <em><strong>A2</strong></em></p>
<p><span class="mjpage"><math alttext="v = 4" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>v</mi> <mo>=</mo> <mn>4</mn> </math></span> (6 cells − 2 restrictions) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> If candidates have combined rows allow FT on their value of <span class="mjpage"><math alttext="v" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>v</mi> </math></span>.</p>
<p>Critical value <span class="mjpage"><math alttext="{\chi ^2} = 13.277" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>13.277</mn> </math></span></p>
<p>Because 5.35 < 13.277, the Poisson law does provide a suitable model. <em><strong>R1 N0</strong></em></p>
<p><em><strong>[14 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Product research leads a company to believe that the revenue (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span>) made by selling its goods at a price (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>) can be modelled by the equation.</p>
<p style="text-align: center;"><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R\left( p \right) = cp{{\text{e}}^{dp}}">
<mi>R</mi>
<mrow>
<mo>(</mo>
<mi>p</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>p</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>d</mi>
<mi>p</mi>
</mrow>
</msup>
</mrow>
</math></span></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}">
<mi>d</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span></p>
<p>There are two competing models, A and B with different values for the parameters <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>. </p>
<p>Model A has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> = 3, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> = −0.5 and model B has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> = 2.5, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> = −0.6.</p>
<p>The company experiments by selling the goods at three different prices in three similar areas and the results are shown in the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The company will choose the model with the smallest value for the sum of square residuals.</p>
<p>Determine which model the company chose.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><em>(Model A)</em></p>
<p><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R = 3p{{\text{e}}^{ - 0.5p}}">
<mi>R</mi>
<mo>=</mo>
<mn>3</mn>
<mi>p</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>0.5</mn>
<mi>p</mi>
</mrow>
</msup>
</mrow>
</math></span></span> <em><strong>M1</strong></em></p>
<p style="text-align: left;">predicted values</p>
<p style="text-align: left;padding-left:150px;"><img src="data:image/png;base64,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"> <em><strong>(A1)</strong></em></p>
<p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S{S_{res}} = {\left( {1.8196 - 1.5} \right)^2} + {\left( {2.2073 - 1.8} \right)^2} + {\left( {2.0082 - 1.5} \right)^2}">
<mi>S</mi>
<mrow>
<msub>
<mi>S</mi>
<mrow>
<mi>r</mi>
<mi>e</mi>
<mi>s</mi>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.8196</mn>
<mo>−</mo>
<mn>1.5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2.2073</mn>
<mo>−</mo>
<mn>1.8</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2.0082</mn>
<mo>−</mo>
<mn>1.5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 0.5263… <em><strong>A1</strong></em></p>
<p> </p>
<p><em>(Model B)</em></p>
<p><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R = 2.5p{{\text{e}}^{ - 0.6p}}">
<mi>R</mi>
<mo>=</mo>
<mn>2.5</mn>
<mi>p</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>0.6</mn>
<mi>p</mi>
</mrow>
</msup>
</mrow>
</math></span></span></p>
<p>predicted values</p>
<p style="padding-left:150px;"><img src="data:image/png;base64,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"> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S{S_{res}} = ">
<mi>S</mi>
<mrow>
<msub>
<mi>S</mi>
<mrow>
<mi>r</mi>
<mi>e</mi>
<mi>s</mi>
</mrow>
</msub>
</mrow>
<mo>=</mo>
</math></span> 0.170576… <em><strong>A1</strong></em></p>
<p>chose model B <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Method marks can be awarded if seen for either model A or model B. Award final <em><strong>A1</strong></em> if it is a correct deduction from their calculated values for A and B.</p>
<p><em><strong>[7 marks]</strong></em></p>
<p> </p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Mr Burke teaches a mathematics class with 15 students. In this class there are 6 female students and 9 male students.</p>
<p>Each day Mr Burke randomly chooses one student to answer a homework question.</p>
<p>In the first month, Mr Burke will teach his class 20 times.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability he will choose a female student 8 times.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The Head of Year, Mrs Smith, decides to select a student at random from the year group to read the notices in assembly. There are 80 students in total in the year group. Mrs Smith calculates the probability of picking a male student 8 times in the first 20 assemblies is 0.153357 correct to 6 decimal places.</p>
<p>Find the number of male students in the year group.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>P(<em>X</em> = 8) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for evidence of recognizing binomial probability. <em>eg,</em> P(<em>X</em> = 8), <em>X</em> ∼ B<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {20,\,\frac{6}{{15}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>20</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mn>6</mn>
<mrow>
<mn>15</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>= 0.180 (0.179705…) <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> be the number of male students</p>
<p>recognize that probability of selecting a male is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{{80}}">
<mfrac>
<mi>x</mi>
<mrow>
<mn>80</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{set up equation}}\,{}^{20}{{\text{C}}_8}{{\left( {\frac{x}{{80}}} \right)}^8}{{\left( {\frac{{80 - x}}{{80}}} \right)}^{12}} = } \right)\,0.153357">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>set up equation</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<msup>
<mrow>
</mrow>
<mrow>
<mn>20</mn>
</mrow>
</msup>
<mrow>
<msub>
<mrow>
<mtext>C</mtext>
</mrow>
<mn>8</mn>
</msub>
</mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mrow>
<mn>80</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>8</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>80</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mrow>
<mn>80</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>0.153357</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p>number of male students = 37 <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)A0</strong></em> for 27.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A company sends a group of employees on a training course. Afterwards, they survey these employees to gather data on the effectiveness of the training. In order to test the reliability of the survey, they design two sets of similar questions, which are given to the employees one week apart.</p>
</div>
<div class="specification">
<p>The questions in the survey were grouped in different sections. The mean scores of the employees on the first section of each survey are given in the table.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the name of this test for reliability.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State a possible disadvantage of using this test for reliability.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate Pearson’s product moment correlation coefficient for this data.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine, with a reason, if the survey is reliable.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Parallel Forms <em><strong> A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER </strong></p>
<p>The two sets of questions might not be of equal difficulty <em><strong> R1 </strong></em></p>
<p><strong>OR</strong></p>
<p>It is time consuming to create two sets of questions<em><strong> R1 </strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 0.958">
<mi>r</mi>
<mo>=</mo>
<mn>0.958</mn>
</math></span><em><strong> A2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Since the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> is close to +1, <strong> </strong><em><strong> R1</strong></em></p>
<p>The survey is reliable. <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>On Paul’s farm, potatoes are packed in sacks labelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo> </mo><mtext>kg</mtext></math>. The weights of the sacks of potatoes can be modelled by a normal distribution with mean weight <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>49</mn><mo>.</mo><mn>8</mn><mo> </mo><mtext>kg</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>9</mn><mo> </mo><mtext>kg</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a sack is under its labelled weight.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the lower quartile of the weights of the sacks of potatoes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The sacks of potatoes are transported in crates. There are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> sacks in each crate and the weights of the sacks of potatoes are independent of each other.</p>
<p>Find the probability that the total weight of the sacks of potatoes in a crate exceeds <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>500</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the random variable “the weight of a sack of potatoes”</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo><</mo><mn>50</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>588</mn><mo> </mo><mtext>kg</mtext><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>587929</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo><</mo><mi>l</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>25</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>49</mn><mo>.</mo><mn>2</mn><mo> </mo><mtext>kg</mtext><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>49</mn><mo>.</mo><mn>1929</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to sum <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> independent random variables <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><msub><mi>X</mi><mi>i</mi></msub><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>498</mn><mo>,</mo><mo> </mo><mn>10</mn><mo>×</mo><mn>0</mn><mo>.</mo><msup><mn>9</mn><mn>2</mn></msup></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>Y</mi><mo>></mo><mn>500</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>241</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The first part of the question was often answered well but there were a number of candidates who interpreted finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo><</mo><mn>50</mn></mrow></mfenced></math> by finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo><</mo><mn>49</mn><mo>.</mo><mn>9</mn></mrow></mfenced></math> or something similar. Not all candidates, however, understood that the lower quartile is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo><</mo><mi>l</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>25</mn></math>. Part (c) was less well understood. Attempts to sum <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> independent random variables correctly involved multiplication of the mean by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> but the standard deviation and not the variance was incorrectly multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The first part of the question was often answered well but there were a number of candidates who interpreted finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo><</mo><mn>50</mn></mrow></mfenced></math> by finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo><</mo><mn>49</mn><mo>.</mo><mn>9</mn></mrow></mfenced></math> or something similar. Not all candidates, however, understood that the lower quartile is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo><</mo><mi>l</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>25</mn></math>. Part (c) was less well understood. Attempts to sum <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> independent random variables correctly involved multiplication of the mean by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> but the standard deviation and not the variance was incorrectly multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math>.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The first part of the question was often answered well but there were a number of candidates who interpreted finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo><</mo><mn>50</mn></mrow></mfenced></math> by finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo><</mo><mn>49</mn><mo>.</mo><mn>9</mn></mrow></mfenced></math> or something similar. Not all candidates, however, understood that the lower quartile is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>X</mi><mo><</mo><mi>l</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>25</mn></math>. Part (c) was less well understood. Attempts to sum <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> independent random variables correctly involved multiplication of the mean by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> but the standard deviation and not the variance was incorrectly multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math>.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Observations on <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> pairs of values of the random variables <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>,</mo><mo> </mo><mi>Y</mi></math> yielded the following results.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</mo><mi>x</mi><mo>=</mo><mn>76</mn><mo>.</mo><mn>3</mn><mo>,</mo><mo> </mo><mo>∑</mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>563</mn><mo>.</mo><mn>7</mn><mo>,</mo><mo> </mo><mo>∑</mo><mi>y</mi><mo>=</mo><mn>72</mn><mo>.</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>∑</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>460</mn><mo>.</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>∑</mo><mi>x</mi><mi>y</mi><mo>=</mo><mn>495</mn><mo>.</mo><mn>4</mn></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, the product moment correlation coefficient of the sample.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming that the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>,</mo><mo> </mo><mi>Y</mi></math> is bivariate normal with product moment correlation coefficient <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math>, calculate the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value of your result when testing the hypotheses <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub><mo> </mo><mo>:</mo><mo> </mo><mo> </mo><mi>ρ</mi><mo>=</mo><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><msub><mi>H</mi><mn>1</mn></msub><mo> </mo><mo>:</mo><mo> </mo><mo> </mo><mi>ρ</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State whether your <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value suggests that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> are independent.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given a further value <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>2</mn></math> from the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math>, predict the corresponding value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>. Give your answer to one decimal place.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>use of</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mo>∑</mo><mi>x</mi><mi>y</mi><mo>-</mo><mi>n</mi><mover><mi>x</mi><mo>¯</mo></mover><mo> </mo><mover><mi>y</mi><mo>¯</mo></mover></mrow><msqrt><mfenced><mrow><mo>∑</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>n</mi><msup><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><mo>∑</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mi>n</mi><msup><mover><mi>y</mi><mo>¯</mo></mover><mn>2</mn></msup></mrow></mfenced></msqrt></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>495</mn><mo>.</mo><mn>4</mn><mo>-</mo><mn>12</mn><mo>×</mo><mstyle displaystyle="true"><mfrac><mrow><mn>76</mn><mo>.</mo><mn>3</mn></mrow><mn>12</mn></mfrac></mstyle><mo>×</mo><mstyle displaystyle="true"><mfrac><mrow><mn>72</mn><mo>.</mo><mn>2</mn></mrow><mn>12</mn></mfrac></mstyle></mrow><msqrt><mfenced><mrow><mn>563</mn><mo>.</mo><mn>7</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mn>12</mn><mo>×</mo><mn>76</mn><mo>.</mo><msup><mn>3</mn><mn>2</mn></msup></mrow><msup><mn>12</mn><mn>2</mn></msup></mfrac></mstyle></mrow></mfenced><mfenced><mrow><mn>460</mn><mo>.</mo><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mn>12</mn><mo>×</mo><mn>72</mn><mo>.</mo><msup><mn>2</mn><mn>2</mn></msup></mrow><msup><mn>12</mn><mn>2</mn></msup></mfrac></mstyle></mrow></mfenced></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>809</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept any answer that rounds to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>81</mn></math>.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>80856</mn><mo>…</mo><msqrt><mfrac><mn>10</mn><mrow><mn>1</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>80856</mn><msup><mo>…</mo><mn>2</mn></msup></mrow></mfrac></msqrt></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>.</mo><mn>345</mn><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>7</mn><mo>.</mo><mn>27</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept any answer that rounds to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup></math>.</p>
<p><strong>Note:</strong> Follow through their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>this value indicates that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>,</mo><mi>Y</mi></math> are not independent <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mover><mi>y</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><mo>∑</mo><mi>x</mi><mi>y</mi><mo>-</mo><mi>n</mi><mover><mi>x</mi><mo>¯</mo></mover><mo> </mo><mover><mi>y</mi><mo>¯</mo></mover></mrow><mrow><mo>∑</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>n</mi><msup><mover><mi>x</mi><mo>¯</mo></mover><mn>2</mn></msup></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mover><mi>x</mi><mo>¯</mo></mover></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mfrac><mrow><mn>72</mn><mo>.</mo><mn>2</mn></mrow><mn>12</mn></mfrac><mo>=</mo><mfenced><mfrac><mrow><mn>495</mn><mo>.</mo><mn>4</mn><mo>-</mo><mn>12</mn><mo>×</mo><mstyle displaystyle="true"><mfrac><mrow><mn>76</mn><mo>.</mo><mn>3</mn></mrow><mn>12</mn></mfrac></mstyle><mo>×</mo><mstyle displaystyle="true"><mfrac><mrow><mn>72</mn><mo>.</mo><mn>2</mn></mrow><mn>12</mn></mfrac></mstyle></mrow><mrow><mn>563</mn><mo>.</mo><mn>7</mn><mo>-</mo><mn>12</mn><mo>×</mo><mstyle displaystyle="true"><mfrac><mrow><mn>76</mn><mo>.</mo><msup><mn>3</mn><mn>2</mn></msup></mrow><msup><mn>12</mn><mn>2</mn></msup></mfrac></mstyle></mrow></mfrac></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mrow><mn>76</mn><mo>.</mo><mn>3</mn></mrow><mn>12</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>2</mn></math> gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable <em>X</em> has a probability density function given by</p>
<p style="padding-left: 120px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \left\{ {\begin{array}{*{20}{l}} {k\sin \left( {\frac{{\pi x}}{6}} \right),}&{0 \leqslant x \leqslant \,6} \\ {0,}&{{\text{otherwise}}} \end{array}} \right.">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π<!-- π --></mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0</mn>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<mtext>otherwise</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <em>f </em>write down the mean of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <em>f </em>write down the median of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <em>f </em>write down the mode of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(0 \leqslant X \leqslant 2) = \frac{1}{4}">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>⩽</mo>
<mi>X</mi>
<mo>⩽</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state the interquartile range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(X \leqslant 4|X \geqslant 3)">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>⩽</mo>
<mn>4</mn>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>X</mi>
<mo>⩾</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to equate integral to 1 (may appear later) <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k\int\limits_0^6 {\sin \left( {\frac{{\pi x}}{6}} \right){\text{d}}x = 1} ">
<mi>k</mi>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>6</mn>
</munderover>
<mrow>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
</math></span></p>
<p>correct integral <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k\left[ { - \frac{6}{\pi }\cos \left( {\frac{{\pi x}}{6}} \right)} \right]_0^6 = 1">
<mi>k</mi>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>6</mn>
<mi>π</mi>
</mfrac>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>6</mn>
</msubsup>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p>substituting limits <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{6}{\pi }( - 1 - 1) = \frac{1}{k}">
<mo>−</mo>
<mfrac>
<mn>6</mn>
<mi>π</mi>
</mfrac>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>k</mi>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{\pi }{{12}}">
<mi>k</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>mean <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3">
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1A0A0 </em></strong>for three equal answers in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}6)">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>median <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3">
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1A0A0 </em></strong>for three equal answers in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}6)">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>mode <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3">
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1A0A0 </em></strong>for three equal answers in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}6)">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}\int\limits_0^2 {\sin } \left( {\frac{{\pi x}}{6}} \right){\text{d}}x">
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>2</mn>
</munderover>
<mrow>
<mi>sin</mi>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{{12}}\left[ { - \frac{6}{\pi }\cos \left( {\frac{{\pi x}}{6}} \right)} \right]_0^2">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>6</mn>
<mi>π</mi>
</mfrac>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>2</mn>
</msubsup>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept without the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}">
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> at this stage if it is added later.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}\left[ { - \frac{6}{\pi }\left( {\cos \frac{\pi }{3} - 1} \right)} \right]">
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>6</mn>
<mi>π</mi>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{4}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>from (c)(i) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{Q_1} = 2">
<mrow>
<msub>
<mi>Q</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>as the graph is symmetrical about the middle value <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 3 \Rightarrow {Q_3} = 4">
<mi>x</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>Q</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>4</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>so interquartile range is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 - 2">
<mn>4</mn>
<mo>−</mo>
<mn>2</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2">
<mo>=</mo>
<mn>2</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(X \leqslant 4|X \geqslant 3) = \frac{{P(3 \leqslant X \leqslant 4)}}{{P(X \geqslant 3)}}">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>⩽</mo>
<mn>4</mn>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>X</mi>
<mo>⩾</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mi>P</mi>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>⩽</mo>
<mi>X</mi>
<mo>⩽</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>⩾</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\frac{1}{4}}}{{\frac{1}{2}}}">
<mo>=</mo>
<mfrac>
<mrow>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of coffees sold per hour at an independent coffee shop is modelled by a Poisson distribution with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>22</mn></math> coffees per hour.</p>
<p>Sheila, the shop’s owner wants to increase the number of coffees sold in the shop. She decides to offer a discount to customers who buy more than one coffee.</p>
<p>To test how successful this strategy is, Sheila records the number of coffees sold over a single <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>-hour period. Sheila decides to use a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> level of significance in her test.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the null and alternative hypotheses for the test.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Sheila will make a type I error in her test conclusion.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sheila finds <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>126</mn></math> coffees were sold during the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>-hour period.</p>
<p>State Sheila’s conclusion to the test. Justify your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo>:</mo><mo> </mo><mi>m</mi><mo>=</mo><mn>110</mn><mo>,</mo><mo> </mo><msub><mtext>H</mtext><mn>1</mn></msub><mo>:</mo><mo> </mo><mi>m</mi><mo>></mo><mn>110</mn></math> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept other appropriate variables for the mean. <br> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>22</mn></math> in place of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>110</mn></math>.</p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≥</mo><mn>128</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>05024</mn></math> <em><strong>(M1)(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≥</mo><mn>129</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>04153</mn></math> <em><strong>(M1)</strong></em></p>
<p>(probability of making a type I error is) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0415</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> If other probabilities are seen, the final <em><strong>A1</strong></em> cannot be awarded unless <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0415</mn></math> is clearly identified as the final answer.</p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>Po</mtext><mfenced><mn>110</mn></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≥</mo><mn>126</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>072</mn><mo>></mo><mn>0</mn><mo>.</mo><mn>05</mn></math> <strong>OR </strong>recognizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>126</mn><mo><</mo><mn>129</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≤</mo><mn>128</mn></math> <em><strong>R1</strong></em></p>
<p>so there is insufficient evidence to reject <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math> <em><strong>A1</strong></em></p>
<p>(<em>ie</em> there is insufficient evidence to suggest that the number of coffees being sold has increased)</p>
<p><strong><br>Note:</strong> Accept ‘Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math>’.<br> Do not award <em><strong>R0A1</strong></em>.</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The heights, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> metres, of the 241 new entrants to a men’s college were measured and the following statistics calculated.</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\sum {x = 412.11,\,\,\sum {{x^2} = 705.5721} } ">
<mo>∑<!-- ∑ --></mo>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mn>412.11</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>∑<!-- ∑ --></mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>705.5721</mn>
</mrow>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>The Head of Mathematics decided to use a <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\chi ^2}">
<mrow>
<msup>
<mi>χ<!-- χ --></mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> test to determine whether or not these heights could be modelled by a normal distribution. He therefore divided the data into classes as follows.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Calculate unbiased estimates of the population mean and the population variance.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">State suitable hypotheses.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Calculate the value of the <span class="mjpage"><math alttext="{\chi ^2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> </math></span> statistic and state your conclusion using a 10% level of significance.</p>
<div class="marks">[11]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math alttext="\bar x = \frac{{412.11}}{{241}} = 1.71" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>412.11</mn> </mrow> <mrow> <mn>241</mn> </mrow> </mfrac> <mo>=</mo> <mn>1.71</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math alttext="{s^2} = \frac{{705.5721}}{{240}} - \frac{{{{412.11}^2}}}{{240 \times 241}} = 0.0036" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>s</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>705.5721</mn> </mrow> <mrow> <mn>240</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>412.11</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>240</mn> <mo>×</mo> <mn>241</mn> </mrow> </mfrac> <mo>=</mo> <mn>0.0036</mn> </math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>H<sub>0</sub>: Data can be modelled by a normal distribution</p>
<p>H<sub>1</sub>: Data cannot be modelled by a normal distribution <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The expected frequencies are</p>
<p><img src="data:image/png;base64,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"> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math alttext="{\chi ^2} = \frac{{{5^2}}}{{8.04}} + \frac{{{{34}^2}}}{{30.19}} + \ldots + \frac{{{{12}^2}}}{{16.10}} - 241 = 3.30\,{\text{/}}\,3.29" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mn>5</mn> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>8.04</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>34</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>30.19</mn> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>16.10</mn> </mrow> </mfrac> <mo>−</mo> <mn>241</mn> <mo>=</mo> <mn>3.30</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>/</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3.29</mn> </math></span> <em><strong>M1</strong></em><em><strong>A1</strong></em></p>
<p>Degrees of freedom = 3 <em><strong>A1</strong></em></p>
<p>Critical value = 6.251 or <em>p</em>-value = 0.35 <em><strong>A1</strong></em></p>
<p>The data can be modelled by a normal distribution. <em><strong>R1</strong></em></p>
<p><em><strong>[11 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The graph below shows a small maze, in the form of a network of directed routes. The vertices <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math> show junctions in the maze and the edges show the possible paths available from one vertex to another.</p>
<p>A mouse is placed at vertex <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and left to wander the maze freely. The routes shown by dashed lines indicate paths sprinkled with sugar.</p>
<p>When the mouse reaches any junction, she rests for a constant time before continuing.</p>
<p>At any junction, it may also be assumed that</p>
<ul>
<li> the mouse chooses any available normal path with equal probability</li>
<li> if the junction includes a path sprinkled with sugar, the probability of choosing this path is twice that of a normal path.</li>
</ul>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the transition matrix for this graph.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>If the mouse was left to wander indefinitely, use your graphic display calculator to estimate the percentage of time that the mouse would spend at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment on your answer to part (b), referring to at least one limitation of the model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>transition matrix is <img src="data:image/png;base64,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"> <em><strong>M1A1A1</strong></em></p>
<p><strong><br>Note:</strong> Allow the transposed matrix. <br> Award <em><strong>M1</strong></em> for a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>6</mn></math> matrix with all values between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, and all columns (or rows if transposed) adding up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, award <em><strong>A1</strong></em> for one correct row (or column if transposed) and <em><strong>A1</strong></em> for all rows (or columns if transposed) correct.<br><em><strong><br><br>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to raise the transition matrix to a large power <em><strong>(M1)</strong></em></p>
<p>steady state vector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfenced><mrow><mn>0</mn><mo>.</mo><mn>157</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0868</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced><mrow><mn>0</mn><mo>.</mo><mn>256</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced><mrow><mn>0</mn><mo>.</mo><mn>241</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0868</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>173</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>so percentage of time spent at vertex <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn><mo>.</mo><mn>3</mn><mo>%</mo></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn><mo>.</mo><mn>2</mn><mo>%</mo></math>.<br><em><strong><br><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the model assumes instantaneous travel from junction to junction, <em><strong>R1</strong></em><br>and hence the answer obtained would be an overestimate <em><strong>R1</strong></em><br><br><strong>OR</strong><br><br>the mouse may eat the sugar over time <em><strong>R1</strong></em><br>and hence the probabilities would change <em><strong>R1</strong></em><br><br><br><strong>Note:</strong> Accept any other sensible answer.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Eggs at a farm are sold in boxes of six. Each egg is either brown or white. The owner believes that the number of brown eggs in a box can be modelled by a binomial distribution. He examines 100 boxes and obtains the following data.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the mean number of brown eggs in a box.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Hence estimate <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span>, the probability that a randomly chosen egg is brown.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">By calculating an appropriate <span class="mjpage"><math alttext="{\chi ^2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> </math></span> statistic, test, at the 5% significance level, whether or not the binomial distribution gives a good fit to these data.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><em>Note:</em> </strong><em>Candidates</em> <em>may</em> <em>obtain</em> <em>slightly</em> <em>different</em> <em>numerical</em> <em>answers</em> <em>depending</em> <em>on</em> <em>the</em> <em>calculator</em> <em>and</em> <em>approach</em> <em>used.</em> <em>Use</em> <em>discretion</em> <em>in</em> <em>marking.</em></p>
<p>Mean <span class="mjpage"><math alttext=" = \frac{{1 \times 29 + \ldots + 6 \times 1}}{{100}} = 1.98" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>×</mo> <mn>29</mn> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mn>6</mn> <mo>×</mo> <mn>1</mn> </mrow> <mrow> <mn>100</mn> </mrow> </mfrac> <mo>=</mo> <mn>1.98</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>Note:</em> </strong><em>Candidates</em> <em>may</em> <em>obtain</em> <em>slightly</em> <em>different</em> <em>numerical</em> <em>answers</em> <em>depending</em> <em>on</em> <em>the</em> <em>calculator</em> <em>and</em> <em>approach</em> <em>used.</em> <em>Use</em> <em>discretion</em> <em>in</em> <em>marking.</em></p>
<p><span class="mjpage"><math alttext="\widehat p = \frac{{1.98}}{6} = 0.33" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1.98</mn> </mrow> <mn>6</mn> </mfrac> <mo>=</mo> <mn>0.33</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>Note:</em> </strong><em>Candidates</em> <em>may</em> <em>obtain</em> <em>slightly</em> <em>different</em> <em>numerical</em> <em>answers</em> <em>depending</em> <em>on</em> <em>the</em> <em>calculator</em> <em>and</em> <em>approach</em> <em>used.</em> <em>Use</em> <em>discretion</em> <em>in</em> <em>marking.</em></p>
<p>The calculated values are</p>
<p><span class="mjpage"><math alttext="{f_0}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> </math></span> <span class="mjpage"><math alttext="{f_e}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>f</mi> <mi>e</mi> </msub> </mrow> </math></span> <span class="mjpage"><math alttext="{\left( {{f_0} - {f_e}} \right)^2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> <mo>−</mo> <mrow> <msub> <mi>f</mi> <mi>e</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></span><br>10 9.046 0.910<br>29 26.732 5.14 <em><strong>(M1)</strong></em><br>31 32.917 3.675 <em><strong>(A1)</strong></em><br>18 21.617 13.083 <em><strong>(A1)</strong></em><br>12 9.688 5.345 <em><strong>(A1)</strong></em></p>
<p><em> <strong>Note:</strong> </em>Award <em><strong>(M1)</strong></em> for the attempt to calculate expected values, <em><strong>(A1)</strong></em> for correct expected values, <em><strong>(A1)</strong></em> for correct <span class="mjpage"><math alttext="{\left( {{f_0} - {f_e}} \right)^2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> <mo>−</mo> <mrow> <msub> <mi>f</mi> <mi>e</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></span> values, <em><strong>(A1)</strong></em> for combining cells.</p>
<p><span class="mjpage"><math alttext="{\chi ^2} = \frac{{0.910}}{{9.046}} + \ldots + \frac{{5.345}}{{9.688}} = 1.56" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>0.910</mn> </mrow> <mrow> <mn>9.046</mn> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mn>5.345</mn> </mrow> <mrow> <mn>9.688</mn> </mrow> </mfrac> <mo>=</mo> <mn>1.56</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math alttext="{\chi ^2} = 1.56" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>1.56</mn> </math></span> <em><strong>(G5)</strong></em></p>
<p>Degrees of freedom = 3; Critical value = 7.815</p>
<p>(or <em>p</em>-value = 0.668 (or 0.669)) <em><strong>(A1)(A1)</strong></em></p>
<p>We conclude that the binomial distribution does provide a good fit. <em><strong>(R1)</strong></em></p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Adesh wants to model the cooling of a metal rod. He heats the rod and records its temperature as it cools.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>He believes the temperature can be modeled by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T\left( t \right) = a{{\text{e}}^{bt}} + 25">
<mi>T</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>b</mi>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>25</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{,}}\,\,b \in \mathbb{R}">
<mi>a</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Hence</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {T - 25} \right) = bt + {\text{ln}}\,a">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>b</mi>
<mi>t</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>a</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the regression line of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {T - 25} \right)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>predict the temperature of the metal rod after 3 minutes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {T - 25} \right) = {\text{ln}}\left( {a{{\text{e}}^{bt}}} \right)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>b</mi>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {T - 25} \right) = {\text{ln}}\,a + {\text{ln}}\left( {{{\text{e}}^{bt}}} \right)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>a</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>b</mi>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {T - 25} \right) = bt + {\text{ln}}\,a">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>b</mi>
<mi>t</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>a</mi>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {T - 25} \right) = - 0.00870t + 3.89">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>0.00870</mn>
<mi>t</mi>
<mo>+</mo>
<mn>3.89</mn>
</math></span> <em><strong>M1A1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = - 0.00870">
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.00870</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = {e^{3.89...}} = 49.1">
<mi>a</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>3.89...</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>49.1</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T\left( {180} \right) = 49.1{e^{ - 0.00870\left( {180} \right)}} + 25 = 35.2">
<mi>T</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>180</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>49.1</mn>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mn>0.00870</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>180</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>25</mn>
<mo>=</mo>
<mn>35.2</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Katie likes to cycle to work as much as possible. If Katie cycles to work one day then she has a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> of not cycling to work on the next work day. If she does not cycle to work one day then she has a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn></math> of not cycling to work on the next work day.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the following transition diagram to represent this information.</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAusAAADgCAYAAAC6oSjXAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAEqeSURBVHhe7d0LeE1n2jfwe7y9OjOoVluHEqdWHJJXJFGHikMVlXHIBG2pUtRhijYjakqkXKYIWhUTxxetQ1RpKzIpJkpLabSlIuWVlihxqFPRMpiZfn2Tb/+frBVbJOy9sw/P2vn/XLl21kqyd+zs/ax73et+7uc3+TZCRERERETaKWfcEhERERGRZhisExERERFpisE6EREREZGmGKwTEREREWnqlgmmtWsHGJ8REREREZG3nThxyvishGDd/huIiIiIiMg7isbiLIMhIiIiItIUg3UiIiIiIk0xWCciIiIi0hSDdSIiIiIiTTFYJyIiIiLSFIN1IiIiIiJNMVgnIiIiItIUg3UiIo398MMPkpOToz7+9a9/GXuJiKis4KJIREQaQCC+b98++eabb+TkyZOyatVKtb9Vq9ZSv3599bm5r169hyUioo20aNFCgoODJTAwUO0nIiLrKxqLM1gnIvKhXbt2yYYNG1Qg3r//84UBeEBAgPz+9783vutmyLZfuHBBDhw4IGlpaXLu3Fnp2/dZ6dOnj9x///3GdxERkRUxWCci0gCC9Dlz5qjPR48eLWFhYSUG53eC4B1B+5o176mM+6hRo6RmzZrGV4mIyEoYrBMR+RBqz+Pj4+WBBx6QMWPGuLWEBaU0mzdvlpiYlyQhYYb07t3b5RMAIiLyjaKxOCeYEhF5yapVq2To0CEyfPgwWbhwodtrzRGYR0dHS1bWfsnOzpaBAweqrDsREVkXg3UiIg9DxnvEiBGSkZEh69enSqdOnY2veAbq1hMSEtRJwWOPtVQlN0REZE0M1omIPAiBOspdQkJCZPbs2V6dAIqTgi+++Eri4sYzYCcisigG60REHmIfqCOz7ov6cUw0Xb36PQbsREQWxWCdiMhD7AN1X2LATkRkXQzWiYg8AJNJYdCgQerW1xCwJyXNVQE7J50SEVkHg3UiIjfDKqRLliyWadOmadU6sWnTphIbO0amTp1q7CEiIt0xWCciciPUqSNInz59hpariaK1I6SmpqpbIiLSG4N1IiI3wqJE9evXl9atWxt79INa+sTE2erEgoiI9MZgnYjITRD8IggePHiwsUdPWIypa9du6sSCiIj0xmCdiMhNMjI+l4iINm5fmdQTevXqxew6EZEFMFgnInKTxYuXSJ8+fYwtveGEIigoWPbt22fsISIiHTFYJyJyA7RDPHfurOq4YhUDBgyQnTt3GltERKQjButERG6wZ88e6dv3WWPLGho1aiTz5881toiISEcM1omI3GD37t2lyKr/Imf3LJKhjQOkdu2GEhn/rmSe/cX4WgnyzsqepCHSuDZ+xvYRGS/JmWclz/iyI9Baslu3HpKTk2PsISIi3TBYJyJyA0wuRabaeXlyNXORDOq9UWos3C25udtk/H+tkecmb5LTJUbeP0vm36ZLau1XZc+JU5K7+30ZG7BF4qNHyN8yfza+xzERERFy8OBBY4uIiHTDYJ2IqJTQUeXYsaOuLYKUlyMfTp4vxwe8ImMfryHlytWQdgN6S+CmWTJ/xwXjm4q4sl8++6/+EhfdUCraNstVby0vjRspobJHVnx2VH4t+C6HVK9eTc6cOWNsERGRbhisExGV0qlTp6R//+eNLefkff+FpGT9ViKaPyKVjH3lHnlMeoX+KClb/leuGPtuUqmdxMY0V4G6qdw990lVqSvdwmrLXcY+R9SpU1dOnjxpbBERkW4YrBMR+UyeXP3hqOTIQxJcu7Kxz6Zcebmv6m/l2oFcOedQEXqeXPkuUzIqPCGdw13I7hMRkbYYrBMR+UyeXP/5klwztm5x4pL805FgPe+UfLrmK2k/fai0q8RhnYjIn3BUJyIqpcuXLxufuVnt++WeO47Sv8jptCR5p+FUmRVdm4M6EZGf4bhOROQATCJFi8OtW7dIamqqTJgwQZ555hnVNvGll0ZJXt7/Gd/pjLukavCjEirX5MKVfxv7bH79UY59fUkqNKkr1W47SufJ1UMfyPz9f5DFf765ht0Zx44dU/+fhQsXqv/bN998oxZ5IiIi32OwTkRUxKVLl1TAagbl7du3k4YNA2XZsmW2gP2I+p4+ffrItGnT5MSJU5KcvEouXy52KugdFTeZNO/Yftl2qa706vzfhZNOi5N3dpvMXiny/JgOUr0Uo3lAQE0ZPHiwBAbWV9tr166V2NhYdSIyYsQImTlzpnoucLKCkxYiIvKe3+TbGJ8rGJxx8CEiKgsQfKKbC3qNY2Ej9EuHiIg20qJFC6lXr548+OCDUrNmTbW/JK6Pnb/I6dS/SMe4/8i41DdkUM0zkjp9rMRdHCKfLIiWGiUE4XlnP5Gpc89Kn7hnpWFF45vyTsv2qSkio0fK4w7WriMIv3r1qvTv39/YczME6MeP58rZs+ckOztbVq1aKa1atZZmzZrZTmAaSnBwsC3IDzS+m4iISqvo8YTBOhGVKQjODx8+LAcOHLAF5hmyceNHahVPLA7UpEkTqVWrlkv90pF9X736vTsG9cW7Ijmps+TlmHckWypI0ICZMjcuSgLNIFxOSurQbhLzcTtJ2j1HouQzmTxopCzPvnVqaoUBq+SraY/fNiNvD6UvyKh36tTZ2HNnKJE5fvy4uvqwf/9+9RyidWVQUJB6Dhs0aCC///3vje8mIiJnMFgnojIHweWePXtk8+bNhcF5ly5d3JoVRqlI27ZtpXXr1sYea0DdPcp5SvM83O4EqGXLlsy8ExE5gcE6Efk9BI/79u1Tmd81a96TatWqS4cOHVQg7amsLyae7t2bKePGjTP26A+1+T17Rstnn+0w9riHffCelpYm586dla5du6mTmUaNGrm20isRURnBYJ2I/JIZoO/cuVPmz5+rMru9e/eSxo2DXCxNcQ4C39DQEDl0KMcyJSDeOsHAc/Pdd9/d9LfBlY3mzZt75W9DRGQlDNaJyK/s2rXrlgA9PLyZT7K36JwyYMAAy5TCoAQmPj5emjZtauzxDlzxwN/NvOoRFRWlrnwwcCciYrBORH6goN/5VhXsBQUF+zRAt4cAdM6cOfL+++8be/SF5xCBuq9/16KB+/Dhw7T4WxIR+QqDdSKyJJS5oK3i4sVL1DaysV27dtUuqPNVttpZ6B+P1pTR0dHGHt8zA/fp06ep7jLdu3eXsLAwdpYhojKFwToRWQo6uWCSIgK4UaNelsjISK0DYStk1xEUowPMihUrtAyEzfkHycnJkp19UPr2fVadnLFMhojKAgbrRGQJCHo3bNigsumxsWOkXbt2limNQO06JlDqlLU2IRAeOHCgjB492hK19ZicumnTJlmyZLFaqArZdqu1xyQicgaDdSKNIZDCapo//vijnD9/Xu07c+aMnDx5Un1uQvkCVKxYQerUqSsPPPCAX9T4mhlVZKbBKgFlUbga8NhjLeWLL77SLhuMRZDwekpISDD2WIN9tv3ixYvSr18/dUJUVkpkMMfg+vXrcuzYMbVd3LiARakqVqyoPscaAuXLl+fVCCILYrBOpBFkDTMz99oOxEdk27Zt8uWXu4yv3FCv3sMqo2j66aef1KIzxUGdLwJ5qy0Bj0AMCxYlJs5W/9c+ffpoX/N9J2iLiPp6nUpNcLUiLm68pKdvtnSQi8A1JSVFNm3aqEpk8HrxpwmpZqtLlCu5a1wwV5e1+vuKqCxgsE5+DVlpZJ9MyCwFBAQYW3owL+ujDts8CJsHXgTa9erVU7+3I8E27gtZxuPHc28J+HGf5kI0umaniwbpgwcP9qvVLpHFxnL8s2fP9nlwjEC9b99ntMz2uwqv/7Vr16r5DHFx8ZYO2s0T93XrUm4Kus0T8KpVq0qVKlVcGhfwGrS/T8z9wLig0+TdomO3CYuYEZU1DNbJr+Cg9Nlnn0l6+j/kgw/WyOOPd7IN7jcOZocP58j27Vtl+PAR0rFjR2nW7FGfHcwRLOESvnnQ9NTCMHhO7BegAQTuw4YN16p7CjLPU6ZM8csg3R7q18GXAbsZqK9Z875f1nvjNb9jxw510me1TLs5N2PVqpVqu1Wr1moybcuWLd36njDLiJCtR5vMY8eOqnHBl5N3d+3KUKvnpqX9XR56qMZNYzfgagHGdXP8btOmraWvCBE56pZYHMG6vVq1ahqfEenr0KFD+ePGvZrfqlXL/Hnz5uXbDkLGV251/fp19XV8X+XK96jbixcvGl/1LDz2+vXr89u1a6veW7i1BexeffyMjIz8uLg49fj4mDFjRv7hw4eN7/A+/D5PP/20+p18+Xt4C/4G+L+++OKL6nNvw/ONvztu/Z39+23BggVee5+5wnwf2I8Lp06dMr7qeVlZWWosMMcFb70f8TdKSUlRY/fw4cNsz8Pnt/074fsxfickJHh9/CbyFbwn7TFYJ0vBwI1BGwM9Bnxngx/zQIFBf+XKFU7/vDO2bPm4MEjHQdnXwRIOcAhgzIOzt4MZBAIICHR4LnwBzzdeD946QcFr23xMbwaBOrB/rSMI9uT73Fn2Qbr5XvDl72c+V+ZYhfeop14vCLp79uypEi1IuDgLvyuCdXP8J/JXeC/aY7BOloGB3sykl/bghkEfBwwcOE6ePGnsdQ8EY0UPxjqxD2TwgUykJ4MF+6ARj1WW4bWA58HTAaT5GkTghb93WYX/O7LHOrz2EADj6oqu4wJej3hdmuOCO1+juB8kRzB+3+4qqKMwZiMrj4+y/Pom/4X3oD0G62QJ5mVTdwz09j7+eLPKsuNSbGmZQSneQwgOkFnXGYIHBHP4fRE8eCLja15dwPPiyeDUSooGkO58XsyrF1Z4/XkTnhcEyp56nd8O/r5mEKzDScOdmK9P8/dFuUxp4P9vBtbuHgPMEwB3J1yIfK1oLM5gnbSHTLonMuAmXI7FgF+agN3MZOL9gwOdlQJTM9uL3x1BhTuYJwK+CI6swj6wxvPuaukBXmv4G5r35ekrJVZmvta99R61z6bj72OlLDCCdHNMc/VkGz+DIB1juKdg3PZEIofIl4rG4gzWSWsY5D2RkSkKJwKuBuwIjvC+cUcWylfw/CKYwP8DwUVpggrz+dA9g6gLBHQI1vH6QXCEwAhZcQTzxf0dsB+vMzy/9n8zBKLeCECtDs+RWZZ1u6sP+D4E9a7C38McF6x6lcN8DszXmDMnlPhZTwfqJozbuELKDDv5C7zn7LF1I2krOXmlrF+fKu+++65X2nXZDkTy9NNPyVtvvSWtW0cYe0uGVmhJSUmqPSLaME6bNs3yC7OgneILLwxWLd2WLn3bqdZxWLVz6tSp6vPXXnuNKye6AM/ht99mq97YtsBDbEGIarFnD6+1ypUrFy5ygz7UbGfnPCysFB8fL/Xr15exY8fe8t6dMGGCaqfoSl969NdH73f8rfzhvWAupgXTp8+4Y/tPjI2jR/9ZQkKayqhRo4y9noU2kK+88op88MGH2q2tQeQstm4kSzAvbXo7U4LHcyRDg6yReXkbWTp/guwZsoH4vyE76AhkDvH9zKaTleB9bF7VsM9+4z2N1zM+nCkNKzouYNtfYFwwy2LuNC6gYxc+vM08bvjT805lE95n9hisk3YQKGPAdaW1lzvcacB35qBlVSi/MIOO2/0f8RyhFAPPhzOXyIl0Ys45QcnHtm2fqte9+YFA3hH2gbq/nrTa/x9LSlKgGYA3ShdLYpZOElkZ3mP2yhkJdiIt4PJpTEyMvPrqOJ8tM40SmP79B8hf/zrZ2HMDyhT69XtWLenvr6tBAkoCsOImLuNj5UtcBi8KZQSRkV2kVq1asmLFCpa9kGWh3Auv4StXrsjzzw8w9hZAGRJW/bwdjFtjxoxRqxNjXIiOjja+4l9QboVxYdSol1WZD8p97NlOeuSNN2bKxImTfFaa9cILL6hb2wmTuiXyBwzWSSsffviBWnK6Z8+exh7fwIB/+HCObNnysbHnRqAOqGP110DdZB6YiwvYEbwMHTpE1a9iOX3WTJPVIeD+/POdxtbN0tPTjc9uVTRQLwvjwrhx4yQuLv6mgB3Pw/jx4+X111/3ac04fj+cLOCkAScPRP6AE0xJGxhYBw583nbQ26TFRE373wcHgIEDB6qMuisTzqysuGAEJy7AbDr5A/vXeEkOHcq55aS0rAXqRZkTaRMSZkh+fp4cPHhQZsyYaXzVt5Bo+fDDD+V//mexsYfIOorG4gzWSRu9evWSESNelM6dnzT2+B460uTm5srp02fK7AEZynpQQv7NDDpv5513lkmnTp2NrQJmx5iy/J7Ac7By5XKVYNm+fYdWHbH+9KfhEhn5B59fqSVyVtFYnGUwpAXUF1ap8qBWgTo89dTT8sUXX5T5IBUZRZTEtGrVWrVwMzPrRFaHkq5t27YZWyVbty7F+KwAAnwE6klJ88r0yevEiROlXbvH5ccff5SLFy8ae/XwyitjVTnMpUuXjD1E1sTMOvkcsrZPPNFBVqxY6bNJpSVJTU2VmJiX1GXe/v37G3vLLrNuv1q16mpCXtGyACIrw6RpBJ3nz5+X3bt3y5EjR1Tpm8ksgcP8DczjQN025myUdRjDMdkcsDaGTtn1+fPnq1tv9XsncgeWwZB2kFX/6qsvtal1NOHA3bFjB1uQ/rwtWE8w9hIykT16dOPzQmWGGcRXqVJFypcvL4891lJNvMbVJp6wFsCJvPm8FO0S40vIqtevX9d24pWr1UkE0e0wWCetmFl1XSaVmuwzRenpm3lALsK84lBcHS+Rv8K4gInm586d1S6DrANzBWTdrkQyu05WUzQWZ806+RRaokVF/VG7g96UKVNUf2Usuc9A/VboI40MGg7MrF+nsmL58uWqLCYpaS4D9WLgxB092CdMGK+uRuji2WeflYkT41i7TpbFYJ18BlkqTP7p3bu3sUcPyA6ZE8ewWAoVb9q0aVKv3sMydepUYw+R/0LwiY4xCEabNm1q7KWisKgdxoX4+Hg1xusAJ1ZTpkyX9957z9hDZC0M1sln9u3LlPbt22s1qRQHF2TVkTX211UI3QUHQHSCQKcclMUQ+SuMCwg+EYQiGKWS4UokrjzgCsS6deuMvb6H7PqqVcnanEAQOYPBOvnMrFlvSVRUlLGlB1zmRvkLeorTneGyNyaaJibO5iVm8lubN28uLH9hWdyd4cqDWQ6jS5kckgtIDpW0Si2Rzhisk09gddAzZ05L69YRxh7fMy9zox0by18ch0lbOMFZsmSJsYfIf+AkFJOpcVLK8hfHDRs2TF2JMCd36gDJIaxqSmQ1DNbJJ7Zs2SIjR440tvSwbNkydXAZNGiQsYccgb7T6P4wf/5crSaVEbnDrFmz1O3YsWPVLTkGmezY2DFq/g/aveoAyaH9+/fLqVPseEfWwmCdfAK1g926dTe2fA+LnOCggoMLL3M7D5OEcaKDEx4if4GTT4wLOBll9xfndenSRa16jMnoukCSaNu2T40tImtgsE5et2tXhoSEhGh18JszZ446qODgQs7DCY6ZRWN2nfyFebVNt45VVoFxYfTo0areHwkRHXTo8IQsWLDA2CKyBgbr5HWffbZDnnrqKWPL93AQwcEEBxVm1V2HEx1m18lfmFl1Xm0rndatW6tECBIiOggICJCHHqohWVlZxh4i/TFYJ69LS/u7NGv2qLHlexs2bFAHExxUyHXMrpM/SUlJUSefvNpWerpl13v2jJYDB/YbW0T6Y7BOXoVshk4lMGb2rF+/fsYeKg0zu45Ah8iq0AEGE6aHDRvOrLobmNl1JEZ0gFKY9eu5NgRZB4N18qqMjAyJjPyDseV7W7duZfbMjRDY9O37rAp02HedrGrt2rXqtmvXruqWSg8JESRGdOi7jlIYtA5mVxiyCgbr5FWffPKJNG/e3NjyLQST6KuO4JLZM/fp06ePut2xY4e6JbISrHC5Zs17alEfdoBxHzMhkpaWpm59rX//AfLtt9nGFpHeGKyT1yA4RjYDWQ0dZGbuVbe6raJqdQhwsIDM6tWrjT1E1rFv3z61yFdkZKSxxw1+zZTE0ACpXfvWj8ZDZ0tq5lnJM771hjy5mrNDUhKHSOPC728okfHLZXvOFeN7TCcldWjITfdb9CM0MVN+Nb4772ympN50v7f7PdwDCREsOIcTIR2W/A8LC5VPP92mSiGLfuiy6iqRicE6uR0GOgx4mEyUmpqqPrZu3aI+IiLaGN/le4sXL5Fu3XqoRX3Ivbp3764mlOmyGAqRo3bu3Knqqz2yWumT82T3iVNywvzI/VpWhhyUuOgoGZl6wi5Q/kXObk+QpzoOk8UXnpB3dx8t+P7sDfKXB3fKiI6dZOjyg3LV+O5C94+R1KN292/3kRUbLnfZviXvdKqM7PCsLLrQQ9KyTxR8vcTfw71Qu44TIZwQ+QLGo4ULF8qIESOkb98+apIpulcV/YiNjVUnMPg+fD9+TocTDCq7GKxTqSEwX7VqlRrYMMBhoMOAh4Oeae/eTMnK+kaOHv2+cBDEz/gqmMMJBYLJ3r17GXvIncLCwtStLt0fiBxhTiz12tW2ctWlecxb8m5sTdkUt1R2XCkIk/NOb5LJI5JFYlfJh9Oek/Dqd6v9UjFQOsYmSurr4ZIx6TVZkvlzwX6H/Vu+//gD2XSts7z4cncJrGiEAPg9XnpVxoVelk3vfCrfeyhaxwkQToTsjw2ehrEeAXf79u1k0aJF8tBDD8mYMWPUScqGDRslISHhlo/3339ffR3fh+/HzzVsGKjuB8c7Iq/LL6JWrZrGZ0Qlu379ev769evzn376afWRnJycn5WVpfY74vDhw+pn8LPt2rVV93Xx4kXjq56Hx8Zr3ZuPWdbMmDFD/W2JrGLLlo/VuHDq1Cljj5v8v735s5vWzK81ZH3+GWOXvf87vCy/R60m+UPWn7Bt/Sv/8LJ++bUaxeSv/+E/Bd9Q1OVt+RMa1cxvNGFb/mW140T++iFN8ms1fSt/7/9TO0rwz/y9s5+0/R/75S87/C9jn3eZY6+jxwpX4W8YFxenxiA8ZmnHevw87gf3h/vFMYzIU4rG4sysk1NwKRBlLZGRXeTQoUNqGWlkIfr376+yJo5O1AwMDFQ/g59duvRtOXPmjISGhqhsuzcuN6IrDUpgOIHMc9q2basuebMUhqwCVwCR+fV2aVy5anWlSYVLkrHne7mSd1wyUr62DZJhEmxm1Iuq9N/SuVdduZbyqWQa2XjHlJcG7btJkHwmk6JGSWJKumSe/cX4mne0bNlS3XqqFAbHDxxHHnuspbRo0ULS0zerY01px/qCuTj91f3hfocOHeK14xURg3VyGIIuBOm7d++W1avfk3Hjxqmgu7RwHyiLycrary5Z4jE8GeDhUvfGjR+xBMbDzFKYAwcOqFsinSHo8moJjL3yleTB34pcO/ezXM+7Jj+duCZS9T65p8Qj9O+k0oMVbD9wSX6+bhesX5ot0Q/fPLG04KOLJGaiwr2cVAwfJHOXvSpPymZJHD1Uols8bPt6D4lfniIfeXCCqQnjPdrleqIUBscPlK5kZ2er40l0dLTbO33h/nC/6NOOxxk4cKB6XCJPYrBODkGtXkzMy5KUNFfV9Hki84TMBU4A8Bh4LDymJ7IW3333nbpt3DhI3ZJn4KCGrjC6tGojup3Dhw+r2yZNmqhbSypxgulmiQ2vaHxTJQnsGCNLvz0qu1OX2sbbv8qAoMOSPClGRkVHyfDiJq66Gdrlbtq00dhyDyR4+vV7VrWIxDHK01dNcf94nOHDh6nH5fwc8iQG63RbCJaR9d6/f7/KJHikQ0IReAxcajx58qTKkrg7YEdGB5kddoHxvCee6KAm8nKBJNKdeQXIG2PcLa5fkQv/sQWAwbXlwXIVpHLtCiLnf5Z/lpjm/rdcuXBNpML9cl95Vw/jd0v18EiJjh4i09K/lexP5tmC9svy8cy18rVTpTXOw3OMEjl3ZaQRKPfo0U2mT5+hst7e1KlTZ5Vgiosbz4CdPIbBOpUIQTKC5ZCQEJXl9nSmwh6yssha4LHxO7gz2ENGp2vXbsYWeVKdOnXVrXk1g0hXmMeCK0G+kHcuVw5cqyvdwmrLXeXqSESvR0Vy9snBkurJr/yvbEnJlQq9npDwSu44jJeTioGRMqiv7XGvfSX7jlw39ntGo0aN1K07FiVCgIxA+YsvvlKtIX0BJx8oDWXATp7CYJ2KZR+oI7PuK3hs/A7x8fFuybAj6EdGp1mzcGMPeZI5p+Ho0aPqlkhHGFswjwUTB73vguxYvlSyKjwhncOREPmdPPLk09JV0mXRop1y9pYk9xU5lLJGUq41l+G9Q6WSsdchV7ZLfOMAaRy/3XYvRd0l99xXWaRCSwmrX97Y5xlI/GAiLyb0lgYy8wiQkVH39ZVSPD4C9r59n2F7R3I7ButUrClTpkjdunV9GqibzIAdJw+lxXp170O2EllLIl2dOnVK3darV0/des3VHPkkcZyMSK4sgxa+KO2MLHm5Gl1l8sIBIstHyqCJ797o2KK+P1aiJ2VKxOtTZVj4fQX7HVWptYya3lsk+VUZk5hq1wnmFzmbuVbmLsqSJ8f1kUfdkq2/vWbNmpWqbh0nWFjTY+LEiT7LqBeFgH3NmvdVpxh3JJeITAzW6RZozfjTTz9JTEyMscf3zJMGtMoqDTPDy3p170G2EllLIl0dPHhQ3TZo0EDdeszHL0kL+y4tQd3lzQut5Y3UJTL58Rp2B+S7pfrjr8mm3e/Jiw9+Ks+pji3m97eVhZ9slaWDgsWcMlqoxG4wto/Q2ZL5691SI/pN2ZY6WdpceNvoBIOvPywdFpyTZuOXyZzi7tcDGjZsqK5yulrimJSUpAJ+1IzrBCcOKLPE70fkLr9Bs3XjcwVvXMwep7IJlxXRnxb1f7oFtBjUe/aMVn3ZXW0ZOWHCBHWLenjyDtRw4tKwjq8pIpg5c6bs3btXrftA3oFSkY4dO8gnn2xzejzHzyJ7jUYE7m7N6A7IqqMFMSae+mTCMlle0VicmXW6ydSpU20DzDwtgyrUOeKSJ+rXXZWR8bnUqlXL2CJvqFOnjro9fvy4uiXSTW5urtSvX9/YIm8ICAhQt+ZVDWfMnj1bHQt0DNQBvxfq6LFoIJE7MFinQsiAXrx40eutr5yBS54PPPCASzPuke3AZdeHHnrI2EPeYJ74nT9/Xt0S6cZ3k0vLLgS0aKGL1audgX7qOE7pVv5SlFlHz+4w5A4M1qnQnDlzZPTo0caWvjDRFL+rs8xJZMHBweqWvAedHw4dOmRsEenDrJmuWLGCuiXviYhoo9bTcMbatWstcZwC/J7JycnGFpHrGKyTYi7vr8us+tsx6xudzVhcv+7Z3sFUMpQYXL582dgi0geytGCuCUDec++998qRI0eMrTvDiRVKGcPCwow9esPxNDv7oNsWf6Kyi8E6Kenp6dKvXz9jS3/IWGAlUmccO3ZM3bo6OZVch3kCOMgS6YYn8b6DjjBY4dhRmZl7pW/fZ7WtVS8Oft9t27YZW0SuYbBOyvz5c6Vdu3bGlv6QWcHvzF621oB5ApgvQKQbnsRbx6efbrNcdxVk17nOBJUWg3VSJTDduvVQ3VasApkVLLazb98+Y8+dYSITJjSVXp5czdkhKYlDpLHZw7h2Cxma+IFszylYFzDvdKq82Ni2PzJJMq/esgShyNU9khjZsMjX73y/yq+Zkhhq+9rQVDlr7LrF2VQZWngfxX10kcTMq8Y32x75bKak3vS4AdJ46GxJzTxr+62IqFTM92zt52R5zr+NnTf8mjlbQmuHyNBU+/rt4saDhhIZv/zm8UBOSurQkML3bUkfoYmZ8qvxE7owF6FydMXPVatWlqIE5hc5u2eRDMW4rJ5Hu8WmHJB39hOZZBuznX0ecXKBCcxMLFFpMFgnlVmKiIgwtqwD3RucWcYeE5kwoal0bAP+9gR5quNY2SR9JS37hOqFmrt7joQcfFOej3pVlh+6olYgnDS9t1TIni8TluyVG2Ex/CyZSxIkMTtIYhOel/CKeBs6dr/Ouj82TY7a7gf3dfPHZokNL1j6BCcWIzs8K4su9Ch83BO5X8vKkIMSFx0lI1NPlDpgr1q1qrp1dQEUIv/wmcx8K11O3/ENZY4Hw2TxhSfk3d1HC96X2RvkLw/ulBEdO8nQ5QeNcaWWRC/df+O9fTRNYpF3eXKe7Db32T6yYsPlLvX9+ihfvrzx2Z0hoMdEdddKYGwnPpmLZFDvjVJj4W7Jzd0m4/9rjTw3eZMDfwubvBOSNjlOlmdfM3Y4B8kws8EBkSsYrJPs3r1bmjRpYmxZB7q6ZGdnG1vekXd6k0wekSwSO1/mxHaWQBVo295I1VtLzOw3ZIBskEl/SZGcvLulRtQrMr3rvZK9eJVsPW1mcHDQWCkTErMlKHZC4XLhjt+v2u1G/5bvP/5ANl3rLC++3L3wcW0PLM1felXGhV6WTe98Kt+X8nGrVKmibs3JfES6QJciBFPecm1TgryedvsT4BvjwSr5cNpzEl797oIvVAyUjrGJkvp6uGRMek2WZP5csL8MwNwCl3vh5+XIh5Pny/EBr8hYrBRbroa0G9BbAjfNkvk7LhjfVJJf5HTaWzI5t7K4ur5tSEiIHD+ea2wROY/BOqnZ+M5kOHSBfuvenbRoBraPSt/uTW5dkrtSa3n53aWSNCxE7sF2udoSNWmCdJV1Eve6kcExDhrZQaMkYVgz4z6cvF+3+lX++RMOVj/Jz/8scnG3XEMZlHZITqQNkkCOFOSn0KWocuXKxpaHBXWSJ4NsJ8Bxb0la4Ql8UcZ4IJHyYp+mxSz9X0ka9uorvSrskcXrssT5623WhCvArvbCz/v+C0nJ+q1ENH/E9uwVKPfIY9Ir9EdJ2fK/t3kO8+TqobUy6Z1H5J25fxJX+wVhzs7Vq65l5YmAh2BSs/GtOLkKNfZenbSYd1wyUr4WCe0sEY/8zthp726pHh4p0T3CpbqZoDbKYURl03LkZNoCmZllX/5i48L9uk95adC+mwTJZzIpapQkpqQ7VcdJRE4IiJJxCaMk6JrdCXxR5ngQGCbBZka9qEr/LZ171ZVrKZ9K5hW3X27zM7aA+4ejkiMPSXBtu5OycuXlvqq/lWsHcuVcSU/h1b2y5C9fSZs3B0n4PQyXyHf46iNyVN41+enENZGq94nj4zbKYUYWlJPEdJCImHUiA/4sQ4zyF8Wl+3XMpcQoebiYyWa1Q2dLpkqkl5OK4YNk7rJX5UnZLImjh0p0i4dt39ND4penyEecYErkRuXknvDnJSG2ecnlMA6NB7+TSg9WELl2SX6+znfo7eXJ9Z8vSYl57ROX5J/FPoWYW7RKfhozQZ5vaObjiXyDwTqRp5ULlKcmj5IgfF6ht0wf1brwUqynlTjBNGuMhBfONqskgR1jZOm3R2V36lJJSvqrDAg6LMmTYmRUdJQML5zIRkSld5+ED5sgsXcshyGvqH1/MSdFmOC7SBZIf3kVNe7GXiJf4WuQyFF3VZF6j94vcv7nEjIxJSknFUPaSBd0aIjoIK1qFLm07fL9uptRbhM9RKalfyvZn8yzBe2X5eOZa+VrXmoncp+KzWSYKodJl0Vrv7n5ZLhcBalcu8IdxoN/y5UL12wn//fLfeV5GL+9u6Rq8KMSKtfkwhW7tpm//ijHvr4kFZrUlWq3PIXn5MtVq+XjxJ4SZF6NbPGSfGz7SsHVypck9axujTDJn/FdTuSwKhLcpolI1hbJ+P7WXsnK1RzZnups3ben7rc0bCcYgZEyqO+jIte+kn1HuMojkfug/AzlMEGSnZggiz47bey3KVdHInrZ3nc5++RgSe/3K/8rW1JypUKvJyS8Utk4jFesWEGtleGK4iaT5h3bL9su1ZVenf+7mCudRdph4mP3PHnS9pWCq5XzJLq6bo0wyZ8xWCe1UJAV+197f5GJ38kjTz4tXSt8LWs2HLi1NOTqQVk++jl5ftFh25HFmYHcU/frgCvbJb5xgDSO315MR4S75J77KotUaClh9a3XLYjIEUFBQV7uKmW6T8KH/FkGVNgjCxLXyI0R2BgPJF0WLdopZ2/Jrl+RQylrJOVacxneO9RrJXW+VqdOXbVWhkvK1ZMnX4gUwfOG9SquHpK05eskp+tYGdXuQeObPAftkc0FoIhcwWCd1EJBVux/jUUmnO2P/NNPPxmfuQbdXSYvHCCSOEpGJ26RHLX6KFYa3CKJowfLpIxwef1vg250enGQp+73jiq1llHoVpP8qoxJTLXL3P8iZzPXytxFWfLkuD7yaCmzdwcPHlS3AQEB6pZIFxUrVvRuVyl7xvuvgrFpKhwPlo+UQRPtVtq8miOfJMZK9KRMiXh9auE6DWUBWvViBVPX3C01oqdI2vSqsqZzkNQOipJFMkTSZkVJDXNoM1d9vt3K0C7CyeCDD3r+pID8F4N1UpklM5iyEiwyUbeu451v0aMXyz6Xzt1S/fFXZXnqZGlzIUk6BtWW2rVrS1DHeDkY/BdZmfaGDHKpc4AL9/vxS9LCrKcs/Lh5yfISu8HYPgqWzcZB7E3Zph73baMTDL7+sHRYcE6ajV8mcwYFF9Pr2TWurT5I5K/MxdNqGNsmjAevyabd78mLD34qz5nvy6Du8uaFtrLwk62y1I3vS18xFwpCIH4naNWLFUx/+OEHY4+zKklg9OuSrspaDkn6tOgbi8BB9WhZiq8tjZbqxq6bGF93diVY86p1zZo11S2RK36Tb2N8rmBAQH0WlR27du2SDRs2SEJCgrHHGmbOnCnNmoVLp06djT23l5qaKjExL/H17QN87klXW7dukRdeGCxZWftVQEje4+y4sHDhQgkMrO/wmK8DvL727s2UcePGGXuI7qxoLM7MOkmjRo3U5UXv14CXzqZNG6VxY9UQ0SGYoARWrM+3OtRsenNJdyJHoRYarFgKaHVXrzrXFLZ169aybl2KsWUN+H3btm1rbBG5hsE6qWwSAql9+/YZe/T3zTffSLVq1Z26tMiDsm95bUl3Ihdcv86OR96WnZ0t/fs/b2zdWdOmTW0/c7AUpTDehcQQft+wsDBjD5FrGKyTMmDAAFUKYxXp6ekyfPgwY8sx5csXdDT58ccf1S15D67cYG4EkW4CAwPV7bFjx9QteY8rE/6HDRsuaWlpxpbe1q5dK337Psu5OlRqDNZJwZk/Zqzn5OQYe/SFbMX8+XNVFxtnmFn48+fPq1vyDrO8Cl03iHSE9rWu9vAm12HCPyb+O6Nr164yffo07csZ8fvh9+zTp4+xh8h1DNZJwZl/bOwYWbZsmbFHX8hWxMXFu5StQLkP6qfJe9BiE4KDg9UtkW5w4r9//35ji7zBLGWpWrWqunUUyjYTEmbIrFmzjD16WrJkiTpOcdIyuQODdSrUpUsX7bPr+N3WrHnP5WwFWj0eOXLE2CJvMNuCOtKejcgXUKJV+rau5IwLFy6o2ypVqqhbZ/Tu3VsdqzB3SUf4vdAAYdCgQcYeotJhsE6FkKmeOHGixMfHa9sZZvbs2eoKgKvZioYNG8qXX+6yXOcbK0N5AcoMmGEiXT388MPq1vMTF69IzvYPJHFoC2M9A9tH4yGSmLLDWAjtFzmd+mdpXLuhRCbuuXU1YyyUlpkkkbd8/U73W+DXzNkSWmQthpv9KmdTX7pxH8V9hM6WzF+Nby+FAwcOqFtzzoAzcKxKSporMTEva1cOg2MLfi/8fqxVJ3dhsE43Qf/a+vXry/Lly409+li1apW6jY6OVreuMEsxDh8+rG7J87Zt2+b0/AIib6pTp466/fbbbHXrEXmnZfuk56TjiHSRZ9+V7BNYnOeo7F7ZXA7G95Oo0e/Koat3GYsk3SvZiQmyJPNn44cNV/fKkgnzJTtolCQMa1awKJJD93sjYHdMsMSmfqf6PN/ykTVGwp1ZFagE6ARTmnau6AyDyaa6JZfGjBmjJpXi9yNyFwbrdAtk11FqgsWSdIHfZcmSxTJt2jRjj2vM5e7NrA55Fg6iuJLh7CQyIm/C5HNc/cnJ8VSJ3C9yOm2mjFh+l8S++5bEdgw0Vh+9W6o3Hy6zF74g8vFU+cuHOZJXrrZETZogXSvskcWLt8vpwjj7Z8lckiCJ2UESm/C8hKvVN524X7VfH+gQFRERYWy5pn///qolbFJSkrHHt7BoE7D8hdyNwTrdApfuVq9+T+LixmsRsKP+r2/fZ2Tp0rdLXUqB/xuyORkZGcYe8iTzCgYnl5Luunbtpq4CeUTeMfn4nXS5FvpH6R56n7HTVE4qtRsp786fKcNC7i3YU6OrTJreW2RTgryedkIF2nk5qTI5MVuCYifIsHDjPpy8X12Y86KaNGmibksDyaXc3FwVKPsyw47HxyRllGqy/IXcjcE6FQuZJtTcIUj2ZcCOx+7Ro5usWfO+S7WNxUE2B5PJWLfueeZrx11/OyJPadYsXF0F8kQNdN73X0hKlkhor8fkkeKOuuWqS3iPaOkRXt04KN9tlMPY4vW4tyTtZI6kvbVAsuzLX2ycv189mJPOGzRooG5LA4ExAmQEyihB8fa4jsebMGECA3XyKAbrVCLU3H3yyTaVYU9NTTX2es/WrVvUYyNQxzLT7mJmc1i37nnIVI4a9bKxRaSvxo0LFu3KzNyrbt0p75+X5IT8VqreV97xgy7KYV4ZKaHX1klMRAeJ2XS3DBjfzyh/KeDS/TrkoCRGNyp2gmloYqaUdn7p5s2b1RVOdwW2ZsAeEhIikZFdvNbRDI8zcOBAuffeexmok0cxWKfbQkYUJTGrV6+WmTNneiVrgcfAY02ZMkU9tjsDdcBJCOpTdSjx8WforIFMJTKWRLrD1cRWrVrL3r2Zxh7fKxcYLZNjm6vPK3QdK6PaPag+97ySJ5hmxYZLaeaXYnzHlc3S1qsXhUB5xIgRMn36DBk6dIhHy2Jwv2h4gMfBStrjxo1joE4exWCd7ggHsRUrVqjPkbXwZJCL+8ZjQHr65sJVR90N9amYREues2fPHnUbHt5M3RLpLioqSq2O7O4g764aD8uj8h85//N1Jyd63ich7dvK/bZ/EZHNpEaRI7br9+s7+/btU7cdOnRQt+6G5M769aly5coVdSzBVWF3/T1xP7g/3C+62eBx0EGNyNMYrJNDkDVA9gCTPOfMmSPPPOPeWnbcF+4T941aeU9nKtq2bSvHjh3VdlENf4CrMbjUzf7qZBVmiZwZULpN1cbSJlQkK+UL+b7YqDpPrubskNSPMuWsM1G3p+7Xg3bu3KmuYHgqEQMYc8zj1aFDh6Rhw0CVaXd1vMfP4ecRpGMFbNxvQkICxzbyGgbr5BSUxbz//vsyevRoFVi3b99OXQ50ZTER/Ax+1gzScZ+4b2/0pw0LC1O3LIXxDLMEpnfvXsYeIv2ZJXIIKN2qXD158oVIqZD1d9mQVaR3OgLqQ+/K6KhhsujI/4ldSfqdeep+PQSTd3HlAlcwvAHHKwTtWVn75aGHHpJFixapuntMCEWGHOM/6s6LfmA/vo7vw/ejZfA999yjMukI0jlhnrztN/k2xucKXpioSyNyBIIyTCJMS0uTc+fOqsVv0FO7YsUKUqdOXeO7Chw/nitXr15TmQksFV2tWnV1KbRTp04+GfyQKUEpDMptWG/oXjgJmzBhvDpIMvtEVuKx1y4WL5r8J3n+g6oSO2+8DFM90a9IzifLZeZLb0hGxHRJnfOcNCwSVWPV0Uejl8ujSRtlaXQtY68dJ+73jvelVjAdLS1iciQ2dZ3Ehpt9Z9wDTQNeeGGwfPHFVx7NrN8OThhOnjwpx44dU1n3PXt2y8MPPyJ33XVzJT6OY1WrVpVGjRpxDCOvKxqLM1gnt7EfBLHEPD43nT9/Tg2GkZF/0GYAxKVNsy2kuyexlnW44oJ5AchqEVkJEhCPPdZSkpLmlWq15GLlnZXMjRtl3fw3JDn7WsG+Cl0kdtpA6f5kGwksJv195wDbxsH7Lbiv2VJ8c8poSdo9S1p9OdYWrN+u+xcmn7oWyOMq6gMPPKASJTrAMatbt662k4cvjT1EemCwTj5x6tQpiYmJkZSUFGOPHnQ7ePgDXEJGf36eBJFVoavIxYsXVVkeuQfKSzp27CDvvLNMm0mZWVlZ6urqjBkzjT1Eeigai2tQxUZlAZb53759q8daabmqX79+qo0YDiTkHsnJyWoCGQN1sqoBAwaoORec0+I+SNRgPgBKJXVx4MB+admylbFFpC8G6+Q1w4ePUDWCOunSpYs6gOiW8bcqnPTg5AcnQURWhRNNnHBu2LDB2EOlYU4s7dv3Wa3mB2El1Xr16hlbRPpisE5egwwGMhk6wYEDBxAcSHBAodJZtmyZOvnBSRCRleGEc9Wqlbzq5gZr165Vt3369FG3uli8eKE0bNjQ2CLSF4N18prg4GD5/PPPjS19mAcQ84BCrkFQg+AmNnYMu+uQ5ZlX3XACSq5DEmT69GkSFxevVVcV1Ks//XRfjlVkCQzWyWsaNGggH3ywRru6dRxAcCDBAcWVfvFUgFl18icI4nDiyex66eiaVcdV3jZt9KmfJ7odBuvkVahb37cv09jSBw4kCDTnz59v7CFnMKtO/sjMrsfHxxt7yBlIfuiYVQdc5W3SJMTYItIbg3Xyqo4dO9qC9SxjSx84kJhZNFeXpC7LEMwwq07+BieeEydOVJ1hsKAPOQfJD4wLumXVUZqzf/9+CQ0NNfYQ6Y3BOnlVs2aP2gLiZGNLLwg00QECS0vrVqqjMyzLjWBm+vQZzKqT30FP8G7desiUKVM4LjgBbS/Nq226ZdX37v1aoqL+aGwR6Y/BOnkVBu2HHqqhJvfoBoEmMsQIPDdv3mzspdtBhioxcbb07/88+6qT3xozZowcO3ZUli9fbuyh28FJTVzceJX8cPsqsG7w4YcfqlWWiayCwTp5Xc+e0ZKRkWFs6aVp06Yq8IyJeYmTTR0wa9YsFcSMGjXK2EPkfwIDAwsnoXOy6Z0lJSWpcQFXKXWDBAMaHYSFhRt7iPTHYJ28rkOHJ7QthYGxY8eqOsupU6fysvdtoIYXl7mTkuZJzZo1jb1E/mnQoEFqXBg6dAjHhdvAnB+sW5GQMEOd5OgGJTB/+UscS/bIUhisk9cFBARISEiI7NqlZ3YdpTqov8ZKnOvWrTP2kj1cdUANL2p5dbzMTeRuCO6WLn1bZYyROaZbIWsdE/OyKn/p3bu3sVcvLIEhK2KwTj7x1FNPyWef7TC29IP661GjXpYJE8ariVJ0A7KKuOoAr732mrolKguQKUbGGJljdoe5Feb84GQmMTFRy8z1qVOnVBeY1q0jjD1E1sBgnXyiTZu28uab01UmRlcxMTEqc4yJUqxfvwFZRVx1wNUHlr9QWdO/f381LrzwwmDWr9tZuHChGhfeeWeZtuPC3//+dxk5cqSxRWQdDNbJJ5B1mTJlum1w32Ds0Q9+RzNzHBsbyzpVG7RpNOtR2f2FyiIEpREREYX16zyRL2jTaC5+hFaXOsL4jblS3bp1N/YQWQeDdfKZzp07y4IFC7QOgpEhSkqaq9o5on1bWQ7YcUBGlxx0y0F2kaisQaCOMgrUY69e/Z4q+SjrE9ExLvTt+4y62oBJuLpKT0+X9u3ba9fzncgRDNbJZxo0aKAGz88/32ns0RPaOa5Z8766xItJlWWR/QEZKzoSlSUIxmfOnKkC9dmzZ6urbjiRN8eFsnoi//333xf2UzefF10lJyfLCy8MMbaIrIXBOvkUBs+FCxcZW/q6du2qmpSEVoXIrpUl9oG67gdkIndDEI5g/PLly7e8/lEKVlYDdpT/9Ov3rOTmHlUdYHQeF9B5rEqVB1WCiMiKGKyTT2HwbNAgUA2mOjp8+LD06tVL+vR5SurWrVu4MAoC9rJwYDYDddTnYoET/J8xqa4sZhGp7EFAOnDgQNVqNiEhodiAtCwG7GagDv/85xWJiupmGxf1bRgwa9Zb6u9IZFUM1snnkF1/5ZVXtDrI4aAzfvw4adXqUdm+fauxV2TEiBGFAbu/H5jtM+qoz0Wt58WLF6Vjxw7SsGGg1K4dIM8884xMmDBBnbxg8ina2SGY56Q7sjos7oOAdPjwYep9fztlKWDHuGAG6m+99Za6BXT3ql+/rqxfv16r/z9+HySE2K6RLC2/iFq1ahqfEXnPuHGv5qekpBhbvnP9+vX8lStX5FeufM8tH/gdTVu2fKzeKy+++GL+qVOnjL3+Izk5ufD/h+fE3tNPP62+dqePrKws4yeIrMUW4OW3a9fW6ddwRkaGeu3jZ/1xXDDHPYwB+P8dOnSo2LGyVauWtufic+OnfAdjF34X/J5EVoL3mT1m1kkLL730sgwZMtCnl1FtByJ54okO8uc/v2TsKRnak5mZNGSZ/KXfMjJimEiHxaDQ9aW4GvXRo0cbn5UMrR0xMZfISvD6x5WizZs3q6tJzr6GkWH/6KON6nOMC8hC+wM8L7h6ht7yuNK2YsWK2/ZSP3QoW7p3/4MqIUQpoa98+OEHEhX1R9aqk+UxWCctBAQEyN/+Nk/eeGOmsce7kpNXqrp0HGQchQPzF198pT5HaQjKQKzMrM81+6iXVKMbFhamathLgoM5WzuS1eCEOzKyi9SqVUudpLq6sA8CfAT61apVV2Vkq1atsnRZDMYFlPaYfdSdmWSOEsKBA5/3ScCOx0Rr4D/96U/GHiLrYrBO2njqqadtA2yOynB724ABz8uGDf+Qxx/vZOxxDA7o6embVRYaPchR22rFem2caDz2WEs5d+6sygzeLtjGgXrYsOHG1s3q1KmrJqISWQUCaQTUWOAIayrgPexoMFoSjAvIPo8a9bK6SoVg14rjAuag4AoBriBiZVJnnxskYD79dJvXM9v4m44fP15ef/119lUnv8BgnbSBg8CMGTNk0qRJPimHwQSkd999V95+e4WxxzH4vZGFTkqapw5qCHoR/Fohm4ZsIg7A5mJH69enOnTpv2vXrsZnN/vPf/4jJ0+eNLaI9IbXP64mZWdnO/zadxTGhXHjxqkgNzv7oBoXrJJlx4kFyoFQ9hIUFKyuIDqzMukLLwyT/fuzVRKktCc+rkD5S4MGgdK585PGHiJr+w0K143PFXR4OHHilLFF5H0oSfn888/lf/5nsbHHu3AwbdeurTz66KOydu1qY6/I8OEjbCcTty/TwUFu/vz5qh87FgpBfTfKZXSDk6G1a9eqS9soacFCR84uE47adpTMmFA606RJE9VzuWvXbrbbGJ8cqP0F/kY48Tl27JgcOnRI9fkuTosWLaRq1arSqFEjZhEdhPf4unXrZMmSxS699p2Fv+WsWbPUuID32/TpM7QcF/C8LF++XI0LgAREdHS0+rw4KDVBxywTrkxWqFDetu8xGTVqlLHXu9AGGN3FNm7cxPcDWdYtsbiaZmqn6AxUIl8YPnxY/rx584wt77J/bHQR6Nmzp+pwYN8N5k7QFQIdIfB+QkcVbOvg4sWL+QsWLFC/Fz7wOfa5wnagLrwf/B9N6MAwY8YM9f/X5f9tFfhboOMGnk88r3FxcaozCZ5HPN9FP7AfX8f34flGlw508nH1b1oWoMMLnic8Z95+nszHxt8Wt7q8P/CeNTvg4HfD+9eR58a+Gwy6eeF+zA4svugGYzu5VY+9b98+Yw+RNeF9aI/BOmkJAz6C5I8/3mzs8Q4E6QjW8fj2cCBKSEgwthxT9ABoHpyL3rc3ILCzD9IRqLijtRz+T/j/FXdgx2OaQZE/trFzJzxXCJDwt8HfydW2l/g5/Dz+Jnjecb9UAK9RXU4i8fj2QTtO0HwxLuB9WXRccOY1g2Ad42LR978ZNHszYMfzh8f09jGDyBPwfrTHYJ205e0Bv6RAvbRwfzgYmwdnfJQmIHMUDqDFPa47A2cEHXf6f5gnLHhsR7J1ZQmeDzO4xvPkrtce7sd83n2RQdaJ+Vzg9Y+rDu5+f5eGfdCOD5xMePqE3hwXzKs35uO6+8QO4zbGb4zjnobny5dXY4ncDe9Le6xZJ63ZAkt5+umn1Ep5nlyBDnXm+/d/I3Pm/M2jddZYFTE9Pf2mWm90jGjWLFx1UgkMDDT2Os+scT5w4ICkpaXJl18W9HhG7bztQCbh4c18VsNp1sivWfOexMaOkS5dupT5enb04I6LGy99+z4rgwYN8sjzYdZmoyMJJjp6ujZbN+ZzHBHRRsaOHattDTMmum7dulW9P44dO6r2YcI35iMEBweXalzAa8AWiKtxwXYioCbBA8aFfv36Sbt27Tz2vJj1454cv3GMmDLldQkJaeqzOnkidysaizNYJ+2ZAXv//gPcPhjjQDZnzhzJzT3m8UDdHh533759Kni3P0AD+pRXrlxZHagBkwerVKmiPjcdPHhQ3Z45c0YF6BkZn99yHwiImzdv7nK/aE8wJ+Di90XQfrvJa/4MC8zg77506dulCsQchec9NjbWdlLYrExM/EWQjvf1Aw88oNomeuM5dgczsMbvv23btsITbig6LlSsWEGd4Nszx4WrV6+qDjfFjQsRERHSoUMHr40LZsCONoru7s6C5wp93EeOHKk6zxD5CwbrZEk4iI0e/Wf1OTopuCMThJMABC4IaNG1xZcBDIKp48ePy9GjR9VB9siRIzcdqG8HGbh7771XGjZsWOosnLcgk7hs2TL1/9S1Y44n4HWM4BGcWVzGHfDYU6ZMkZ9++snrj+0tZpAO/vC6whWp7777Ts6fPy+7d+8u1bgQEBDgs7+5J8ba9evXq0X0PH3VlcgXGKyTpaGt44IFC+TVV8dJZGSkS4M+ghb04cX9eCLb424I5K9fv25sFfDlgdedzKDdzLT7c3mMLwN1e8jq79+/368Cdn8L0h1htXEBr3/8jdLS/l6qABvZ9LfemqU+nzhxkvo/E/kbButkecjSzJs31zZo58iAAQMcDtqRpdq4cYMK0qOi/qiWoWYfXj0UDdo9WUfrK+hLn5ubq0WQ7A8BO4I/vF4WL16itsvSFRorQ7CN1UVhxIgXpU2btg69BlFOg7k4n332mSWSLESlwWCd/AYGfUyee/PN6WrBopYtW0m9evWkfPnyxneIXLjwo3z//fdqkSUEJ6h7/+Mf/8hsjKaQLcQBGYuyxMXF206qorSquXcVVq7E5D6dgmOsUFmrVi21gq2V4KR706ZNakEjTBwdPHiwZWrS6YasrCz5xz/+cdP4bV+Hj6sGWBAMZUDIxoeEhKjVZsPCwi17gknkKAbr5HeQYcMKjxjYv/rqS2PvDWYQHxoaauwh3SEg27FjhyQmzlbLneMKSlhYmCUP0rhqMHToEFm9+j2tTjzwvomM7OKVFTzdwb6Tkj+dyJV19uM3AvPLl39W+++99z61Km+1alidtzGvglKZwmCdiCwF9cjJycmSnX1QtTm0UpCGQATZQLTO1DEgNk8k1q9P1TIYMrPouNoCeB6RTWdmlYj8GYN1IrIklMignR3KH5Bt7927l097xzsiNTVVdfFISEgw9ugH9eugSzkMTnDQ1tT+BK1Tp04sdSGiMoPBOhFZnn1JBFrUde/eXbsyGbPMRLfyl6KQve7ZM9qn2XUzQN+5c6f6m2KhsLZt21q29ImIqDQYrBOR3yguyMNqsDpk3JFVRy3uuHHjjD368kV2vejfDgv24GoJy1yIqKxjsE5Efqm44M+Xq7i2b9/OayuUlhay66GhIbaTixyPBsooZfr222xZty5FLXtvZtAxkZATCImICjBYJ6IyAaUy5rLt586dVbXPTZs29UpgiMeeNm2avP/++8Ye/aEPPAJnd/Yqx0kAOnzg+Viz5j2pVq26Wuoej9GgQQNm0ImIisFgnYjKHDOj++mn22TVqpUq6x4RESFNmjTxSNCIspJ77rlH+vfvb+zRH05scFWiNGU79sE5TpKwNL5ZmtS4cRBbLRIROYDBOhGVeWhZePDgQdWpBcF7q1atVcY3MLC+W4LKZ555RmXWrdTBxCyFcWb8N59H1Obv3btXXcHo2rWbNGzYUIKDg9nBhYjIBQzWiYiKQNB5/HiuLeDMVEEnMsLoMhMUFCQPP/ywVKlSxeHAE7XzDRsGWnIcxUlGYmLiLScrCOQvXrxYGJjn5uaqmnNcocDKkigvqlOnDjPnRERuwGCdiOgOEHCfOnXqluAUGfj69etLixYtpGrVqiqIDwgIuKmMBoH/smXLtO6tXhLUrWO137vvvlvOnDkjJ0+elIyMz+XYsaOFJy/Vq1dTS8Iza05E5BkM1omIXIRA/Pr162ppdATxly9fVmU0gGAW7r33XnVrhZaNRaHd5MaNG6V9+/ZSsWJFVcrywAMPsFMLEZEXMVgnInIzMxOPQH7t2rUq8x4dHW181ToQrIMVf3ciIn9RNBYvZ9wSEZGLUAaDshDUbiNQtyqU9uCKARER6YPBOhERKajBR2kPERHpg8E6EREpmFCLSaRERKQPButERG529epV4zPrwcRSIiLSB4N1IiI3QgeV7OxsY8tarHySQUTkrxisExG5Ufny5eXIkSPGlrXgJAMnG0REpA8G60REboRVPLECKlb9tBr0jMciT0REpA8G60REboYFkr777jtjyxqw4BNWaLVfjZWIiHyPwToRkZs98UQH2blzp7FlDV999ZVERUUZW0REpAsG60REbhYe3kzmz5+rVja1iiVLFkuHDh2MLSIi0gWDdSIiN7v//vtVKczmzZuNPXrbtWuXVKtWXdXbExGRXhisExF5wODBg2X16tXGlt6Sk5Nl9OjRxhYREemEwToRkQcEBgZK/fr1JTU11dijJ2TVL168KK1btzb2EBGRThisExF5CLLriYmztW3jiJr6OXPmMKtORKQxButERB6C7PqwYcNl1qxZxh69LF++XGX/mVUnItIXg3UiIg/q3bu3WtFUt3IYlL+sWfOejB071thDREQ6YrBORORBWGQoMTFRlcMgQNbBDz/8IHFx42Xp0rdV5xoiItIXg3UiIg9DS8SkpLnSt+8zPg/YEaj36/esTJ8+Q5XpEBGR3hisExF5QdOmTWXNmvd9GrDn5OQUBuqsUycisgYG60REXoIA+aOPNqoSFG/XsG/dukU6duzAQJ2IyGIYrBMReREy7KtXv6dWN50wYYLH2zri/mfOnCmLFy+RL774ioE6EZHFMFgnIvIy1LDPnj1bWrRoIaGhISrLjp7n7oT7w/327BktlSpVkhUrVqjHJSIia2GwTkTkA+gSEx0dLVlZ+2X37t0SGdlFVq1aVepMO34eQTruD/eLji8jRoxQj0dERNbzm3wb43Oldu0AOXHilLFFRETegCB77dq1qvd5UFCwdOnSRYKDgx3q2IKJowcPHlSlNRs3fiRxcfHSqVMndnshIrKgorE4g3UiIs188803cuDAAcnIyFDBd6tWrdVKo0VhsaUvv9wl3br1kJCQEFUPHxYWxiw6EZGFMVgnIrIY9Ea/fv26sXVD+fLlWYdORORnGKwTEREREWmqaCzOCaZERERERJpisE5EREREpCkG60REREREmmKwTkRERESkKQbrRERERESaYrBORERERKQpButERERERJoqts86ERERERH5xm0XRSIiIiIiIj2wDIaIiIiISFMM1omIiIiINMVgnYiIiIhISyL/H+wJUTEs97Q3AAAAAElFTkSuQmCC"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Katie works for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn></math> days in a year.</p>
<p>Find the probability that Katie cycles to work on her final working day of the year.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>4</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mn>180</mn></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>75</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This question was often done well. Many textbooks teach the method of multiplying the transition matrix by an initial state vector. This was often seen in candidates’ responses. eg <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>4</mn></mtd></mtr></mtable></mfenced><mn>180</mn></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd></mtr></mtable></mfenced></math>. Errors were often due to the figures being incorrectly placed in the transition matrix; not just the transpose, but other combinations of the four values as well.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question was often done well. Many textbooks teach the method of multiplying the transition matrix by an initial state vector. This was often seen in candidates’ responses. eg <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>4</mn></mtd></mtr></mtable></mfenced><mn>180</mn></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd></mtr></mtable></mfenced></math>. Errors were often due to the figures being incorrectly placed in the transition matrix; not just the transpose, but other combinations of the four values as well.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A psychologist records the number of digits (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math> that a sample of IB Mathematics higher level candidates could recall.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The psychologist has read that in the general population people can remember an average of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>4</mn></math> digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math>. The psychologist wants to perform a statistical test to see if IB Mathematics higher level candidates can remember more digits than the general population.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo> </mo><mo>:</mo><mo> </mo><mi>μ</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>4</mn></math> is the null hypothesis for this test.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an unbiased estimate of the population mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an unbiased estimate of the population variance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the alternative hypothesis.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that all assumptions for this test are satisfied, carry out an appropriate hypothesis test. State and justify your conclusion. Use a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> significance level.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mn>4</mn><mo>.</mo><mn>63</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>4</mn><mo>.</mo><mn>62686</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>098702</mn></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>s</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mn>1</mn><mo>.</mo><mn>21</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>207146</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>A0A0</strong> </em>for an answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>19</mn></math> from biased estimate.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub><mo> </mo><mo>:</mo><mo> </mo><mi>μ</mi><mo>></mo><mn>4</mn><mo>.</mo><mn>4</mn></math> <strong><em>A1</em></strong></p>
<p><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>using a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>-test <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>0454992</mn><mo>…</mo></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo><</mo><mn>0</mn><mo>.</mo><mn>05</mn></math> <strong><em>R1</em></strong></p>
<p>reject null hypothesis <strong><em>A1</em></strong></p>
<p>(therefore there is significant evidence that the IB HL math students know more digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math> than the population in general)</p>
<p><br><strong>Note:</strong> Do not award <em><strong>R0A1</strong></em>. Allow <em><strong>R1A1</strong> </em>for consistent conclusion following on from their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>using a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>-test <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>0478584</mn><mo>…</mo></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo><</mo><mn>0</mn><mo>.</mo><mn>05</mn></math> <strong><em>R1</em></strong></p>
<p>reject null hypothesis <strong><em>A1</em></strong></p>
<p>(therefore there is significant evidence that the IB HL math students know more digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math> than the population in general)</p>
<p><br><strong>Note:</strong> Do not award <em><strong>R0A1</strong></em>. Allow <em><strong>R1A1</strong> </em>for consistent conclusion following on from their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value.</p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>In parts (a) and (b), candidates used the 1-Var Stats facility to find the estimates of mean and variance although some forgot to include the frequency list so that they just found the mean and variance of the numbers 2, 3, …6, 7. Candidates who looked ahead realized that the answers to parts (a) and (b) would be included in the output from using their test. In part (c), the question was intended to use the <em>t</em>-test (as the population variance was unknown), however since the population could not be assumed to be normally distributed, the Principal Examiner condoned the use of the <em>z</em>-test (with the estimated variance from part (b)). As both methods could only produce an approximate <em>p</em>-value, either method (and the associated <em>p</em>-value) was awarded full marks.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A manager wishes to check the mean mass of flour put into bags in his factory. He randomly samples 10 bags and finds the mean mass is 1.478 kg and the standard deviation of the sample is 0.0196 kg.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{n - 1}}">
<mrow>
<msub>
<mi>s</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</math></span> for this sample.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a 95 % confidence interval for the population mean, giving your answer to 4 significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The bags are labelled as being 1.5 kg mass. Comment on this claim with reference to your answer in part (b).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{n - 1}} = \sqrt {\frac{{10}}{9}} \times 0.0196 = 0.02066 \ldots ">
<mrow>
<msub>
<mi>s</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<msqrt>
<mfrac>
<mrow>
<mn>10</mn>
</mrow>
<mn>9</mn>
</mfrac>
</msqrt>
<mo>×</mo>
<mn>0.0196</mn>
<mo>=</mo>
<mn>0.02066</mn>
<mo>…</mo>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(1.463, 1.493) <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> If <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_n}">
<mrow>
<msub>
<mi>s</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> used answer is (1.464, 1.492), award <em><strong>M1A0</strong></em>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>95 % of the time these results would be produced by a population with mean of less than 1.5 kg, so it is likely the mean mass is less than 1.5 kg <em><strong>R1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>As part of the selection process for an engineering course at a particular university, applicants are given an exam in mathematics. This year the university has produced a new exam and they want to test if it is a valid indicator of future performance, before giving it to applicants. They randomly select 8 students in their first year of the engineering course and give them the exam. They compare the exam scores with their results in the engineering course.</p>
</div>
<div class="specification">
<p>The results of the 8 students are shown in the table.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the name of this test for validity.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate Pearson’s product moment correlation coefficient for this data.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine, with a reason, if the new exam is a valid indicator of future performance.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>criterion-related <em><strong> A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 0.414">
<mi>r</mi>
<mo>=</mo>
<mn>0.414</mn>
</math></span> <em><strong> A2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Since the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> is low (closer to 0 than +1), <em><strong> R1</strong></em></p>
<p>The new exam is not a valid indicator of future performance. <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>In a coffee shop, the time it takes to serve a customer can be modelled by a normal distribution with a mean of 1.5 minutes and a standard deviation of 0.4 minutes.</p>
<p>Two customers enter the shop together. They are served one at a time.</p>
<p>Find the probability that the total time taken to serve both customers will be less than 4 minutes.</p>
<p>Clearly state any assumptions you have made.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> be the time to serve both customers and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_i}">
<mrow>
<msub>
<mi>T</mi>
<mi>i</mi>
</msub>
</mrow>
</math></span> the time to serve the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span>th customer</p>
<p>assuming independence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_1}">
<mrow>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_2}">
<mrow>
<msub>
<mi>T</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> is normally distributed and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T = {T_1} + {T_2}">
<mi>T</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>T</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="E\left( T \right) = 1.5 + 1.5 = 3">
<mi>E</mi>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1.5</mn>
<mo>+</mo>
<mn>1.5</mn>
<mo>=</mo>
<mn>3</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}\left( T \right) = {0.4^2} + {0.4^2} = 0.32">
<mrow>
<mtext>Var</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>0.4</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>0.4</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0.32</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( {T < 4} \right) = 0.961">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo><</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.961</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 \times 2">
<mn>2</mn>
<mo>×<!-- × --></mo>
<mn>2</mn>
</math></span> transition matrix for a Markov chain will have the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf{M}} = \left( {\begin{array}{*{20}{c}} a&{1 - b} \\ {1 - a}&b \end{array}} \right){\text{,}}\,\,0 < a < 1{\text{,}}\,\,0 < b < 1">
<mrow>
<mrow>
<mi mathvariant="bold">M</mi>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo><</mo>
<mi>a</mi>
<mo><</mo>
<mn>1</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo><</mo>
<mi>b</mi>
<mo><</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 1">
<mi>λ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> is always an eigenvalue for <strong>M</strong> and find the other eigenvalue in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the steady state probability vector for <strong>M</strong> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\begin{array}{*{20}{c}} {a - \lambda }&{1 - b} \\ {1 - a}&{b - \lambda } \end{array}} \right| = 0 \Rightarrow \left( {a - \lambda } \right)\left( {b - \lambda } \right) - \left( {1 - b} \right)\left( {1 - a} \right) = 0">
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\lambda ^2} - \left( {a + b} \right)\lambda + a + b - 1 = 0 \Rightarrow \left( {\lambda - 1} \right)\left( {\lambda + \left( {1 - a - b} \right)} \right) = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>λ</mi>
<mo>+</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>λ</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>λ</mi>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \lambda = 1\,\,{\text{or}}\,\,\lambda = a + b - 1">
<mo stretchy="false">⇒</mo>
<mi>λ</mi>
<mo>=</mo>
<mn>1</mn>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>or</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>λ</mi>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong> AGA1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&{1 - b} \\ {1 - a}&b \end{array}} \right)\,\left( {\begin{array}{*{20}{c}} p \\ {1 - p} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} p \\ {1 - p} \end{array}} \right) \Rightarrow ap + 1 - b - p + bp = p">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>p</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>p</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mi>p</mi>
<mo>+</mo>
<mn>1</mn>
<mo>−</mo>
<mi>b</mi>
<mo>−</mo>
<mi>p</mi>
<mo>+</mo>
<mi>b</mi>
<mi>p</mi>
<mo>=</mo>
<mi>p</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 1 - b = \left( {2 - a - b} \right)p \Rightarrow p = \frac{{1 - b}}{{2 - a - b}}">
<mo stretchy="false">⇒</mo>
<mn>1</mn>
<mo>−</mo>
<mi>b</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>p</mi>
<mo stretchy="false">⇒</mo>
<mi>p</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
</mrow>
</mfrac>
</math></span> <em><strong> M1</strong></em></p>
<p>So vector is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{{1 - b}}{{2 - a - b}}} \\ {\frac{{1 - a}}{{2 - a - b}}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span> has the Poisson distribution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Po}}(m)"> <mrow> <mtext>Po</mtext> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X > 0) = \frac{3}{4}"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m"> <mi>m</mi> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\ln a"> <mi>ln</mi> <mo></mo> <mi>a</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> is an integer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y"> <mi>Y</mi> </math></span> has the Poisson distribution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Po}}(2m)"> <mrow> <mtext>Po</mtext> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> </math></span>. Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y > 1)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{b - \ln c}}{c}"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>ln</mi> <mo></mo> <mi>c</mi> </mrow> <mi>c</mi> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> are integers.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P(}}X > 0) = 1 - {\text{P(}}X = 0)"> <mrow> <mtext>P(</mtext> </mrow> <mi>X</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mtext>P(</mtext> </mrow> <mi>X</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 1 - {{\text{e}}^{ - m}} = \frac{3}{4}"> <mo stretchy="false">⇒</mo> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mrow> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span> or equivalent <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow m = \ln 4"> <mo stretchy="false">⇒</mo> <mi>m</mi> <mo>=</mo> <mi>ln</mi> <mo></mo> <mn>4</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y > 1) = 1 - {\text{P}}(Y = 0) - {\text{P}}(Y = 1)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 - {{\text{e}}^{ - 2\ln 4}} - {{\text{e}}^{ - 2\ln 4}} \times 2\ln 4"> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mi>ln</mi> <mo></mo> <mn>4</mn> </mrow> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mi>ln</mi> <mo></mo> <mn>4</mn> </mrow> </msup> </mrow> <mo>×</mo> <mn>2</mn> <mi>ln</mi> <mo></mo> <mn>4</mn> </math></span> <strong><em>A1</em></strong></p>
<p>recognition that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\ln 4 = \ln 16"> <mn>2</mn> <mi>ln</mi> <mo></mo> <mn>4</mn> <mo>=</mo> <mi>ln</mi> <mo></mo> <mn>16</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y > 1) = \frac{{15 - \ln 16}}{{16}}"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mn>15</mn> <mo>−</mo> <mi>ln</mi> <mo></mo> <mn>16</mn> </mrow> <mrow> <mn>16</mn> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A robot moves around the maze shown below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Whenever it leaves a room it is equally likely to take any of the exits.</p>
<p style="text-align: left;">The time interval between the robot entering and leaving a room is the same for all transitions.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the transition matrix for the maze.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A scientist sets up the robot and then leaves it moving around the maze for a long period of time.</p>
<p>Find the probability that the robot is in room <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> when the scientist returns.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMAAAACACAYAAABdhGZrAAAUOklEQVR4Ae1dyXLcOBLlWpL7q1yL7D465ktci3zVYvd5ui+9nKyS/R+16ANG6ojp60hh3/ogHfuGiQcqq0AKBAkQJKtYqAgFKS5A5ks8bAQyPeZ+DoEDRsA7YN2d6g4B5gjgCsFBI+AIcNDmd8p3hgCjQZ8tV+tGLbpaLtjF+Tk7Pz9nV/M5m8/n7OLigt3d3TUqhyyzu9tbNhgO2eXHT1weyPSvd++Y50fs4eFB9krt154eH9n86orjdXFxyRaLBcfs8vKC3d62g1knCPDn3R3zPK9xAqDEHMcR84JtofrxZMhlubv7s/YClZfBsN9nQXT0goh//fVfFkZHea81cv1mveL4DEdvN/mBnKPBa/a6399ca+pk7wnwcH/PoiBgnuez1fqmKdw2+USBlypUZOA4Pt480+TJqN9nfhC9KPyQ4fv3b2wkFLwm5aK8vlzPpbZ6eLhnoe+xwfCEHm3kuPcEGA1H7FUvZp4XsHXDBEBh9z2PXc2/bIyVGNhj8+uvm2tNnXyZo3B5bHjyY1NZaufTiwLmh7H0vaMo5ORosou21wS4nl+xkzc/suM45n3bpgmQFLgt8dCUB37AxOZdaumaLqI18vyQ3bfUxy9SCwU79DwW9V5JH+XdSd6Vba4l31sCoOvTHyTNJa85AnmtIkXa0sVe6DM/CNnnz5/ZaDhgvh+yVcMDcVKFul5t9/FJHtlxvVryFiqvdUxaAK/RruzeEgBdH6rpeEEMezLMa7uG2izw0Gd9s8njzbDPm/D7++ZnWahwxTm160bIFk9OhgNlCxW30ILtJQFORkM2GAz5NBqm0gI/PRBtwsbU1xcH3jfPNdxwtCVFE7Igj30gQKSwE1p0jKfChicP9o4AmPIcCDMZqIkBbNQwcEm3azv9KRbCpmVB3jQg39Uu0Hr1cvpTrBzwHcfzA3bb8DeUvSNAHKW7OjfrNe+KNN30Y8ouO5h7MxrwLlBeH1c0eB3nxzFmUQKW1wX7/fc/6si2VJpvR/g+sp0wEF+6u7tNCn8LH8P2hgCYYen3X7PJ7IOIHaOuSJNNJ003irJ8vLzghb/NeXZg1AtD1pd8UDo5ecv+/vvvFHZN/oMKwwti9vT0lMr28uKShXFP+t0i9WBN/+wFAVarJTs7O3v+O9+AuFqK18/YYrmqCaZtsi/yXCzY2dk5w6d9FMBd+P308SNfboDZqaurOfv06afWxELXB3KQ/XCOcdvZ+W5gthcEaM16LuPOI+AI0HkTOwVVCDgCqNBx9zqPgCNA503sFFQh4AigQsfd6zwCjgCdN7FTUIWAI4AKHXev8wg4AnTexE5BFQKOACp03L3OI+AI0HkTOwVVCDgCqNBx9zqPgCNA503sFFQh4AigQsfd6zwCjgCdN7FTUIWAI4AKHXev8wjsDAGwlv7Nm931Z9P5krDnCvLyM9J3qrUTBBiNRuw/LWyH23ObO/ElCIwGA6bjlrJ1AsC51WiHPZlJMHaXdhgBOEnoRVFpCVslADw8tOFBoTQ67sG9RODr9TUbDkelZG+VAHEYsusvzfvQLIWMe2ivEYCHjDLeOVojADZLRy276t5rCzvhlQh8vZ6zMONCR/ZCawSAn5i2nMjKgHDXuocAehgYE6h+rREgDuROklTCunsOAR0EXvWiwqAprRAArAzDtFtBHcXcsw6BMgh8wQyj4EZT9k4lApjG5YIfy8CSN2fMJIFM+BDy9mTEBv1yo38ZGGWuYZ4ZXTfkhya2zJzziHuN9rhrcB7AosAoZeTQfQZy//bbH41gVCRbUzZDOSvylWpMAChhGpcLnoxtTX/C5d7VfDuTFId+KmJLkTF07qNGEaObfJ1fcXffqjR4CKeWB/tZuevESIUF3WvKZvAbG+ZEoyFZjAhQNS7XerlgNmJoJT46/VRoJB5lxA9JP6tH7hBX8EINgOHSey6ESMpmCJ/44/Eke7nR/7Ny14lRkWJN2ow7Ti4InGJEgKpxuWwRgLso96MUAX44Qrww+1FGKAJLfPTDxsYAWOnz/iHxeQ958Hdx8XHzblMnMrnrwqiMTs3abM0C2wSwEZfLFgF4RJGMgl/mn427ZioDUgCKz5naHjKI3aJsGhgrfLq8ZD0eAM5j/f4w+0it/8vkrgujMoo0aTPrLYCtuFx1EgBpm45NVAakdBfLdDDuIgKIab4djbhs8+ttVEnxfh3nMrnpWtOBxaGfjAB1yWOdALbickFhG2MADObgc178YSaoFgI8hz8SCUDRaXyNGS2ECbU1ASDqnXdOLYAod10Y5ckgXm/SZlYJYDMu1/39/9jlZfX+cBJWM/21L4kZnB4YiwYwPQeYGPCK88q4xsN+CgPjovQR0rVJAsjkrgujIt1xv0mbIfj2+fmlUqxSg2BMee5CXK6sJljvgdpeDFSHQZatbwzZ/HphkJpXpgFmmUVXlNZRFJVapEXP2zhm5a4ToyJ5m7ZZkTylCLArcblkysC4YqT20PdrK2AwHmIB0w/z6zpkQyyswXNsY0qjiWNW7joxKqNPkzYrkkdJAMxg1BGXy9YYAMpBRmyAuL29Y8PBiCEWVp0/DGSx1vz29pZFmdhWaBF4yNbnLhEW/KHb9Pr1a9bv4wty8+FTCQvI/euvvzaCEeWZd2zKZpXGAHXG5bJJgDyQ3XWHQCUC1AmfI0Cd6Lq0CQFHAELCHQ8SAUeAgzS7U5oQcAQgJNzxIBFwBDhIszulCQFHAELCHQ8SAUeAgzS7U5oQcAQgJNzxIBFwBDhIszulCQFHAELCHQ8SgYMiADzNLVer1gz99PTEVsslWyyWbLVKb5ppUigTOZ4eH9lisdj8LZf2cOR2WSwY0oRsZX625LFGAAg0Hr/nS4+DIOJAxVHIwihmj4/llBIVt7kU4sNsxuIw4LI1uc5e1AeGhYuUyfSUGxnn09mp+Egj56ZyYJMOlpXTH/Sw8TudTbnLGsjFz4O4FAlsyWONAAADiWFlI4HDwQ585pdUSgTUJgFINmxOb4sAs8k4tS+Y78Lyw1LGFnGpem4iR10+WlE+4I1C3HaJ3WDjyUyppk15rBLgdDphnue/2HziefouDrtEADK0SD6qLIqMrSwJmjdN5cDmGLhtsd1tAxlRXtbrm40m5I4Fsub9bMpjlQAQTNx/S4DrbAghpVfLhdXaGoq21QLQrjAddymEg82jiRx4B606dX3CsMceFYVTR15eXjRd1tiWh6fnq4NlKDfEkMJU2KPeK34JzRTfZmcIWJdaANmmc4Ck4y2CcK5yNJUDtuX2fHbb4geRla6bqfcHm/JYawEI3PfjCZuMxwxb6sbjqTFQnSLAsxsW0etCKwSwIMfpdMpbg8lU3U8vQ1RTAohpV5XHGgHgZtoTBnXUIph0f6BgpwggcZdCXTIddymi4U3OqZISiWgihy23LTL3J+SRThwYF+laRR4rBEBhjwIv5Q0BQidTVelBcZEydL9LBADI6EfHz91D6Ihruu5SCBvToy05bLltIfcn4iDYxB1LFXmsEIAGJjT9SQbqwSlVZpRP94qOXSIAdM26HQFmGFjq1HRFmJW5b0MOuG2xITe1SFVd1lSRxwoBkums9FQnBk2B52m5BBEN2DUCAA+4S0FriR8wM+0eijjpnhfJ8f3bN7ZYLjdyrvD1/PkLLWSH3ONx9f4/yQ1C0lRw0m32U9OtkLdOeSoTIBE6cf6Kcw7SbMJdfwRhua96BIZ4tEmA5Cs15pw95vlBpcG5KKPuOQZsURSz8XjM3aUAqzZ+Kjl++flnhu82VCvj2w5Ng/oBvmTbK/zQHRjAZQ2+M0RhzGaZr+NJN6k+eSoRILseg9aKLJfV17rYJEAbhczluR8IVCJAnSo6AtSJrkubEHAEICTc8SARcAQ4SLM7pQkBRwBCwh0PEgFHgIM0u1OaEHAEICTc8SARcAQ4SLM7pQmBnSUA9gOEGmGFSCF3dAjoIJDsB9gGNJG9W2o/gOzFKtfcd4Aq6Ll3yyKwsy2AI0BZE7rnqiBglQBY1zGdjFkv7vG1HZPJhPWimG+Q6QlLgcsIbJsAWFS11HS9UUZOPKObNnBK3KMsuJuWvDVB6AbS8hIcbW1FzNMLcmAZCy1+y3vO9nVd/JB/dhmOqZsWawTAAqvAD14sZgKo2Z3/ZQC0SQBT1xtl5NRNG3hEAZaJb/fZyhYNolCIz3iWtiHm6QS54KoFG9913JPkpVf2ui5+lO5OuUU5nSa7+2ezDyRf6ngUx6md/6mbOf/YIoCMgGVcb+SIlbpskjZ2zmE5Md7FHxmSlgRTBljjbtsLA6UtO2KZsx9uA4rbwkiWF10zwQ/vonKIoiNKptKxcgsAYbBcVjVjY2JIWwRI9irou94og6pJ2se9tOG4Aby0vyKOqR801hWhgii6bSnjnqQMRqpnTPBDejvlFqXKtkcVOLYIYOJ6QyWXeM9G2ih8cNcitgA8Xeoi+cHG0ZiYt81z2p0mum2hvbm0L8BmfpSWCX60+5C6h1XdtFRqAaj2EptOUq7q0RYBbHgeyNPFRtqJQbcbPiiv9XrNZpMJ964BY2e3m9JzNo60NVHcLA/8ka+NrY95Mprih0oDrWTPgpuWSgQg4MSmM09Z3euHQoDjOEzV/lmcYOw4CFIeN7LPVP2fCvu+EEDUt1W3KASc2HSKwlU5t0YAbMwPtoM7yETNe9XarapbD9RiYYnBHJ7Lupysgm32XarIRALYwiibl/h/VfwoLXTDTSthKy1AXMOSBVsEsOV6g8AWj1XSxjx2HPX4TJCYpuw86WqmB/Ky50yvIf2s2xYT9yS6+VfBT8yrNbcoNHug8v6MGSA8p/uzRQCq3cTBHAZfNjwymKb99ITCn3YYAIzyPnTxliLs6UKo9XzWXYotjFRCmOKXTbNVtyjwvY/aQ5zFIAFnsxnL+zZAz+QdbREA6Re53siTocz1orRReEW3Hij8+BAGF5L0lZe7kgxC/q0ERFguF5tvAHg/DiOjGAtl5KdnkA/ctvzzzz+8woJrS5Ppa0qv7FEXP9tuWip1gUhJbiReqybuLSaTKTs+Oq4EoE0CoFCpXG+QHibHorSTZj6Z5UHhD/ztF2CaysMxiJIaHulRMA/f91kY2fPGXKQfBpTv3r2Tuicpetf0vg5+yMO2mxYrBDBVXvWeTQKo8nH3DhuBnSWA7fgAh21mp30eAsl+AAvxAfIyML3uWgBT5Nx7OgjsbAvgCKBjRvesKQKOAKbIufc6gYAjQCfM6JQwRcARwBQ5914nEHAE6IQZnRKmCDgCmCLn3usEAjtLAOcXqBPla+eVSL4D7KBfIEeAnS87nRDQCgGyLiqSRV7V3Gs4AnSifO28ElYIAC2xIA6LuihSPBY5zaaIFeZLV4oWIWOTAHzBVZws1MMy5GwcqiJZVPdN0saCLloUh7hbMnmQLtzJbBfMvdw2qZJLde90NuPLsSeIy1VyT0KygX0rj8oJgirvsvdMcC2btvicRQIspXtI+TJbgz2tNglQtORWBET3XDftD7Mp83mgvgkPlpcEoHu59BiVx/vxeLNk2tTxU1YfWvaMAoYfCnbR3gg8G4XhRhbupOtRf49HVhbV/7q4qtJS3bNGgGykeDFTvvVNCBEq3ss7t0UAW5suZHKapA0fSeLGFxCCt5zCrrqkwEVGG4lkcorXULDEbZgoAMhftT0UZLQdHVKUKXtugms2jbL/WyEAN5gkUjwJkayJ9zahN+m66oi1QGFcfReUrW13MllN0kYNLP44dn7aLxAKHAolD0s6SZxoie+YnmPKL7v1EddUEeshH3XFwiiutMejrNwmuJZNO/vczXrJPL/ialDOIkU3hwiwXN1k88/9HwTwLWwD5NHq/XRtmniyU9d6uYIJN2ykTQVQrIHhNxRjle04IUq1GoIIWqdUs46np5v3iIB5WGMTD49rHIaclCCmbPffJkELJzZwLSsGxyTjNCH7bqF7dJmHLzERrpCnt6kbBAAzYaAqP1PfM2XytJH2BwQVD+Wb46E7fAOh1ha7LWVkkz3DMfU8Jnp/wHPQo4xvJ7Re3EWL93LMIsvP9JoNXMvmzTf/VyUA+vh5AFITmnc/T1BMrUZBULnJlYFpqwWomjavfcOwcL9vUQWTh2H2OrUAIgG4DH45AiA9sqepG5KsTLL/q+IqSzPvGpxrFbVoyhaAf0r2PBbnuD+n/uwkx3FunmC4junCIuFU7+Med92YaUnI543oKaIoHdn9qmnPxuNSTgOSvruvNYaSyYsanHdhhC4Q0oZrRp0WBl3aOglQFVeZ7rJrSQVU3MtQEgCFFKDKPAjQJnB4Qzb58f5xWCygKm1Z7cnHJJqzUrI8qqSNd6dCQZSlT9d4JWNBXlntnZBLr18PPzxVKybSTXasgqssvbxrKLv03SrvGVxXEkDGVryE2iYIgspAwXWgSetBCpHRxUEmumw2DGia9ulkzLIDUfT1V+s1iZ06okDYkBeJIi2xO8q7RRrkKltrphTQ/McUV51skAePh7AunpiREgAFfDx+n8wMPH/YwQeSOO6xKEy+uspaBR0h8awNwBGEgYJQoEsWhmmnVLoyic8XpQ0Ce972K25Su22/qKL15H/PATCoRoaMiNYSR0k8ATHPKufc8EHAprPTZ2xDfk5p/vLzv1PyHvci5sFD9WTCI/3Ato81fwSDLEW4krymR51KRUoA04xN3gPZojjtV183HaQBgtYR/keVNr+3XPLChi5hsk4qHfqIyyV8H4Dzp0TWZW6roKu/+DxIAHItFssXXdfv375lHHnRs+pwTmL6ts5VuFbJA+nqlKfWCQBl4YGOIqtUUd69e9gIoPD34iNeIZVFYicIAGExNfrq6Act4csq6Z7rPgKoRGdT/QmZnSFA903kNNxFBBwBdtEqTqbGEHAEaAxql9EuIuAIsItWcTI1hoAjQGNQu4x2EYH/A3Bp3uGsePm6AAAAAElFTkSuQmCC"> <strong>(M1)A1A1</strong> </p>
<p> </p>
<p><strong>Note:</strong> Award <strong>A1A0</strong> if there is one error in the matrix. <strong>A0A0</strong> for more than one error.</p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Steady state column matrix is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>4</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <strong>(M1)</strong> </p>
<p>Probability it is in room <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of cars arriving at a junction in a particular town in any given minute between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>:</mo><mn>00</mn><mo> </mo><mtext>am</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>:</mo><mn>00</mn><mo> </mo><mtext>am</mtext></math> is historically known to follow a Poisson distribution with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>4</mn></math> cars per minute.</p>
<p>A new road is built near the town. It is claimed that the new road has decreased the number of cars arriving at the junction.</p>
<p>To test the claim, the number of cars, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>, arriving at the junction between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>:</mo><mn>00</mn><mo> </mo><mtext>am</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>:</mo><mn>00</mn><mo> </mo><mtext>am</mtext></math> on a particular day will be recorded. The test will have the following hypotheses:</p>
<p style="padding-left: 90px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo> </mo><mo>:</mo></math> the mean number of cars arriving at the junction has not changed,<br><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub><mo> </mo><mo>:</mo></math> the mean number of cars arriving at the junction has decreased.</p>
<p>The alternative hypothesis will be accepted if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>≤</mo><mn>300</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming the null hypothesis to be true, state the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of a Type I error.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of a Type II error, if the number of cars now follows a Poisson distribution with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn></math> cars per minute.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>Po</mtext><mfenced><mn>324</mn></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Both distribution and mean must be seen for <em><strong>A1</strong></em> to be awarded.</p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≤</mo><mn>300</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0946831</mn><mo>…</mo><mo>≈</mo><mn>0</mn><mo>.</mo><mn>0947</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(mean number of cars =) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo>×</mo><mn>60</mn><mo>=</mo><mn>270</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>></mo><mn>300</mn><mo> </mo><menclose notation="left"><mo> </mo><mi>λ</mi><mo>=</mo><mn>270</mn></menclose></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>M1</strong> </em>for using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>270</mn></math> to evaluate a probability.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≥</mo><mn>301</mn></mrow></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≤</mo><mn>300</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0334207</mn><mo>…</mo><mo>≈</mo><mn>0</mn><mo>.</mo><mn>0334</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) should have been routine as all the information needed to answer it was there in the question but here again a reliance of the use of a calculator’s probability distribution functions has meant that simply stating a distribution is too frequently neglected. Many candidates failed to progress beyond part (a). In parts (b) and (c), a lack of knowledge of Type I and Type II errors prevented candidates from tackling what was otherwise a relatively straightforward question to answer. Some had difficulty with the mechanics of using their own GDC model where <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mi>X</mi><mo>></mo><mn>300</mn><mo>)</mo></math> must be interpreted as either <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mi>X</mi><mo>≥</mo><mn>301</mn><mo>)</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mtext>P</mtext><mo>(</mo><mi>X</mi><mo>≤</mo><mn>300</mn><mo>)</mo></math> to be able to perform the calculation.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider two events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> defined in the same sample space.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = \frac{4}{9},{\text{ P}}(B|A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B|A') = \frac{1}{6}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span>;</p>
<p>(ii) hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A \cap B) + {\text{P}}(A' \cap B) - {\text{P}}(A \cap B)">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A' \cap B)">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A|B) \times {\text{P}}(B)">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + \left( {1 - {\text{P}}(A|B)} \right) \times {\text{P}}(B)">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A'|B) \times {\text{P}}(B)">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(A) + {\text{P}}(A' \cap B)">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B) = {\text{P}}(B|A'){\text{P}}(A')">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{9} = {\text{P}}(A) + \frac{1}{6}\left( {1 - {\text{P}}(A)} \right)">
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 = 18{\text{P}}(A) + 3\left( {1 - {\text{P}}(A)} \right)">
<mn>8</mn>
<mo>=</mo>
<mn>18</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p>(ii) <strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B) = {\text{P}}(A \cap B) + {\text{P}}(A' \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(B|A){\text{P}}(A) + {\text{P}}(B|A'){\text{P}}(A')">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3} \times \frac{1}{3} + \frac{1}{6} \times \frac{2}{3} = \frac{2}{9}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cap B) = {\text{P}}(B|A){\text{P}}(A) \Rightarrow {\text{P}}(A \cap B) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B) = {\text{P}}(A \cup B) + {\text{P}}(A \cap B) - {\text{P}}(A)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B) = \frac{4}{9} + \frac{1}{9} - \frac{1}{3} = \frac{2}{9}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Sue sometimes goes out for lunch. If she goes out for lunch on a particular day then the probability that she will go out for lunch on the following day is 0.4. If she does not go out for lunch on a particular day then the probability she will go out for lunch on the following day is 0.3.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the transition matrix for this Markov chain.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>We know that she went out for lunch on a particular Sunday, find the probability that she went out for lunch on the following Tuesday.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the steady state probability vector for this Markov chain.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {0.4}&{0.3} \\ {0.6}&{0.7} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0.6</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.7</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\begin{array}{*{20}{c}} {0.4}&{0.3} \\ {0.6}&{0.7} \end{array}} \right)^2}\left( {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {0.34} \\ {0.66} \end{array}} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0.6</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.7</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.34</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0.66</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p>So probability is 0.34 <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {0.4}&{0.3} \\ {0.6}&{0.7} \end{array}} \right)\left( {\begin{array}{*{20}{c}} p \\ {1 - p} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} p \\ {1 - p} \end{array}} \right) \Rightarrow 0.4p + 0.3\left( {1 - p} \right) = p \Rightarrow p = \frac{1}{3}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0.6</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.7</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>p</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>p</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mn>0.4</mn>
<mi>p</mi>
<mo>+</mo>
<mn>0.3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>p</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>p</mi>
<mo stretchy="false">⇒</mo>
<mi>p</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>M1A1</strong></em></p>
<p>So vector is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\tfrac{1}{3}} \\ {\tfrac{2}{3}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p>[or by investigating high powers of the transition matrix]</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of telephone calls received by a helpline over 80 one-minute periods are summarized in the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Find the exact value of the mean of this distribution.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Test, at the 5% level of significance, whether or not the data can be modelled by a Poisson distribution.</p>
<div class="marks">[12]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Mean <span class="mjpage"><math alttext="\lambda = \frac{{\left( {9 \times 0 + 12 \times 1 + 22 \times 2 + 10 \times 3 + 11 \times 4 + 8 \times 5 + 8 \times 6} \right)}}{{80}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>λ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>9</mn> <mo>×</mo> <mn>0</mn> <mo>+</mo> <mn>12</mn> <mo>×</mo> <mn>1</mn> <mo>+</mo> <mn>22</mn> <mo>×</mo> <mn>2</mn> <mo>+</mo> <mn>10</mn> <mo>×</mo> <mn>3</mn> <mo>+</mo> <mn>11</mn> <mo>×</mo> <mn>4</mn> <mo>+</mo> <mn>8</mn> <mo>×</mo> <mn>5</mn> <mo>+</mo> <mn>8</mn> <mo>×</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>80</mn> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p> <span class="mjpage"><math alttext=" = 2.725 = \left( {\frac{{109}}{{40}}} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mn>2.725</mn> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>109</mn> </mrow> <mrow> <mn>40</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em> <strong>Note:</strong></em> Do not accept 2.73.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>H<sub>0</sub>: the data can be modelled by a Poisson distribution <em><strong>A1</strong></em></p>
<p>H<sub>1</sub>: the data cannot be modelled by a Poisson distribution <em><strong>A1</strong></em></p>
<p><img src="data:image/png;base64,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"> <em><strong> A3</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A2</strong> </em>for one error, <em><strong>A1</strong> </em>for two errors, <em><strong>A0</strong> </em>for three or more errors.</p>
<p>Combining last two columns <em><strong>(M1) </strong></em></p>
<p><strong>Note:</strong> Allow <em><strong>FT</strong></em> from not combining the last two columns and / or getting 2.98 for the last expected frequency.</p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math alttext="{\chi ^2} = \frac{{{9^2}}}{{5.244}} + \frac{{{{12}^2}}}{{14.289}} + \frac{{{{22}^2}}}{{19.469}} + \frac{{{{10}^2}}}{{17.684}} + \frac{{{{11}^2}}}{{12.047}} + \frac{{{{16}^2}}}{{11.267}} - 80" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mn>9</mn> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>5.244</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>14.289</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>22</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>19.469</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>17.684</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>11</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>12.047</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>16</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>11.267</mn> </mrow> </mfrac> <mo>−</mo> <mn>80</mn> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p> = 8.804 (accept 8.8) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math alttext="v = 6 - 2 = 4" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>v</mi> <mo>=</mo> <mn>6</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> </math></span>, <span class="mjpage"><math alttext="\chi _{5{\text{% }}}^2 = 9.488" xmlns="http://www.w3.org/1998/Math/MathML"> <msubsup> <mi>χ</mi> <mrow> <mn>5</mn> <mrow> <mtext>% </mtext> </mrow> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <mn>9.488</mn> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>Hence 8.804 is not significant since 8.804 < 9.488 and we accept H<sub>0</sub> <em><strong>R1 </strong></em></p>
<p><strong>OR</strong></p>
<p><em>p</em>-value = 0.0662 (accept 0.066) which is not significant since <strong><em>A5 </em></strong></p>
<p>0.0662 > 0.05 and we accept H<sub>0</sub> <em><strong>R1 N0</strong></em></p>
<p><em><strong>[12 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram below shows part of the screen from a weather forecasting website showing the data for town <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. The percentages on the bottom row represent the likelihood of some rain during the hour leading up to the time given. For example there is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>69</mn><mo>%</mo></math> chance (a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>69</mn></math>) of rain falling on any point in town <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0900</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn></math>.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">Paula works at a building site in the area covered by this page of the website from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0900</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1700</mn></math>. She has lunch from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1300</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1400</mn></math>.</p>
</div>
<div class="specification">
<p>In the following parts you may assume all probabilities are independent.</p>
<p>Paula needs to work outside between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1300</mn></math> and will also spend her lunchtime outside.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the probability it rains during Paula’s lunch break.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability it will not rain while Paula is outside.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability it will rain at least once while Paula is outside.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given it rains at least once while Paula is outside find the probability that it rains during her lunch hour.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><strong>Note:</strong> Accept probabilities written as percentages throughout.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>27</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> Accept probabilities written as percentages throughout.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>22</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>28</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>52</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>73</mn></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0234</mn><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>02338336</mn><mo>)</mo></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> Accept probabilities written as percentages throughout.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>−</mo><mn>0</mn><mo>.</mo><mn>02338336</mn></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>977</mn><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>97661664</mn><mo>)</mo></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> Accept probabilities written as percentages throughout.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mtext>rains during lunch </mtext><mo> </mo><menclose notation="left"><mo> </mo><mtext>rains at least once</mtext></menclose></mrow></mfenced><mi mathvariant="normal">=</mi><mfrac><mrow><mtext>P</mtext><mfenced><mrow><mtext>r</mtext><mi>ains</mi><mo> </mo><mi>during</mi><mo> </mo><mi>lunch</mi></mrow></mfenced></mrow><mrow><mtext>P</mtext><mfenced><mtext>rains at least once</mtext></mfenced></mrow></mfrac></math> <strong>M1A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>0</mn><mo>.</mo><mn>27</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>97661664</mn></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>276</mn><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>276464</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> be a random variable which follows a normal distribution with mean <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < \mu - 5} \right) = 0.2">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mi>μ<!-- μ --></mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.2</mn>
</math></span> , find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X > \mu + 5} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < \mu + 5\,\left| {\,X > \mu - 5} \right.} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>|</mo>
<mrow>
<mspace width="thinmathspace"></mspace>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>use of symmetry <em>eg</em> diagram <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X > \mu + 5} \right) = 0.2">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < \mu + 5\,\left| {\,X > \mu - 5} \right.} \right) = \frac{{{\text{P}}\left( {X > \mu - 5 \cap X < \mu + 5} \right)}}{{{\text{P}}\left( {X > \mu - 5} \right)}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>|</mo>
<mrow>
<mspace width="thinmathspace"></mspace>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
<mo>∩</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{P}}\left( {\mu - 5 < X < \mu + 5} \right)}}{{{\text{P}}\left( {X > \mu - 5} \right)}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.6}}{{0.8}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.6</mn>
</mrow>
<mrow>
<mn>0.8</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> <em><strong>A1</strong></em> for denominator is independent of the previous <em><strong>A</strong></em> marks.</p>
<p><strong>OR</strong></p>
<p>use of diagram <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Only award <em><strong>(M1)</strong></em> if the region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu - 5 < X < \mu + 5">
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</math></span> is indicated and used.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X > \mu - 5} \right) = 0.8">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.8</mn>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\mu - 5 < X < \mu + 5} \right) = 0.6">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.6</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> Probabilities can be shown on the diagram.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.6}}{{0.8}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.6</mn>
</mrow>
<mrow>
<mn>0.8</mn>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong></em><em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{3}{4}\,\,\, = \left( {0.75} \right)">
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.75</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>George goes fishing. From experience he knows that the mean number of fish he catches per hour is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>1</mn></math>. It is assumed that the number of fish he catches can be modelled by a Poisson distribution.</p>
<p>On a day in which George spends <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> hours fishing, find the probability that he will catch more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> fish.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>Po</mtext><mfenced><mrow><mn>8</mn><mo>.</mo><mn>8</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for calculating the mean, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>8</mn></math>, of the distribution</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>></mo><mn>9</mn></mrow></mfenced><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≥</mo><mn>10</mn></mrow></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>></mo><mn>9</mn></mrow></mfenced><mo>=</mo><mn>1</mn><mo>-</mo><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≤</mo><mn>9</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>></mo><mn>9</mn></mrow></mfenced><mo>=</mo><mi mathvariant="normal">0</mi><mo>.</mo><mn>386</mn><mo> </mo><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>386260</mn><mo>…</mo><mo>)</mo></math> <em><strong>(M1)A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)(M0)(M1)A0</strong></em> for finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≥</mo><mn>9</mn></mrow></mfenced><mo>=</mo><mi mathvariant="normal">0</mi><mo>.</mo><mn>518</mn><mo> </mo><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>517719</mn><mo>…</mo><mo>)</mo></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>≤</mo><mn>9</mn></mrow></mfenced><mo>=</mo><mi mathvariant="normal">0</mi><mo>.</mo><mn>614</mn><mo> </mo><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>613740</mn><mo>…</mo><mo>)</mo></math>.</p>
<p><em><strong><br>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> has eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>A switch has two states, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>. Each second it either remains in the same state or moves according to the following rule: If it is in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> it will move to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> with a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn></math> and if it is in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> it will move to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> with a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>7</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an eigenvector corresponding to the eigenvalue of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using your answer to (a), or otherwise, find the long-term probability of the switch being in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>c</mi><mi>d</mi></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>7</mn><mi>y</mi></math> <em><strong>(A1)</strong></em></p>
<p>an eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>8</mn></mtd></mtr></mtable></mfenced></math> (or equivalent with integer values) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>(the long-term probability matrix is given by the eigenvector corresponding to the eigenvalue equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, scaled so that the sum of the entries is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>+</mo><mn>7</mn><mo>=</mo><mn>15</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>p</mi></mtd></mtr><mtr><mtd><mn>1</mn><mo>-</mo><mi>p</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>p</mi></mtd></mtr><mtr><mtd><mn>1</mn><mo>-</mo><mi>p</mi></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>considering high powers of the matrix <em>e.g.</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mn>50</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>7</mn><mn>15</mn></mfrac></mtd><mtd><mfrac><mn>7</mn><mn>15</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>8</mn><mn>15</mn></mfrac></mtd><mtd><mfrac><mn>8</mn><mn>15</mn></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p>probability of being in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>15</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>In part (a), some candidates could correctly use either <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="bold-italic">A</mi><mo>-</mo><mi>λ</mi><mi mathvariant="bold-italic">I</mi><mo>)</mo><mi mathvariant="bold-italic">x</mi><mo>=</mo><mn>0</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">x</mi><mo>=</mo><mi>λ</mi><mi mathvariant="bold-italic">x</mi></math>to find an eigenvector but many did not pay attention to the fact that integer values of the eigenvector were required. Some candidates used the method of finding the steady state by finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mi>n</mi></msup></math> for some high value of n in part (b) but ignored the fact that they needed to express their answer in rational form. Some did try to convert their calculated answer of 0.467 to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>467</mn><mn>1000</mn></mfrac></math> but this could only receive partial credit as an exact answer was required.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The faces of a fair six-sided die are numbered 1, 2, 2, 4, 4, 6. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> be the discrete random variable that models the score obtained when this die is rolled.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the probability distribution table for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
<p><img src="images/Schermafbeelding_2017-02-28_om_11.16.45.png" alt="N16/5/MATHL/HP1/ENG/TZ0/02.a"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the expected value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><img src="images/Schermafbeelding_2017-02-28_om_11.18.41.png" alt="N16/5/MATHL/HP1/ENG/TZ0/02.a/M"> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for each correct row.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X) = 1 \times \frac{1}{6} + 2 \times \frac{1}{3} + 4 \times \frac{1}{3} + 6 \times \frac{1}{6}">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mn>4</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mn>6</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
</math></span> <strong>(<em>M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{19}}{6}{\text{ }}\left( { = 3\frac{1}{6}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mn>19</mn>
</mrow>
<mn>6</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>3</mn>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>If the probabilities in (a) are not values between 0 and 1 or lead to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X) > 6">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>6</mn>
</math></span> award <strong><em>M1A0 </em></strong>to correct method using the incorrect probabilities; otherwise allow <strong><em>FT </em></strong>marks.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The strength of earthquakes is measured on the Richter magnitude scale, with values typically between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> is the most severe.</p>
<p>The Gutenberg–Richter equation gives the average number of earthquakes per year, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>, which have a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>. For a particular region the equation is</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>N</mi><mo>=</mo><mi>a</mi><mo>-</mo><mi>M</mi></math>, for some <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>This region has an average of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> earthquakes per year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>The equation for this region can also be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfrac><mi>b</mi><msup><mn>10</mn><mi>M</mi></msup></mfrac></math>.</p>
</div>
<div class="specification">
<p>Within this region the most severe earthquake recorded had a magnitude of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>The number of earthquakes in a given year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> can be modelled by a Poisson distribution, with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>. The number of earthquakes in one year is independent of the number of earthquakes in any other year.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> be the number of years between the earthquake of magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> and the next earthquake of at least this magnitude.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the average number of earthquakes in a year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mi>Y</mi><mo>></mo><mn>100</mn><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mn>100</mn><mo>=</mo><mi>a</mi><mo>-</mo><mn>3</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><msup><mn>10</mn><mrow><mn>5</mn><mo>-</mo><mi>M</mi></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msup><mn>10</mn><mn>5</mn></msup><msup><mn>10</mn><mi>M</mi></msup></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>100000</mn><msup><mn>10</mn><mi>M</mi></msup></mfrac></mrow></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo>=</mo><mfrac><mi>b</mi><msup><mn>10</mn><mn>3</mn></msup></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>100000</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mn>10</mn><mn>5</mn></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfrac><msup><mn>10</mn><mn>5</mn></msup><msup><mn>10</mn><mrow><mn>7</mn><mo>.</mo><mn>2</mn></mrow></msup></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>00631</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0063095</mn><mo>…</mo></mrow></mfenced></math></strong> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Do not accept an answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow><mo>-</mo><mn>2</mn><mo>.</mo><mn>2</mn></mrow></msup></math>.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>></mo><mn>100</mn><mo>⇒</mo></math>no earthquakes in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> years <strong><em>(M1)</em></strong></p>
<p><br><strong>EITHER</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the number of earthquakes of at least magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> in a year</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>Po</mtext><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0063095</mn><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>=</mo><mn>0</mn></mrow></mfenced></mrow></mfenced><mn>100</mn></msup></math> <strong><em>(M1)</em></strong></p>
<p><br><strong>OR</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the number of earthquakes in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> years</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>Po</mtext><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0063095</mn><mo>…</mo><mo>×</mo><mn>100</mn></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>532</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>532082</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>></mo><mn>100</mn><mo>⇒</mo></math>no earthquakes in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> years <strong><em>(M1)</em></strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the number of earthquakes in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> years</p>
<p>since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is large and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> is small</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>100</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>0063095</mn><mo>…</mo></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>531</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>531019</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Parts (a), (b), and (c) were accessible to many candidates who earned full marks with the manipulation of logs and indices presenting no problems. Part (d), however, proved to be too difficult for most and very few correct attempts were seen. As in question 9, most candidates relied on calculator notation when using the Poisson distribution. The discipline of defining a random variable in terms of its distribution and parameters helps to conceptualize the problem in terms that aid a better understanding. Most candidates who attempted this question blindly entered values into the Poisson distribution calculator and were unable to earn any marks. There were a couple of correct solutions using a binomial distribution to approximate the given quantity.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Find the coordinates of the point of intersection of the planes defined by the equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + z = 3,{\text{ }}x - y + z = 5">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + 2z = 6">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>6</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>for eliminating one variable from two equations <strong><em>(M1)</em></strong></p>
<p><em>eg</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {\begin{array}{*{20}{l}} {(x + y + z = 3)} \\ {2x + 2z = 8} \\ {2x + 3z = 11} \end{array}} \right.">
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p>for finding correctly one coordinate</p>
<p><em>eg</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {\begin{array}{*{20}{l}} {(x + y + z = 3)} \\ {(2x + 2z = 8)} \\ {z = 3} \end{array}} \right.">
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p>for finding correctly the other two coordinates <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left\{ {\begin{array}{*{20}{l}} {x = 1} \\ {y = - 1} \\ {z = 3} \end{array}} \right.">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span></p>
<p>the intersection point has coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }} - 1,{\text{ }}3)">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><strong>METHOD 2</strong></p>
<p>for eliminating two variables from two equations or using row reduction <strong><em>(M1)</em></strong></p>
<p><em>eg</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {\begin{array}{*{20}{l}} {(x + y + z = 3)} \\ { - 2 = 2} \\ {z = 3} \end{array}} \right.">
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mo>=</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <strong>or</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&1&1 \\ 0&{ - 2}&0 \\ 0&0&1 \end{array}\left| {\begin{array}{*{20}{c}} 3 \\ 2 \\ 3 \end{array}} \right.} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p>for finding correctly the other coordinates <strong><em>A1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left\{ {\begin{array}{*{20}{l}} {x = 1} \\ {y = - 1} \\ {(z = 3)} \end{array}} \right.">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <strong>or</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\left| {\begin{array}{*{20}{c}} 1 \\ { - 1} \\ 3 \end{array}} \right.} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>the intersection point has coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }} - 1,{\text{ }}3)">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\begin{array}{*{20}{c}} 1&1&1 \\ 1&{ - 1}&1 \\ 1&1&2 \end{array}} \right| = - 2">
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to use Cramer’s rule <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{\left| {\begin{array}{*{20}{c}} 3&1&1 \\ 5&{ - 1}&1 \\ 6&1&2 \end{array}} \right|}}{{ - 2}} = \frac{{ - 2}}{{ - 2}} = 1">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{\left| {\begin{array}{*{20}{c}} 1&3&1 \\ 1&5&1 \\ 1&6&2 \end{array}} \right|}}{{ - 2}} = \frac{2}{{ - 2}} = - 1">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = \frac{{\left| {\begin{array}{*{20}{c}} 1&1&3 \\ 1&{ - 1}&5 \\ 1&1&6 \end{array}} \right|}}{{ - 2}} = \frac{{ - 6}}{{ - 2}} = 3">
<mi>z</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>only if candidate attempts to determine at least one of the variables using this method.</p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>