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<h2>HL Paper 2</h2><div class="specification">
<p>Charlotte decides to model the shape of a cupcake to calculate its volume.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>From rotating a photograph of her cupcake she estimates that its cross-section passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, where all units are in centimetres. The cross-section is symmetrical in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, as shown below:</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>She models the section from <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> as a straight line.</p>
</div>
<div class="specification">
<p>Charlotte models the section of the cupcake that passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> with a quadratic curve.</p>
</div>
<div class="specification">
<p>Charlotte thinks that a quadratic with a maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> and that passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> would be a better fit.</p>
</div>
<div class="specification">
<p>Believing this to be a better model for her cupcake, Charlotte finds the volume of revolution about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis to estimate the volume of the cupcake.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the line passing through these two points.</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 quadratic curve for these four points.</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>By considering the gradient of this curve when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>, explain why it may not be a good model.</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>Find the equation of the new model.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for her estimate of the volume as a sum of two integrals.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of Charlotte’s estimate.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.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"><mi>y</mi><mo>=</mo><mfrac><mn>5</mn><mn>8</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>625</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>625</mn><mi>x</mi></math>, <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>5</mn></math>.<br>Award a maximum of <em><strong>A0A1</strong></em> if not part of an equation.</p>
<p><strong><br></strong><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>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>975</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>.</mo><mn>56</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>.</mo><mn>7</mn></math> <em><strong>(M1)A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>974630</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>.</mo><mn>55919</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>.</mo><mn>6569</mn><mo>…</mo></mrow></mfenced></math></p>
<p><strong><br></strong><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>gradient of curve is positive at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> <em><strong>R1</strong></em></p>
<p><em><br></em><strong>Note:</strong> Accept a sensible rationale that refers to the gradient.</p>
<p><strong><br></strong><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</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"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math></p>
<p>differentiating or using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p>substituting in the coordinates<br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mi>a</mi><mo>+</mo><mn>7</mn><mo>.</mo><mn>5</mn><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math> <em><strong>(A1)<br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>4</mn><mn>2</mn></msup><mi>a</mi><mo>+</mo><mn>4</mn><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>6</mn></math> <em><strong>(A1)</strong></em></p>
<p>solve to get<br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>192</mn><mn>49</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>90</mn><mn>49</mn></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>490</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>.</mo><mn>92</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>.</mo><mn>84</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Use of quadratic regression with points using the symmetry of the graph is a valid method.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>7</mn><mo>.</mo><mn>5</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>490</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</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>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mfenced><mrow><mfrac><mn>5</mn><mn>8</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>4</mn><mrow><mn>7</mn><mo>.</mo><mn>5</mn></mrow></msubsup><msup><mfenced><mrow><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi></math> <em><strong>(M1)(M1) (M1)A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)(M1)(M1)A0</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math> is omitted but response is otherwise correct. Award <em><strong>(M1)</strong></em> for an integral that indicates volume,<em><strong> (M1)</strong></em> for their part (a) within their volume integral, <em><strong>(M1)</strong></em> for their part (b)(i) within their volume integral, <em><strong>A1</strong></em> for their correct two integrals with all correct limits.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>501</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>501</mn><mo>.</mo><mn>189</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.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>
<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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>It is known that the weights of male Persian cats are normally distributed with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>1</mn><mo> </mo><mtext>kg</mtext></math> and variance <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo> </mo><msup><mtext>kg</mtext><mn>2</mn></msup></math>.</p>
</div>
<div class="specification">
<p>A group of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn></math> male Persian cats are drawn from this population.</p>
</div>
<div class="specification">
<p>The male cats are now joined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn></math> female Persian cats. The female cats are drawn from a population whose weights are normally distributed with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</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>45</mn><mo> </mo><mtext>kg</mtext></math>.</p>
</div>
<div class="specification">
<p>Ten female cats are chosen at random.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a diagram showing the above information.</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 proportion of male Persian cats weighing between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>kg</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>kg</mtext></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>Determine the expected number of cats in this group that have a weight of less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>3</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that exactly one of them weighs over <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>62</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> be the number of cats weighing over <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>62</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<p>Find the variance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A cat is selected at random from all <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>160</mn></math> cats.</p>
<p>Find the probability that the cat was female, given that its weight was over <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>7</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <strong><em>A1A1</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for a normal curve with mean labelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>1</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math>, <em><strong>A1</strong></em> for indication of SD <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>)</mo><mo>:</mo></math> marks on horizontal axis at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>6</mn></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>6</mn></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> on the correct side and approximately correct position.<br><br></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>X</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>6</mn><mo>.</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mn>5</mn><mo>.</mo><mn>5</mn><mo><</mo><mi>X</mi><mo><</mo><mn>6</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> <strong>OR </strong>labelled sketch of region <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>673</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>673074</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><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"><mfenced><mrow><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo><</mo><mn>5</mn><mo>.</mo><mn>3</mn></mrow></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mn>0</mn><mo>.</mo><mn>0547992</mn><mo>…</mo></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0547992</mn><mo>…</mo><mo>×</mo><mn>80</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>.</mo><mn>38</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>4</mn><mo>.</mo><mn>38393</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><em><strong>[3 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"><mi>Y</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>4</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><msup><mn>45</mn><mn>2</mn></msup></mrow></mfenced></math>,</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>P</mtext><mfenced><mrow><mi>Y</mi><mo>></mo><mn>4</mn><mo>.</mo><mn>62</mn></mrow></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mn>0</mn><mo>.</mo><mn>394862</mn><mo>…</mo></math> <strong><em>(A1)</em></strong></p>
<p>use of binomial seen or implied <strong><em>(M1)</em></strong></p>
<p>using <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mfenced><mrow><mn>10</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>394862</mn><mo>…</mo></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0430</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0429664</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>p</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>p</mi></mrow></mfenced><mo>=</mo><mn>2</mn><mo>.</mo><mn>39</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>2</mn><mo>.</mo><mn>38946</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</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>F</mi><mo>∩</mo><mfenced><mrow><mi>W</mi><mo>></mo><mn>4</mn><mo>.</mo><mn>7</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>3284</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>1642</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>attempt use of tree diagram <strong>OR</strong> use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>F</mi><mo> </mo><menclose notation="left"><mi>W</mi><mo>></mo><mn>4</mn><mo>.</mo><mn>7</mn></menclose></mrow></mfenced><mo>=</mo><mfrac><mrow><mtext>P</mtext><mfenced><mrow><mi>F</mi><mo>∩</mo><mfenced><mrow><mi>W</mi><mo>></mo><mn>4</mn><mo>.</mo><mn>7</mn></mrow></mfenced></mrow></mfenced></mrow><mrow><mtext>P</mtext><mfenced><mrow><mi>W</mi><mo>></mo><mn>4</mn><mo>.</mo><mn>7</mn></mrow></mfenced></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>3284</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>9974</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>3284</mn></mrow></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>248</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>247669</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>A city has two cable companies, X and Y. Each year 20 % of the customers using company X move to company Y and 10 % of the customers using company Y move to company X. All additional losses and gains of customers by the companies may be ignored.</p>
</div>
<div class="specification">
<p>Initially company X and company Y both have 1200 customers.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a transition matrix <strong><em>T</em></strong> representing the movements between the two companies in a particular year.</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 eigenvalues and corresponding eigenvectors of <strong><em>T</em></strong>.</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>Hence write down matrices <em><strong>P</strong></em> and <em><strong>D</strong></em> such that <em><strong>T</strong></em> = <em><strong>PDP</strong></em><sup>−1</sup>.</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>Find an expression for the number of customers company X has after <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> years, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{N}">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down the number of customers that company X can expect to have in the long term.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</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}} {0.8}&{0.1} \\ {0.2}&{0.9} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.8</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0.2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.9</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}} {0.8 - \lambda }&{0.1} \\ {0.2}&{0.9 - \lambda } \end{array}} \right| = 0">
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.8</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0.2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.9</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
<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="\lambda = 1">
<mi>λ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and 0.7 <em><strong>A1</strong></em></p>
<p>eigenvectors <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right)">
<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>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Accept any scalar multiple of the eigenvectors.</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><em><strong>EITHER</strong></em></p>
<p><em><strong>P</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&1 \\ 2&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>D</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{0.7} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mn>0.7</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1A1</strong></em></p>
<p><em><strong>OR</strong></em></p>
<p><em><strong>P</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&1 \\ { - 1}&2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>D</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {0.7}&0 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.7</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</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><em><strong>P</strong></em><sup>−1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{3}\left( {\begin{array}{*{20}{c}} 1&1 \\ 2&{ - 1} \end{array}} \right)">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</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="\frac{1}{3}\left( {\begin{array}{*{20}{c}} 1&1 \\ 2&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{{{0.7}^n}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&1 \\ 2&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {1200} \\ {1200} \end{array}} \right)">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mrow>
<msup>
<mrow>
<mn>0.7</mn>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1200</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1200</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p>attempt to multiply matrices <em><strong>M1</strong></em></p>
<p>so in company A, after <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> years, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="400\left( {2 + {{0.7}^n}} \right)">
<mn>400</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>0.7</mn>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em> A1</em></strong></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>400 × 2 = 800 <strong><em> A1</em></strong></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>In a small village there are two doctors’ clinics, one owned by Doctor Black and the other owned by Doctor Green. It was noted after each year that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>5</mn><mo>%</mo></math> of Doctor Black’s patients moved to Doctor Green’s clinic and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> of Doctor Green’s patients moved to Doctor Black’s clinic. All additional losses and gains of patients by the clinics may be ignored.</p>
<p>At the start of a particular year, it was noted that Doctor Black had <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2100</mn></math> patients on their register, compared to Doctor Green’s <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3500</mn></math> patients.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a transition matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> indicating the annual population movement between clinics.</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 prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.</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 matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math>, with integer elements, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">D</mi><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math> is a diagonal matrix.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that the long-term transition matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mo>∞</mo></msup></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mo>∞</mo></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</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 mathvariant="bold-italic">T</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>965</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>05</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>035</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>95</mn></mtd></mtr></mtable></mfenced></math> <em><strong>M1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>95</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>035</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>05</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>965</mn></mtd></mtr></mtable></mfenced></math>.<br>Award the <em><strong>A1</strong></em> for a transposed <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> if used correctly in part (b) i.e. preceded by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>×</mo><mn>2</mn></math> matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2100</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>3500</mn></mrow></mfenced></math> rather than followed by a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>1</mn></math> matrix.</p>
<p><br><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"><msup><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>965</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>05</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>035</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>95</mn></mtd></mtr></mtable></mfenced><mn>2</mn></msup><mfenced><mtable><mtr><mtd><mn>2100</mn></mtd></mtr><mtr><mtd><mn>3500</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2294</mn></mtd></mtr><mtr><mtd><mn>3306</mn></mtd></mtr></mtable></mfenced></math></p>
<p>so ratio is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2294</mn><mo>:</mo><mn>3306</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>1147</mn><mo>:</mo><mn>1653</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>693889</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mi>x</mi><mo>=</mo><mi>λ</mi><mi>x</mi><mo> </mo><mo>:</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>965</mn><mo>-</mo><mi>λ</mi></mtd><mtd><mn>0</mn><mo>.</mo><mn>05</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>035</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>95</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>965</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>.</mo><mn>95</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>-</mo><mn>0</mn><mo>.</mo><mn>05</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>035</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>915</mn></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn></math> <em><strong>(A1)</strong></em></p>
<p>attempt to find eigenvectors for at least one eigenvalue <em><strong>(M1)</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>915</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> (or any real multiple) <em><strong>(A1)</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd></mtr></mtable></mfenced></math> (or any real multiple) <em><strong>(A1)</strong></em></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced></math> (accept integer valued multiples of their eigenvectors and columns in either order) <em><strong>A1</strong></em></p>
<p><br><em><strong>[6 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"><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mn>17</mn></mfrac><mfenced><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> This mark is independent, and may be seen anywhere in part (d).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>915</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</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">T</mi><mi>n</mi></msup><mo>=</mo><mi mathvariant="bold-italic">P</mi><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><msup><mn>915</mn><mi>n</mi></msup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msup><mn>1</mn><mi>n</mi></msup></mtd></mtr></mtable></mfenced><mfrac><mn>1</mn><mn>17</mn></mfrac><mfenced><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)A0</strong></em> for finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mi mathvariant="bold-italic">P</mi></math> correctly.</p>
<p><br>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>→</mo><mo>∞</mo><mo>,</mo><mo> </mo><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><msup><mn>915</mn><mi>n</mi></msup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msup><mn>1</mn><mi>n</mi></msup></mtd></mtr></mtable></mfenced><mo>→</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mi>n</mi></msup><mo>→</mo><mfrac><mn>1</mn><mn>17</mn></mfrac><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>7</mn></mtd><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>AG</strong></em> line must be seen for the final <em><strong>A1</strong></em> to be awarded.</p>
<p><br><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD ONE</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>2100</mn></mtd></mtr><mtr><mtd><mn>3500</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3294</mn></mtd></mtr><mtr><mtd><mn>2306</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>so ratio is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3294</mn><mo>:</mo><mn>2306</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>1647</mn><mo>:</mo><mn>1153</mn><mo>,</mo><mo> </mo><mo> </mo><mn>1</mn><mo>.</mo><mn>42844</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>700060</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD TWO</strong></p>
<p>long term ratio is the eigenvector associated with the largest eigenvalue <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>:</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The masses in kilograms of melons produced by a farm can be modelled by a normal distribution with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>6</mn><mtext> kg</mtext></math> and a standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn><mtext> kg</mtext></math>.</p>
</div>
<div class="specification">
<p>Find the probability that two melons picked at random and independently of each other will</p>
</div>
<div class="specification">
<p>One year due to favourable weather conditions it is thought that the mean mass of the melons has increased.</p>
<p>The owner of the farm decides to take a random sample of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math> melons to test this hypothesis at the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> significance level, assuming the standard deviation of the masses of the melons has not changed.</p>
</div>
<div class="specification">
<p>Unknown to the farmer the favourable weather conditions have led to all the melons having <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> greater mass than the model described above.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a melon selected at random will have a mass greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>0</mn><mo> </mo><mtext>kg</mtext></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>both have a mass greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>0</mn><mo> </mo><mtext>kg</mtext></math>.</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>have a total mass greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>0</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the null and alternative hypotheses for the test.</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 the critical region for this test.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the mean and standard deviation of the mass of the melons for this year.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of a Type II error in the owner’s test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</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> represent the mass of a melon</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>></mo><mn>3</mn><mo>.</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>212</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>2118</mn><mo>…</mo></mrow></mfenced></math> <strong>(M1)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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2118</mn><mo>…</mo><mo>×</mo><mn>0</mn><mo>.</mo><mn>2118</mn><mo>…</mo></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0449</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>04488</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> represent the total mass</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>T</mi></mfenced><mo>=</mo><mn>5</mn><mo>.</mo><mn>2</mn></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mi>T</mi></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo>+</mo><mn>0</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <strong>(M1)A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>5</mn><mo>.</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>5</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>6</mn><mo>.</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>129</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>1289</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> be the mean mass of the melons produced by the farm.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo>:</mo><mo> </mo><mi>μ</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>6</mn><mo> </mo><mtext>kg</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub><mo>:</mo><mo> </mo><mi>μ</mi><mo>></mo><mn>2</mn><mo>.</mo><mn>6</mn><mo> </mo><mtext>kg</mtext></math> only <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo>:</mo></math> The mean mass of melons produced by the farm is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>6</mn><mo> </mo><mtext>kg</mtext></math><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 mass of melons produced by the farm is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>6</mn><mo> </mo><mtext>kg</mtext></math></p>
<p><strong>Note:</strong> Award <strong>A0</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>6</mn><mo> </mo><mtext>kg</mtext></math> does not appear in the hypothesis.</p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Under <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>X</mi><mo>¯</mo></mover><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>2</mn><mo>.</mo><mn>6</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>0</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup></mrow><mn>16</mn></mfrac></mrow></mfenced></math> <strong>A1</strong></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><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>05</mn></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>81</mn><mo> </mo><mfenced><mrow><mn>2</mn><mo>.</mo><mn>805606</mn><mo>…</mo></mrow></mfenced></math> <strong>(A1)</strong></p>
<p>Critical region is <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>X</mi><mo>¯</mo></mover><mo>></mo><mn>2</mn><mo>.</mo><mn>81</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math> represent the new mass of the melons</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>W</mi></mfenced><mo>=</mo><mn>1</mn><mo>.</mo><mn>1</mn><mo>×</mo><mn>2</mn><mo>.</mo><mn>6</mn><mo>=</mo><mn>2</mn><mo>.</mo><mn>86</mn></math> <strong>A1</strong></p>
<p>Standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>1</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>55</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> award <strong>M1A0</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mi>W</mi></mfenced><mo>=</mo><mn>1</mn><mo>.</mo><msup><mn>1</mn><mn>2</mn></msup><mo>×</mo><mn>0</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>.</mo><mn>3025</mn></math></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML" class="wrs_chemistry"><mi mathvariant="normal">P</mi><mo>(</mo><mi>Type</mi><mo> </mo><mi>II</mi><mo> </mo><mi>error</mi><mo>)</mo><mo>=</mo></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>2</mn><mo>.</mo><mn>81</mn><mo> </mo><menclose notation="left"><mo> </mo><mi>μ</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>86</mn><mo>,</mo><mo> </mo><mi>σ</mi><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>55</mn></mrow><mn>4</mn></mfrac></menclose></mrow></mfenced></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>346</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>346204</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>358</mn></math> from use of the three‐figure answer to part (d)</p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">f.</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">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variable <em>X</em> is thought to follow a binomial distribution B (4, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>). In order to investigate this belief, a random sample of 100 observations on <em>X</em> was taken with the following results.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>An automatic machine is used to fill bottles of water. The amount delivered, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y">
<mi>Y</mi>
</math></span> ml, may be assumed to be normally distributed with mean <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> ml and standard deviation 8 ml. Initially, the machine is adjusted so that the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> is 500. In order to check that the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> remains equal to 500, a random sample of 10 bottles is selected at regular intervals, and the mean amount of water, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\bar y}">
<mrow>
<mrow>
<mover>
<mi>y</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
</math></span>, in these bottles is calculated. The following hypotheses are set up.</p>
<p style="text-align: center;">H<sub>0</sub>: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> = 500; H<sub>1</sub>: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> ≠ 500</p>
<p style="text-align: left;">The critical region is defined to be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\bar y < 495} \right) \cup \left( {\bar y > 505} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mover>
<mi>y</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
<mo><</mo>
<mn>495</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>∪<!-- ∪ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mover>
<mi>y</mi>
<mo stretchy="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
<mo>></mo>
<mn>505</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">State suitable hypotheses for testing this belief.</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;">Calculate the mean of these data and hence estimate the value of <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span>.</p>
<div class="marks">[5]</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;">Calculate an appropriate value of <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> and state your conclusion, using a 1% significance level.</p>
<div class="marks">[13]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Find the significance level of this procedure.</p>
<div class="marks">[5]</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;">Some time later, the actual value of <span class="mjpage"><math alttext="\mu " xmlns="http://www.w3.org/1998/Math/MathML"> <mi>μ</mi> </math></span><em> </em>is 503. Find the probability of a Type II error.</p>
<div class="marks">[3]</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 class="indent1"><em>H</em><sub>0 </sub>: The data are B (4, <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span>); H<sub>1</sub> : The data are not B (4, <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span>) <em><strong>A1</strong></em></p>
<p class="indent1"><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="indent1">Mean <span class="mjpage"><math alttext=" = \frac{{1 \times 32 + \ldots + 4 \times 14}}{{100}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>×</mo> <mn>32</mn> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mn>4</mn> <mo>×</mo> <mn>14</mn> </mrow> <mrow> <mn>100</mn> </mrow> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p class="indent1">= 1.8 <em><strong>A1</strong></em></p>
<p class="indent1"><span class="mjpage"><math alttext="4\widehat p = 1.8 \Rightarrow \widehat p = 0.45" xmlns="http://www.w3.org/1998/Math/MathML"> <mn>4</mn> <mrow> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> <mo>=</mo> <mn>1.8</mn> <mo stretchy="false">⇒</mo> <mrow> <mover> <mi>p</mi> <mo>^</mo> </mover> </mrow> <mo>=</mo> <mn>0.45</mn> </math></span> <em><strong>M1A1</strong></em></p>
<p class="indent1"><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="indent1">The expected frequencies are</p>
<p class="indent1"><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></p>
<p class="indent1">The last two classes must be combined because the expected frequency for <span class="mjpage"><math alttext="x" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> </math></span> = 4 is less than 5. <em><strong>R1</strong></em></p>
<p class="indent1"><span class="mjpage"><math alttext="{\chi ^2} = \frac{{{{17}^2}}}{{9.15}} + \frac{{{{32}^2}}}{{29.95}} + \frac{{{{19}^2}}}{{36.75}} + \frac{{{{32}^2}}}{{24.15}} - 100" 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> <mn>17</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>9.15</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>32</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>29.95</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>19</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>36.75</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>32</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>24.15</mn> </mrow> </mfrac> <mo>−</mo> <mn>100</mn> </math></span> <em><strong>M2</strong></em></p>
<p class="indent1">= 18.0 <em><strong>A2</strong></em></p>
<p class="indent1">DF = 2 <em><strong>(A1)</strong></em></p>
<p class="indent1">Critical value = 9.21 <em><strong>A1 </strong></em></p>
<p class="indent1">We conclude, at the 1% significance level, that <em>X</em> does not fit a binomial model. <em><strong>R1 </strong></em></p>
<p class="indent1"> </p>
<p class="indent1"><strong>Special case:</strong> award the following marks to candidates who do not combine classes.</p>
<p class="indent1"><span class="mjpage"><math alttext="{\chi ^2} = \frac{{{{17}^2}}}{{9.15}} + \frac{{{{32}^2}}}{{29.95}} + \frac{{{{19}^2}}}{{36.75}} + \frac{{{{18}^2}}}{{20.05}} + \frac{{{{14}^2}}}{{4.1}} - 100" 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> <mn>17</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>9.15</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>32</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>29.95</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>19</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>36.75</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>18</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>20.05</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>14</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>4.1</mn> </mrow> </mfrac> <mo>−</mo> <mn>100</mn> </math></span> <em><strong>M2</strong></em></p>
<p class="indent1">= 39.6 <em><strong>A0</strong></em></p>
<p class="indent1">DF = 3 <em><strong>(A1)</strong></em></p>
<p class="indent1">Critical value = 11.345 <em><strong>A1 </strong></em></p>
<p class="indent1">We conclude, at the 1% significance level, that <em>X</em> does not fit a binomial model. <em><strong>R1</strong></em></p>
<p class="indent1"> </p>
<p class="indent1"><em><strong>[13 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="indent1">Under H<sub>0</sub>, the distribution of <span class="mjpage"><math alttext="{\bar y}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </math></span> is N (500, 6.4). <em><strong>(A1)</strong></em></p>
<p class="indent1">Significance level = P <span class="mjpage"><math alttext="{\bar y}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </math></span> < 495 or > 505 | H<sub>0</sub> <em><strong>M2</strong></em></p>
<p class="indent1"> = 2 × 0.02405 <em><strong>(A1)</strong></em></p>
<p class="indent1"> = 0.0481 <em><strong>A1 N5 </strong></em></p>
<p class="indent1"><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="indent1">The distribution of <span class="mjpage"><math alttext="{\bar y}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </math></span> is now N (503, 6.4). <em><strong>(A1)</strong></em></p>
<p class="indent1">P(Type ΙΙ error) = P(495 < <span class="mjpage"><math alttext="{\bar y}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mover> <mi>y</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </math></span> < 505) <em><strong>(M1)</strong></em></p>
<p class="indent1"> = 0.785 <em><strong>A1 N3</strong></em></p>
<p class="indent1"><em><strong>[3 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.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.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 student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math></p>
<p>This equation links a variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> with the temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> are positive constants and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The Arrhenius equation predicts that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> is a straight line.</p>
</div>
<div class="specification">
<p>Write down</p>
</div>
<div class="specification">
<p>The following data are found for a particular reaction, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is measured in Kelvin and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>mol</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>:</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Find an estimate of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac></math> is always positive.</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mo>∞</mo></mrow></munder><mi>k</mi><mo>=</mo><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>k</mi><mo>=</mo><mn>0</mn></math>, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></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>(i) the gradient of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>;</p>
<p>(ii) the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the regression line for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use chain rule, including the differentiation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac><mo>=</mo><mi>A</mi><mo>×</mo><mfrac><mi>c</mi><msup><mi>T</mi><mn>2</mn></msup></mfrac><mo>×</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>this is the product of positive quantities so must be positive <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>R1</strong> </em>may be awarded for correct argument from <strong>their</strong> derivative. <em><strong>R1</strong> </em>is not possible if their derivative is not always positive.</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><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAToAAADZCAYAAACw5fzEAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAABkFSURBVHhe7d0HeFRV+gbwF9JIDwESUkggBQKCkAjSQTCAK6KCBRFQEdAVBWERXQtrXVZXpSju+lcXfMQFpK/SJIKUCAQIoQkBUyE9pEJ64X/O5ewiYoaUmcnN5P09zzyT881MHiXk5TvnnntviysCiIgsWEv1TERksRh0RGTxGHREZPFqvUbHpTwi0qMWLVqor2rW4I6usrJSfUVEpE/17uiij8Zg/boNyEhLwbIvl6sqEZF5mbSjuy0sFP7+HeDj66sqRET61KCp66GoQwjp2lWNiIj0qd5BV3jpEqIOHEBoaKiqEBHpU72DbvN3m9Ha3R1eXu2xZ/durPhqBSIiIpCamqreQUSkD/UKuurqauz4/nsMGToEjk5OOHDwkHh2xp133gkfHx/1LiIifajXUdeSkhIM7D8A7yxYgIKCAvTo0QO33NKtVkc/iIiMyWRHXQ8cOIjysjL88ksctm/bhu7db2HIEZFu1Svotm3dhlatWqFDhw5IuZCiqjWTHWBRUZEaERGZV52Drkx0ckcOH8KMmTMxbtz9sLK2QmFhoXr1RnLK+/Zbb2PKE1O0zxIRmVudgy4pMQlZmZno06ePNr6lWzds3rwFxcXFiIo6pNV+LSLiB6xbuxYx0dHYuHGTqhIRmU+dg+5MbCz6DxyIrl1DtHHf/v2wds0aHDp0GKFh1++py8jIwJuvvw5fMcX18vHB+++9h8jISPUqEZF5mPTqJfNfm4+Kykp4enoiPi4OQUHB+Gb1amz8dhPaixoRUUOZ5eolNbmYk4N9e/dhzpzZqgI8+9wMeHh4YME7f9X24hERmYNJgk6G2FtvvInwEeFaN/dfNjY2+PMrL+OHiAjEnj2rqkREpmWSoDtw8KB25sRDDz+sKtf0798PEydNwvJ/LVMVIiLTMnrQFRQWYv4rr2LCxIno0qWzql7vmRnP4Mjhw0hNTVMVIiLTMXrQZWdlw8vbG7NmzVSVG7m7u2P8hAlISbn5ZmMiooYyyVFXuUbXsuW1DF20aLF21HXpJ0tV5dr346ljRNQQjXbU9dchVxP5H8eQIyJzMEnQERHpCYOOiCweg46ILB6DjogsHoOOiCweg46ILB6DjogsHoOOiCweg46ILB6DjogsHoOOiCweg46ILB6DjogsHoOOiCweg46ILB6DjogsHoOOiCweg46ILB6DjogsntGCbt26ddi6dbsaERHph1GCbs+evZj/ymtwb+2qKkRE+tHgoMvLy8Pbb72F6uoquLZurapERPrR4KDbunWbdjPqllZWcHVhR0dE+tOgoEtPz8DlS5dgbdVSPKzg5sagIyL9qXfQVVZW4tN/foqHHn4IGSLwnF1c4ODgoF4lItKPFlcE9bVBv35beXkF3n//ffTu0we3hYVh3gvzUFZejpUrv1bvuOaS6PimPTkNefl5GDlylKoCAYEBGDdurBoZlpKSitWrVquRYSFdQ3DPPaPVyLAjR6Kx+8fdamRYYFAgxo69X40M2/H9Dpw4cVKNDPPx9cWECePVyLCfftqPA/sPqJFhY+4dgy5dOquRYfvF99wvvndtjBgZjp49e6qRYYsWLUZVZZUaGTb8zuEICwtVI8M+//wLFOQXqJFh4x4Yi4CAADUybMWKfyMzI0ONDJs953lYW1urkWHLl3+JnIs5amTY9Kemw9XVRY1qlpaWhpX/XqVGNzdp8iS0b++pRjXLy8vH8mXLUF19fSz07NUTI0aEq5G+tGjRQn1VszoHXXV1NT744EPtB+fk6KjV1q5dg9H3jMHf3l2gjX8tJycX9997L0pLSxEQFKSqEH+pw/DSSy+qkWFnzpzFG2+8rkaGDRwwALOen6VGhm3ZshVfffWVGhl222298eKLL6iRYZ999hl27tylRoaFhHTFm2/W7v9tzZq1WL9+vRoZNnv2bPTv30+NDFu3br34Ga5VI8OmT5+G8PDa/YWfNHESKkTnXxtTpkzBXXdd+4fQkOdmzkJ2VpYaGfbSSy/VOkBffvlVJCTEq5FhK1Z8BVtbWzUybO7cF8Q/1ilqZNjSpUvRrl1bNarZuXPnMH/+X9To5hYsWIBA0VzcTLoI+rlz5qKq+vp/oMJHjMD0aVPVSF9MEnQ/ig4oOTkZTzzxuDaWU9iQ4M54fs4czJw1U6v9lvyXPT4uDks/WaoqRETGUZugq9MaXXx8PDZt2oRHH52gKmJqevmy9iynYEREelTroJNz9xdfmIfRo++5rmVPuXC1Jb906WrgERHpTa2DbuWqVegSEoKYmBgUFFxdCL4gQi4i4gc8/MgjSExIwPoNm7Q6EZGe1Ouoa11xjY6ITMUkByPqg0FHZFylpWUoKi5CcXExysvKUVwinsvLkZeTq+1wKCsrRUVllXguQ2VFBcorylFaUnL1veIz8ve5WIzlLooSWRefla/Lg4vy9UrxWa0uPhcc3BnLv1yGli0bdH6ByTDoiHRChkdxcYm2r1RuuZHniJeLEMrLzUF+foEIlAoUXb6MUlGTQVVYkC/GRSgU75djGTpyXFFZgZzsbC2gbkaupbu1doetnZ22md/BwR72rVrBzc0NVvJMJnd3bS+gi7Oz9l4nJyfY2NjA2dkJVqLu3qatVndv3Rq+vj61CpTGwKAjaqCqqqudTYkIm8qKSu3ry5cKtW6oRARXQWGh1g0VFRUhPzcXWSKEZDClZ2Rq3VRBfh6yMjO1bskQGTwOjo4ieGxEuNjAToSTDBl7e3stpBxFCMlnGxtbbUNxKxFYTk6OaCPCyFEElYuLi/beNm3aaMHl7t58LrDBoKNmT/69rRRhVSW6KNkFVVWJh+iKsi9eREZmlhZQsstKT0nRuic5vnhRdll5yBHvkQElOzA5pasWn5UdlezE/kt2RC3ElM5KPkRYyYcc+3boIILKDi4ilFq7tdbOA5cB1M7DA9aia3IXgWQrnmV31aaNu/Y5GxFs176PtfjeoibeQ4Yx6MiiySlfvpjiZYjuSZ4pkZ2VKcaF2oUmZGDJ6WJWRgaKxPNlMe2THVZpqZg+ii6sJnLKJveEuru3gavokpxFOMkgcnF1hb3oqGTn5Cq/tm+ldVCurm5aGMkOy9FRdF6iK2M4mReDjpoMOf3LLyhAXm6eeOQiJydHmxbKbqtQPGemp4suLEdb28rNzdHWqSoqKtSnryeneU4ioORUz00EkZub6KpEcMkQcnBwFGF1darn5e199b2OTtpUT9ZleFHTwqCjRiP/vsj1LG1xXVvXuoRcLaTykCFCS577KTuxjMwMpKWkIj8vV/uMPJn8ypWrC+0tW1qhleic5HqVDCC5kN6ufXut05JBpa1FiW6rg39HeIgpYbu2bcVzu/91VPIXQD70erSQjINBR0YjtzOUlBSLR6kWWnI96/LlS8jKzBJhlal1YLLLkpvJL1++rHVe8llbs1KBI4NKdlpyGiinhLKDkkf2ZGjJI3tyHcvVrbV29E8utGuL77a2sBHTSbkwLx9Ev8Wgo1qTl96S00c5TUxKTkZSUiKSEhKRlpaufZ0twkweeZQL8XJhXnZJciFedk9t27WDj28HtPdqD18fb/h08IOX6LzklgQZVnJR3s7ualDV5i8lUV0w6OgGcr9WYmIS4uJ+QWJ8wtXpY3oaLoippNwG8duFemcXVwR37oygwAC09/ZC+/ZeaC26L09PD9GZuYnp4tW1L6LGwqBrpuQWiqzsLKSLbiw5KQmpaWki1OIRe/YczsWeue5n6ermBg9PT9F9+cLPzx9eoivr4OcHT9GRdfT307Y/EOkZg87CyX1h8qikXA9LTj6PhIQE/HzqFA5FRWlXypWvy5+b7LicnJzh69cBgYGBCAwK0rq0YPHcVnRkchoqH5xWUlPEoLMg8s9fBpfczBodHY3Ifftw6uRJZGVlIT8/XzufUf7A5ZpZaO/e6Hv77QgQoeYnwk1uUnV2ctICj2FGloZB18TJu6wdPHgAp38+jbOxsUhITESGmIZKcstFt+7dERwcjKCgYPiLaaZ/p47w8fbRNrMSNRcMuiZC7i87ceyYNv08f148xHN8Qry2v6xTQCfxCIC3t5d4+Gj7xAICg9CpUycGGpHAoNMpGWzHRbAdP34Chw5G4Wj0EW3LRus2bRASEoLuolO7tWdP9OvXVzvCSUQ1Y9DphFxbk7ds3LNnDyJ27NAOFsg1NXt7B3iJTm3I0Dtw/7ix6BwcpK2xcR2NqPYYdI1I/nldzMnFfzZuxK6dO3Hq1CkUFxVpV60YddddGDxkCLp166btR+NJ4ET1x6AzM/lnJA8YbN+6HXt2/4iTJ0+iorwc3j4+GHPvvRg5ahS6du0Keb0xIjIOBp2ZlJWV48SJE/h6xdfY8t232vmd8gBCvwEDcPfdd+P22/uodxKRsTHoTEiuu2VmZeE/m/6DNatXI+XCBe3AwX1jx+K+++9HUHAQ7HgSOpHJMeiMTP4ZXMzJwdbNW7Bp40YkJSahvZcXhtwxFIMGDUSvXqHa9faJyHwYdEYi/9/PnInFx0s+wk8/ReKK6OaenjEDo0ffDX9/f17vjKgR1Sbo+Bt6E8eOHcfM52biATEdjdy3F/fedx82b9uG5557Vtu0y5Aj0j/+ltbg/PkLmDXreTw4diwO7N+P8RMmYPsPEXjnr+9op1sRUdPBoPsNeYZCRMQPGP/QQ9gVEYEnpk7F1u3b8fobr8PH21u9i4iaEgbdr8THJ2Dqk1Px7B//qF3K6NstW/Daa69qm3prsw5ARPrEoBPk7fAWL16CMaNH41hMDJ6e8Qw+/+ILBAR0Uu8goqasWQddUVExvvjiXxgVPgLr167FU08/jZ0/7sLcuXN5ZRAiC9Jsg05emXfGM8/gg7//HSNGjsT3ERGYPWc22rZtq95BRJaiWQadPKPh8cmP4eTxE1j25ZeY/5f5cHDgjYuJLFWzC7rU1DQ8NnESYmNj8dEnSzFgQH/uhSOycM3qN1yG3JNPTNFu67fk44+107aIyPI1m6CTN5F5bPJkpKenYeGSJQgPv1O9QkSWrlkE3dmzZ/HoI49q56h+vXIlhg27g/viiJoRiw+63Nw8THnsCVRUlGPVN9/g1lt7qFeIqLmw+KD7/LPPUFhYgMUffaSd4UBEzY9FB11aWjrWrP4Gf5o3D6GhvVSViJobiw06eXL+K39+GX5+fpgy5QlVJaLmyGKDbs2atfgpch+enDaVBx6ImjmLDLqUlBQs/OAD3NK9O0aOGqmqRNRcWVzQXbp0GfNemAdraxt8uGgh7Ozs1CtE1FxZXNB9tGQJYqKj8bf33kVgYKCqElFzZlFBl5x8Hv9e8TUGDhqMIUMGqyoRNXcWE3TyPqvvvP025HGHV//ymnYTaSIiyWKCLikpCZF792o3sQnoxCsDE9E1FhN0GzdshI2trbadhIjo1ywi6CorqxCxYweefuYZ+Pr4qCoR0VUWEXQbNmzQLo0+adJEVSEiuqbJB11WVjb+/u67eGj8eLi6uqoqEdE1TT7otm7ZgsKCAowcOUJViIiu16SD7vLlInzyySfo3KULgjt3VlUious16aDb/N13yMvJwR9nzICNtbWqEhFdr0kH3apVq+DfsSOGDx+mKkREN2qyQRcZGYmfT57EfWPHwsHBQVWJiG7UJIOurKwcixcu1rq56dOnqSoR0e9rkkG3f/9+HIs5iqnTp8PennfYJyLDmlzQXblyBcv+tQxOzs4Yfc9oVSUiqlmTC7r4+HgcPhSFRx59FK4uLqpKRFSzJhd027Zug7wDxD1j7rlaICK6iSYVdPLOXmvWrNE2CIeEhKgqEZFhTSroTp06hfTUVEyYOBHWvLAmEdVSkwq6vXv2ws/fH2M4bSWiOmhSQbdz5y6Me+ABODo6qgoR0c01maBLEVPWM6d/xgMPPqgqRES102SCbtuWrejWrRu8vNqrChFR7TSZoNu1cyd69uypRkREtdckgi75/AUcjY7GLT26qwoRUe01iaBbv3atdt/WQYN4U2oiqjvdB11JSQm2bd2KgYMHw9vbS1WJiGpP90F3+PARJCYkYOy4capCRFQ3ug86eWNqW1tb3HHHUFUhIqobXQddQUEh9u7+EcPvDOetDImo3nQddMePH0NhYSH+MPpuVSEiqjtdB508t9XB0RFht4WpChFR3ek26GQnt2H9egwcNAjtPT1VlYio7nQbdNu3bdfuwD/58cfRooW81CYRUf3oMujkfSF+ioxEO9HJ9by1h6oSEdWPLoOuoqIChw8fRq9evXiXLyJqMF0GXfTRGGRlZmLQ4MGcthJRg+ky6LZs/g5WVla46w93qQoRUf3pLujkDXAORx1Cr7AwtHF3V1UiovrTXdClp2cgOTmZN6cmIqPRXdAdO3YM1VVVGDZsuKoQETWM/oIuJgZBnbvA19dHVYiIGkZ3QXf69GmMHj2aR1uJyGh0FXRlZWVITkpCv/79VIWIqOF0FXQZGRla2AUGBqgKEVHD6SrooqIOwcPTk9eeIyKj0k3QyfNb5dVKwsLCuD5HREalm6BLS0vHyRMnMGDAAFUhIjIO3QTdqVMntfW5TgGdVIWIyDh0E3QxR2Pg6OgIbx/unyMi49JN0O3btw/+Hf3h6uKiKkRExqGLoJPnt549cwb9+vXngQgiMjpdBF3UwYPa89Bhw7RnIiJjavSgk9tKIiMj0badB3r1vFVViYiMp9GDrrq6GoeiotBThJw8GEFEZGyNHnQJiUlIS03Frb16qgoRkXE1etDt3bNHex44aLD2TERkbI0edNGHj8DVzQ09ut+iKkRExtWoQSfPhDhz+jRCw8K0m+EQEZlCowZddnY2MjMz0Ss0VFWIiIyvUYMuOvooysvL0Lfv7apCRGR8jRp0O3bsgIODI3r04P45IjKdRgu6yqoqHDp4EH3790erVnaqSkRkfI0WdBezs1GQn49+/fuqChGRaTRa0KWkpGpnRXDaSkSm1mhBJ8+GcHJ2hr+/v6oQEZlGowVdUnIyPDw84O7eWlWIiEyj0YLuzM8/o3efPrCxsVEVIiLTaJSgk5dmOiWCbnh4uKoQEZlOowRdbm6edsS1T+/bVIWIyHQaJeiSkhLh7OzM688RkVk0StDFxcXD09OTJ/ITkVk0StDJK5Z4eXnxRjhEZBZmD7rqK1ewd+8+dAkJURUiItMye9BdOH8B55MSERwcrCpERKZl9qA7cfy49twpIEB7JiIyNbMHXVxcHGzt7ODj46MqRESmZdagkxuFz507B29vb16aiYjMxuxBFx8Xjw5+frC2tlZVIiLTMmvQFRcX48KF8wgMCuLWEiIyG7MG3YGDUagoL0efPn1UhYjI9MwadJF796KllRX69e+nKkREpmfWoEtISETHTgFwdXFRFSIi0zNb0JWUlOJ8chKCgwJVhYjIPMwWdAUFBcjLy0OXrl1VhYjIPMwWdFmZmSguKkJH3iOCiMzMbEGXnpaqPXcK5KlfRHQ9eUdAOeszFbMFndws3MreHoE8x5WIfiM+IRGPPPwIliz5CNnZF7W8MCazBZ3UKywMDg4OakREdJU8SPnZF5/jyOHDGBUejn/+81OUl5erVxvOrEF3a48e6isiout16OCLZcuXY/bcufhy2TJMnDARR4/GGKW7M2vQBXfurL4iIrqRjY01Jk+ehO+2boGbmxsmT5yIDz9ciJKSEvWO+mkh0rJWcWnobevXb0DqhQtqdKMDBw/icFQU7hs7Fv5+fqpKRFSzyqoqREZG4sSxYxg6bBg+/b9Pf/c+0LU5b94oQffee+/jXGysGt0oKTERqakp6HN7X9ja2qoqEZFhqWmpiDt3TrtQ77ebv4O9vb165RqzBd3NLFq0GMePH8fnn3/GO/MT0U2VlZVj/bp1+OD997WrHS1ctEhbw/s9tQk6s63ROTk6MuSIyKDc3Fx8veJr3DfmXuzYsQMfLFyIVatX1RhytWXWgxFERDWJOhiFUSNGYtXKlVjw3t+w/MvlGD58mFEu0sugI6JGd/z4CTz91FOYOv0prBFT1rDQUKNenNdsa3TxcXFY+slSVSEiuib74kXE/RKH/r+5VmVWdrb66vfZ2tiidWs3NaoZg46IdOncL7/g7pGj4O3jA09PD5wQXV87Dw/4+vpqZ03EnjmDJ6dNw7wX56lP1IxTVyLSpb27d+P5OX/C9xE7sPQf/9DW6mY9Pxur13yDdRvW4+Hx49F/wAD1bsPMEnS9e/dG+IgRakREdHNW1jZ49rkZ2t65pKRk7QonoWGh2mstW7ZEe29vhKnxzZhl6kpEVBcyb0pLS/+3QXjpx0uxa+curF2/FlZWVlrt0qXLcHZ20tc+OiKi2pLh9euzIPbt24cHH37ofyEnyZCrLQYdEemavN/MhfPnMXToEFWpOwYdEelaUnKSto2kXbt2qlJ3DDoi0rWYo0fh19G/QRcEYdARka5t37YdPRp40V4GHRHpVnpGBvZHRiLstjBVqR8GHRHpTklpKWJjY/HR4iVXx8Ul2naS+uI+OiLSnYqKihtuf+js7Aw7Ozs1ukY3F94kIjIVbhgmIhIYdERk8Rh0RGTxGHREZPFqfTCCiKhpAv4fkpXpe7DgpYwAAAAASUVORK5CYII="> <em><strong>A1A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for an increasing graph, entirely in first quadrant, becoming concave down for larger values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, <em><strong>A1</strong></em> for tending towards the origin and <em><strong>A1</strong> </em>for asymptote labelled at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi></math>.</p>
<p> </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>taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi></math> of both sides <strong>OR</strong> substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>T</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>A</mi><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>A</mi></math> <em><strong>(A1)</strong></em></p>
<p><br>(i) so gradient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p><br>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>A</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The implied <em><strong>(M1)</strong></em> and <em><strong>(A1)</strong></em> can only be awarded if <strong>both</strong> correct answers are seen. Award zero if only one value is correct <strong>and</strong> no working is seen.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>an attempt to convert data to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> <em><strong>(M1)</strong></em></p>
<p>e.g. at least one correct row in the following table</p>
<p><img src="data:image/png;base64,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"></p>
<p>line is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi><mo>=</mo><mo>-</mo><mn>13400</mn><mo>×</mo><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mn>15</mn><mo>.</mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>13383</mn><mo>.</mo><mn>1</mn><mo>…</mo><mo>×</mo><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mn>15</mn><mo>.</mo><mn>0107</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">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>13400</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>13383</mn><mo>.</mo><mn>1</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to rearrange or solve graphically <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>A</mi><mo>=</mo><mn>15</mn><mo>.</mo><mn>0107</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn><mo> </mo><mn>300</mn><mo> </mo><mn>000</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>3</mn><mo> </mo><mn>304</mn><mo> </mo><mn>258</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> <strong>Note</strong>: Accept an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3269017</mn></math>… from use of <math xmlns="http://www.w3.org/1998/Math/MathML" class="wrs_chemistry"><mn>3</mn><mi>sf</mi></math> value.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This question caused significant difficulties for many candidates and many did not even attempt the question. Very few candidates were able to differentiate the expression in part (a) resulting in difficulties for part (b). Responses to parts (c) to (e) illustrated a lack of understanding of linearizing a set of data. Those candidates that were able to do part (d) frequently lost a mark as their answer was given in <em>x</em> and <em>y</em>.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Willow finds that she receives approximately 70 emails per working day.</p>
<p>She decides to model the number of emails received per working day using the random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> follows a Poisson distribution with mean 70.</p>
</div>
<div class="specification">
<p>In order to test her model, Willow records the number of emails she receives per working day over a period of 6 months. The results are shown in the following table.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">From the table, calculate</p>
</div>
<div class="specification">
<p>Archie works for a different company and knows that he receives emails according to a Poisson distribution, with a mean of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ<!-- λ --></mi>
</math></span> emails per day.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using this distribution model, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < 60} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mn>60</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>Using this distribution model, find the standard deviation 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">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>an estimate for the mean number of emails received per working day.</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>an estimate for the standard deviation of the number of emails received per working day.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give one piece of evidence that suggests Willow’s Poisson distribution model is not a good fit.</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>Suppose that the probability of Archie receiving more than 10 emails in total on any one day is 0.99. Find the value of <em>λ</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Now suppose that Archie received exactly 20 emails in total in a consecutive two day period. Show that the probability that he received exactly 10 of them on the first day is independent of<em> λ</em>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</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{P}}\left( {X < 60} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mn>60</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}\left( {X \leqslant 59} \right)">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>⩽</mo>
<mn>59</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p>= 0.102 <em><strong>A1</strong></em></p>
<p> </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>standard deviation = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {70} ">
<msqrt>
<mn>70</mn>
</msqrt>
</math></span> (= 8.37) <em><strong> (M1)</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of midpoints (accept consistent use of 45, 55 etc.) <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{44.5 \times 2 + 54.5 \times 15 + 64.5 \times 40 + 74.5 \times 53 + 94.5 + 104.5 \times 3 + 114.5 \times 6}}{{2 + 15 + 40 + 53 + 0 + 1 + 3 + 6}}">
<mfrac>
<mrow>
<mn>44.5</mn>
<mo>×</mo>
<mn>2</mn>
<mo>+</mo>
<mn>54.5</mn>
<mo>×</mo>
<mn>15</mn>
<mo>+</mo>
<mn>64.5</mn>
<mo>×</mo>
<mn>40</mn>
<mo>+</mo>
<mn>74.5</mn>
<mo>×</mo>
<mn>53</mn>
<mo>+</mo>
<mn>94.5</mn>
<mo>+</mo>
<mn>104.5</mn>
<mo>×</mo>
<mn>3</mn>
<mo>+</mo>
<mn>114.5</mn>
<mo>×</mo>
<mn>6</mn>
</mrow>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>15</mn>
<mo>+</mo>
<mn>40</mn>
<mo>+</mo>
<mn>53</mn>
<mo>+</mo>
<mn>0</mn>
<mo>+</mo>
<mn>1</mn>
<mo>+</mo>
<mn>3</mn>
<mo>+</mo>
<mn>6</mn>
</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{{8530}}{{120}}\left( { = 71.1} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mn>8530</mn>
</mrow>
<mrow>
<mn>120</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>71.1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> If 45, 55, etc. are used consistently instead of midpoints (implied by the answer 71.58…) award <em><strong>M1M1A0</strong></em>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>13.9 <em><strong> (M1)</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid reason given to include the examples below <em><strong>R1</strong></em></p>
<p>variance is 192 which is not close to the mean (accept not equal to) standard deviation too high (using parts (a)(ii) and (b)(ii))</p>
<p>relative frequency of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> ≤ 59 is 0.142 which is too high (using part (a)(i))</p>
<p>Poisson would give a frequency of roughly 14 for 80 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> ≤ 89</p>
<p><strong>Note:</strong> Reasons which do not use values found in previous parts must be backed up with numerical evidence.</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> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {Y > 10} \right) = 0.99">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Y</mi>
<mo>></mo>
<mn>10</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.99</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - {\text{P}}\left( {Y \leqslant 10} \right) = 0.99 \Rightarrow {\text{P}}\left( {Y \leqslant 10} \right) = 0.01">
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Y</mi>
<mo>⩽</mo>
<mn>10</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.99</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Y</mi>
<mo>⩽</mo>
<mn>10</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.01</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p>attempt to solve a correct equation <em><strong>(M1)</strong></em></p>
<p><em>λ </em>= 20.1 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>in 1 day, no of emails is <em>X</em> ~ Po(<em>λ</em>)</p>
<p>in 2 days, no of emails is <em>Y</em> ~ Po(2<em>λ</em>) <em><strong>(A1)</strong></em></p>
<p>P(10 on first day | 20 in 2 days) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}\left( {X = 10} \right) \times {\text{P}}\left( {X = 10} \right)}}{{{\text{P}}\left( {Y = 20} \right)}}">
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>10</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>10</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Y</mi>
<mo>=</mo>
<mn>20</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{{{{\left( {\frac{{{\lambda ^{10}}{e^{ - \lambda }}}}{{10{\text{!}}}}} \right)}^2}}}{{\frac{{{{\left( {2\lambda } \right)}^{20}}{e^{ - 2\lambda }}}}{{20{\text{!}}}}}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>λ</mi>
<mrow>
<mn>10</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mn>10</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>20</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>λ</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mn>20</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</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{{{\lambda ^{20}}{e^{ - 2\lambda }}}}{{{2^{20}}{\lambda ^{20}}{e^{ - 2\lambda }}}} \times \frac{{20{\text{!}}}}{{{{\left( {10{\text{!}}} \right)}^2}}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>λ</mi>
<mrow>
<mn>20</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>λ</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mn>20</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>λ</mi>
<mrow>
<mn>20</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>λ</mi>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mn>20</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>10</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</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{{20{\text{!}}}}{{{2^{20}}{{\left( {10{\text{!}}} \right)}^2}}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>20</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mn>20</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>10</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p>which is independent of <em>λ</em> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</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.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">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Long term experience shows that if it is sunny on a particular day in Vokram, then the probability that it will be sunny the following day is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn></math>. If it is not sunny, then the probability that it will be sunny the following day is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>3</mn></math>.</p>
<p>The transition matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> is used to model this information, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> can be written as a product of three matrices, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">D</mi><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math> is a diagonal matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is sunny today. Find the probability that it will be sunny in three days’ time.</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 eigenvalues and eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math>.</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>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math>.</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 find the long-term percentage of sunny days in Vokram.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mn>3</mn></msup></math> <strong>OR </strong> use of tree diagram <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mn>3</mn></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>65</mn></mtd><mtd><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>525</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>35</mn></mtd><mtd><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>475</mn></mtd></mtr></mtable></mfenced></math></p>
<p>the probability of sunny in three days’ time is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>65</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>attempt to find eigenvalues <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong><br>Note:</strong> Any indication that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mrow><mi mathvariant="bold-italic">T</mi><mo>-</mo><mi>λ</mi><mi mathvariant="bold-italic">I</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> has been used is sufficient for the <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn><mo>-</mo><mi>λ</mi></mtd><mtd><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>7</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>8</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>.</mo><mn>7</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>.</mo><mn>5</mn><mi>λ</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p>attempt to find either eigenvector <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>3</mn><mi>y</mi><mo>=</mo><mi>x</mi><mo>⇒</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math> so an eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</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>3</mn><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>⇒</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math> so an eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept multiples of the stated eigenvectors.</p>
<p> </p>
<p><em><strong>[5 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 mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mo> </mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mo> </mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo> </mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mo> </mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Examiners should be aware that different, correct, matrices <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> may be seen.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></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 mathvariant="bold-italic">D</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math> must be consistent with each other.</p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><msup><mn>5</mn><mi>n</mi></msup><mo>→</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo> </mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>only if their <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup></math> corresponds to their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><msup><mi mathvariant="bold-italic">D</mi><mi>n</mi></msup><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>6</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>4</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo>%</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 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 continuous random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> has probability density function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> given by</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \left\{ {\begin{array}{*{20}{l}} {\frac{{{x^2}}}{a} + b,}&{0 \leqslant x \leqslant 4} \\ 0&{{\text{otherwise}}} \end{array}} \right.{\text{where }}a{\text{ and }}b{\text{ are positive constants.}}">
<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>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>a</mi>
</mfrac>
<mo>+</mo>
<mi>b</mi>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mrow>
<mtext>otherwise</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
<mrow>
<mtext>where </mtext>
</mrow>
<mi>a</mi>
<mrow>
<mtext> and </mtext>
</mrow>
<mi>b</mi>
<mrow>
<mtext> are positive constants.</mtext>
</mrow>
</math></span></p>
<p>It is given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X \geqslant 2) = 0.75">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.75</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Eight independent observations of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> are now taken and the random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y">
<mi>Y</mi>
</math></span> is the number of observations such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \geqslant 2">
<mi>X</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>2</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="a = 32">
<mi>a</mi>
<mo>=</mo>
<mn>32</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{1}{{12}}">
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</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="{\text{E}}(X)">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</math></span>.</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</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>Find 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">[3]</div>
<div class="question_part_label">d.</div>
</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="{\text{E}}(Y)">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</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="{\text{P}}(Y \geqslant 3)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>⩾</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</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="\int\limits_0^4 {\left( {\frac{{{x^2}}}{a} + b} \right){\text{d}}x = 1 \Rightarrow \left[ {\frac{{{x^3}}}{{3a}} + bx} \right]_0^4 = 1 \Rightarrow \frac{{64}}{{3a}} + 4b = 1} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>4</mn>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>a</mi>
</mfrac>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mi>a</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mi>b</mi>
<mi>x</mi>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>4</mn>
</msubsup>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mn>64</mn>
</mrow>
<mrow>
<mn>3</mn>
<mi>a</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>4</mn>
<mi>b</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_2^4 {\left( {\frac{{{x^2}}}{a} + b} \right){\text{d}}x = 0.75 \Rightarrow \frac{{56}}{{3a}} + 2b = 0.75} ">
<munderover>
<mo>∫</mo>
<mn>2</mn>
<mn>4</mn>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>a</mi>
</mfrac>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>0.75</mn>
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mn>56</mn>
</mrow>
<mrow>
<mn>3</mn>
<mi>a</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>=</mo>
<mn>0.75</mn>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^2 {\left( {\frac{{{x^2}}}{a} + b} \right)} \,dx = 0.25 \Rightarrow \frac{8}{{3a}} + 2b = 0.25">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>2</mn>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mi>a</mi>
</mfrac>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<mn>0.25</mn>
<mo stretchy="false">⇒</mo>
<mfrac>
<mn>8</mn>
<mrow>
<mn>3</mn>
<mi>a</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>=</mo>
<mn>0.25</mn>
</math></span> could be seen/used in place of either of the above equations.</p>
<p> </p>
<p>evidence of an attempt to solve simultaneously (or check given <em>a</em>,<em>b </em>values are consistent) <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 32,{\text{ }}b = \frac{1}{{12}}">
<mi>a</mi>
<mo>=</mo>
<mn>32</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[5 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}}\left( X \right) = \int\limits_0^4 {x\left( {\frac{{{x^2}}}{{32}} + \frac{1}{{12}}} \right){\text{d}}x} ">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>4</mn>
</munderover>
<mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X) = \frac{8}{3}\,\,\,( = 2.67)">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>8</mn>
<mn>3</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>2.67</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></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{E}}\left( {{X^2}} \right) = \int\limits_0^4 {{x^2}\left( {\frac{{{x^2}}}{{32}} + \frac{1}{{12}}} \right){\text{d}}x} ">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>4</mn>
</munderover>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = {\text{E}}({X^2}) - {[{\text{E}}(X)]^2} = \frac{{16}}{{15}}\,\,\,( = 1.07)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mo stretchy="false">[</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">]</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>16</mn>
</mrow>
<mrow>
<mn>15</mn>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>1.07</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></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="\int\limits_0^m {\left( {\frac{{{x^2}}}{{32}} + \frac{1}{{12}}} \right){\text{d}}x = 0.5} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mi>m</mi>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>0.5</mn>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{m^3}}}{{96}} + \frac{m}{{12}} = 0.5\,\,\,( \Rightarrow {m^3} + 8m - 48 = 0)">
<mfrac>
<mrow>
<mrow>
<msup>
<mi>m</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>96</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>m</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.5</mn>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>m</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>8</mn>
<mi>m</mi>
<mo>−</mo>
<mn>48</mn>
<mo>=</mo>
<mn>0</mn>
<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="m = 2.91">
<mi>m</mi>
<mo>=</mo>
<mn>2.91</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y \sim B(8,{\text{ }}0.75)">
<mi>Y</mi>
<mo>∼</mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0.75</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="{\text{E}}(Y) = 8 \times 0.75 = 6">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>8</mn>
<mo>×</mo>
<mn>0.75</mn>
<mo>=</mo>
<mn>6</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">e.</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 \geqslant 3) = 0.996">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>⩾</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.996</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">f.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Arianne plays a game of darts.</p>
<p style="text-align: center;"><img 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"></p>
<p>The distance that her darts land from the centre, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, of the board can be modelled by a normal distribution with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mtext>cm</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mtext>cm</mtext></math>.</p>
</div>
<div class="specification">
<p>Find the probability that</p>
</div>
<div class="specification">
<p>Each of Arianne’s throws is independent of her previous throws.</p>
</div>
<div class="specification">
<p>In a competition a player has three darts to throw on each turn. A point is scored if a player throws <strong>all</strong> three darts to land within a central area around <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>. When Arianne throws a dart the probability that it lands within this area is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8143</mn></math>.</p>
</div>
<div class="specification">
<p>In the competition Arianne has ten turns, each with three darts.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>a dart lands less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mo> </mo><mtext>cm</mtext></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</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>a dart lands more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo> </mo><mtext>cm</mtext></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</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>Find the probability that Arianne throws two consecutive darts that land more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo> </mo><mtext>cm</mtext></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></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 probability that Arianne does <strong>not</strong> score a point on a turn of three darts.</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>Find Arianne’s expected score in the competition.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Arianne scores at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> points in the competition.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Arianne scores at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> points and less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> points.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that Arianne scores at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> points, find the probability that Arianne scores less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> points.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.iv.</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 “distance from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>”.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>10</mn><mo>,</mo><mo> </mo><msup><mn>3</mn><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>13</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>841</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>841344</mn><mo>…</mo></mrow></mfenced></math> <strong><em>(M1)(A1)</em></strong></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>></mo><mn>15</mn></mrow></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>0478</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0477903</mn></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.ii.</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>15</mn></mrow></mfenced><mo>×</mo><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>></mo><mn>15</mn></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>00228</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00228391</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>8143</mn></mrow></mfenced><mn>3</mn></msup></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>460</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>460050</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>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> be the random variable “number of points scored”</p>
<p>evidence of use of binomial distribution <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>10</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>539949</mn><mo>…</mo></mrow></mfenced></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>E</mtext><mfenced><mi>Y</mi></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mn>10</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>539949</mn><mo>…</mo></math> <strong><em>(M1)</em></strong></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>5</mn><mo>.</mo><mn>40</mn></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>P</mtext><mfenced><mrow><mi>Y</mi><mo>≥</mo><mn>5</mn></mrow></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mn>0</mn><mo>.</mo><mn>717</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>716650</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">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mn>5</mn><mo>≤</mo><mi>Y</mi><mo><</mo><mn>8</mn></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>628</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>627788</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for a correct probability statement or indication of correct lower and upper bounds, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math>.<br><br></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>P</mtext><mfenced><mrow><mn>5</mn><mo>≤</mo><mi>Y</mi><mo><</mo><mn>8</mn></mrow></mfenced></mrow><mrow><mtext>P</mtext><mfenced><mrow><mi>Y</mi><mo>≥</mo><mn>5</mn></mrow></mfenced></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>627788</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>716650</mn><mo>…</mo></mrow></mfrac></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>876</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>876003</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">d.iv.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Most candidates started this question well and applied the normal distribution correctly in part (a). The same goes for part (b) where the candidates were able to combine probabilities correctly. Part (c) was not very well done, and there were a surprising number of incorrect approaches on a seemingly straightforward problem. This suggests that candidates were not interpreting the problem correctly and there was a lack of careful reading to be sure of the scenario being described. In part (d) most candidates recognized the need to model the situation with the binomial distribution. However, many candidates did not choose a correct probability for the Bernoulli trial in this question and oversimplified the problem. This again seems to be a problem of “interpretation” rather than “conceptual understanding”.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates started this question well and applied the normal distribution correctly in part (a). The same goes for part (b) where the candidates were able to combine probabilities correctly. Part (c) was not very well done, and there were a surprising number of incorrect approaches on a seemingly straightforward problem. This suggests that candidates were not interpreting the problem correctly and there was a lack of careful reading to be sure of the scenario being described. In part (d) most candidates recognized the need to model the situation with the binomial distribution. However, many candidates did not choose a correct probability for the Bernoulli trial in this question and oversimplified the problem. This again seems to be a problem of “interpretation” rather than “conceptual understanding”.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates started this question well and applied the normal distribution correctly in part (a). The same goes for part (b) where the candidates were able to combine probabilities correctly. Part (c) was not very well done, and there were a surprising number of incorrect approaches on a seemingly straightforward problem. This suggests that candidates were not interpreting the problem correctly and there was a lack of careful reading to be sure of the scenario being described. In part (d) most candidates recognized the need to model the situation with the binomial distribution. However, many candidates did not choose a correct probability for the Bernoulli trial in this question and oversimplified the problem. This again seems to be a problem of “interpretation” rather than “conceptual understanding”.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates started this question well and applied the normal distribution correctly in part (a). The same goes for part (b) where the candidates were able to combine probabilities correctly. Part (c) was not very well done, and there were a surprising number of incorrect approaches on a seemingly straightforward problem. This suggests that candidates were not interpreting the problem correctly and there was a lack of careful reading to be sure of the scenario being described. In part (d) most candidates recognized the need to model the situation with the binomial distribution. However, many candidates did not choose a correct probability for the Bernoulli trial in this question and oversimplified the problem. This again seems to be a problem of “interpretation” rather than “conceptual understanding”.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates started this question well and applied the normal distribution correctly in part (a). The same goes for part (b) where the candidates were able to combine probabilities correctly. Part (c) was not very well done, and there were a surprising number of incorrect approaches on a seemingly straightforward problem. This suggests that candidates were not interpreting the problem correctly and there was a lack of careful reading to be sure of the scenario being described. In part (d) most candidates recognized the need to model the situation with the binomial distribution. However, many candidates did not choose a correct probability for the Bernoulli trial in this question and oversimplified the problem. This again seems to be a problem of “interpretation” rather than “conceptual understanding”.</p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates started this question well and applied the normal distribution correctly in part (a). The same goes for part (b) where the candidates were able to combine probabilities correctly. Part (c) was not very well done, and there were a surprising number of incorrect approaches on a seemingly straightforward problem. This suggests that candidates were not interpreting the problem correctly and there was a lack of careful reading to be sure of the scenario being described. In part (d) most candidates recognized the need to model the situation with the binomial distribution. However, many candidates did not choose a correct probability for the Bernoulli trial in this question and oversimplified the problem. This again seems to be a problem of “interpretation” rather than “conceptual understanding”.</p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates started this question well and applied the normal distribution correctly in part (a). The same goes for part (b) where the candidates were able to combine probabilities correctly. Part (c) was not very well done, and there were a surprising number of incorrect approaches on a seemingly straightforward problem. This suggests that candidates were not interpreting the problem correctly and there was a lack of careful reading to be sure of the scenario being described. In part (d) most candidates recognized the need to model the situation with the binomial distribution. However, many candidates did not choose a correct probability for the Bernoulli trial in this question and oversimplified the problem. This again seems to be a problem of “interpretation” rather than “conceptual understanding”.</p>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates started this question well and applied the normal distribution correctly in part (a). The same goes for part (b) where the candidates were able to combine probabilities correctly. Part (c) was not very well done, and there were a surprising number of incorrect approaches on a seemingly straightforward problem. This suggests that candidates were not interpreting the problem correctly and there was a lack of careful reading to be sure of the scenario being described. In part (d) most candidates recognized the need to model the situation with the binomial distribution. However, many candidates did not choose a correct probability for the Bernoulli trial in this question and oversimplified the problem. This again seems to be a problem of “interpretation” rather than “conceptual understanding”.</p>
<div class="question_part_label">d.iv.</div>
</div>
<br><hr><br><div class="specification">
<p>A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</math></span> be the probability that Kati obtains her third voucher on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n{\text{th}}">
<mi>n</mi>
<mrow>
<mtext>th</mtext>
</mrow>
</math></span> bar opened.</p>
<p>(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)</p>
</div>
<div class="specification">
<p>It is given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n) = \frac{{{n^2} + an + b}}{{2000}} \times {0.9^{n - 3}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mi>n</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×<!-- × --></mo>
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 3,{\text{ }}n \in \mathbb{N}">
<mi>n</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Kati’s mother goes to the shop and buys <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> chocolate bars. She takes the bars home for Kati to open.</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}}(X = 3) = 0.001">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.001</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 4) = 0.0027">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.0027</mn>
</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>Find the values of the constants <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>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{0.9(n - 1)}}{{n - 3}}">
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.9</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 3">
<mi>n</mi>
<mo>></mo>
<mn>3</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> has two modes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_1}">
<mrow>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_2}">
<mrow>
<msub>
<mi>m</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
<p>(ii) State the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_1}">
<mrow>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_2}">
<mrow>
<msub>
<mi>m</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> such that the probability Kati receives at least one free gift is greater than 0.5.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</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 = 3) = {(0.1)^3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo stretchy="false">(</mo>
<mn>0.1</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.001">
<mo>=</mo>
<mn>0.001</mn>
</math></span> <strong><em>AG</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 4) = {\text{P}}(VV\bar VV) + {\text{P}}(V\bar VVV) + {\text{P}}(\bar VVVV)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>V</mi>
<mi>V</mi>
<mrow>
<mover>
<mi>V</mi>
<mo stretchy="false">¯</mo>
</mover>
</mrow>
<mi>V</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>V</mi>
<mrow>
<mover>
<mi>V</mi>
<mo stretchy="false">¯</mo>
</mover>
</mrow>
<mi>V</mi>
<mi>V</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mover>
<mi>V</mi>
<mo stretchy="false">¯</mo>
</mover>
</mrow>
<mi>V</mi>
<mi>V</mi>
<mi>V</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=" = 3 \times {(0.1)^3} \times 0.9">
<mo>=</mo>
<mn>3</mn>
<mo>×</mo>
<mrow>
<mo stretchy="false">(</mo>
<mn>0.1</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>0.9</mn>
</math></span> (or equivalent) <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.0027">
<mo>=</mo>
<mn>0.0027</mn>
</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><strong>METHOD 1</strong></p>
<p>attempting to form equations in <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> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{9 + 3a + b}}{{2000}} = \frac{1}{{1000}}{\text{ }}(3a + b = - 7)">
<mfrac>
<mrow>
<mn>9</mn>
<mo>+</mo>
<mn>3</mn>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1000</mn>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>7</mn>
<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="\frac{{16 + 4a + b}}{{2000}} \times \frac{9}{{10}} = \frac{{27}}{{10\,000}}{\text{ }}(4a + b = - 10)">
<mfrac>
<mrow>
<mn>16</mn>
<mo>+</mo>
<mn>4</mn>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>9</mn>
<mrow>
<mn>10</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>27</mn>
</mrow>
<mrow>
<mn>10</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p>attempting to solve simultaneously <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 3,{\text{ }}b = 2">
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>=</mo>
<mn>2</mn>
</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}}(X = n) = \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) \times {0.1^3} \times {0.9^{n - 3}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<msup>
<mn>0.1</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>×</mo>
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{(n - 1)(n - 2)}}{{2000}} \times {0.9^{n - 3}}">
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{n^2} - 3n + 2}}{{2000}} \times {0.9^{n - 3}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 3,b = 2">
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mo>,</mo>
<mi>b</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Condone the absence of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{0.9^{n - 3}}">
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> in the determination of the values 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>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n) = \frac{{{n^2} - 3n + 2}}{{2000}} \times {0.9^{n - 3}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n) = \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) \times {0.1^3} \times {0.9^{n - 3}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<msup>
<mn>0.1</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>×</mo>
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{(n - 1)(n - 2)}}{{2000}} \times {0.9^{n - 3}}">
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = n - 1) = \frac{{(n - 2)(n - 3)}}{{2000}} \times {0.9^{n - 4}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mn>0.9</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{(n - 1)(n - 2)}}{{(n - 2)(n - 3)}} \times 0.9">
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>×</mo>
<mn>0.9</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.9(n - 1)}}{{n - 3}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.9</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
</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="\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{\frac{{{n^2} - 3n + 2}}{{2000}} \times {{0.9}^{n - 3}}}}{{\frac{{{{(n - 1)}^2} - 3(n - 1) + 2}}{{2000}} \times {{0.9}^{n - 4}}}}">
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>0.9</mn>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>0.9</mn>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
</msup>
</mrow>
</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{{0.9({n^2} - 3n + 2)}}{{({n^2} - 5n + 6)}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.9</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mi>n</mi>
<mo>+</mo>
<mn>6</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for a correct numerator and <strong><em>A1 </em></strong>for a correct denominator.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.9(n - 1)(n - 2)}}{{(n - 2)(n - 3)}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.9</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{0.9(n - 1)}}{{n - 3}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.9</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) attempting to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0.9(n - 1)}}{{n - 3}} = 1">
<mfrac>
<mrow>
<mn>0.9</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 21">
<mi>n</mi>
<mo>=</mo>
<mn>21</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0.9(n - 1)}}{{n - 3}} < 1 \Rightarrow n > 21">
<mfrac>
<mrow>
<mn>0.9</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
<mo><</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>n</mi>
<mo>></mo>
<mn>21</mn>
</math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0.9(n - 1)}}{{n - 3}} > 1 \Rightarrow n < 21">
<mfrac>
<mrow>
<mn>0.9</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
<mo>></mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>n</mi>
<mo><</mo>
<mn>21</mn>
</math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> has two modes <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>R1R1 </em></strong>for a clearly labelled graphical representation of the two inequalities (using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}}">
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span>).</p>
<p> </p>
<p>(ii) the modes are 20 and 21 <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">d.</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="Y \sim {\text{B}}(x,{\text{ }}0.1)">
<mi>Y</mi>
<mo>∼</mo>
<mrow>
<mtext>B</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0.1</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempting to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y \geqslant 3) > 0.5">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>⩾</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0.5</mn>
</math></span> (or equivalent <em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - {\text{P}}(Y \leqslant 2) > 0.5">
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>⩽</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0.5</mn>
</math></span>) for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for attempting to solve an equality (obtaining <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 26.4">
<mi>x</mi>
<mo>=</mo>
<mn>26.4</mn>
</math></span>).</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 27">
<mi>x</mi>
<mo>=</mo>
<mn>27</mn>
</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="\sum\limits_{n = 0}^x {{\text{P}}(X = n) > 0.5} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>x</mi>
</munderover>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0.5</mn>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<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="x = 27">
<mi>x</mi>
<mo>=</mo>
<mn>27</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>102</mn><mo> </mo><mtext>g</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mtext>g</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a randomly selected packet has a weight less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mtext>g</mtext></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>The probability that a randomly selected packet has a weight greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> grams is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>444</mn></math>. Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</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>A packet is randomly selected. Given that the packet has a weight greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>105</mn><mo> </mo><mtext>g</mtext></math>, find the probability that it has a weight greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>110</mn><mo> </mo><mtext>g</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>From a random sample of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>500</mn></math> packets, determine the number of packets that would be expected to have a weight lying within <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> standard deviations of the mean.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Packets are delivered to supermarkets in batches of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn></math>. Determine the probability that at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> packets from a randomly selected batch have a weight less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mo> </mo><mtext>g</mtext></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Χ</mi><mo>~</mo><mtext>N</mtext><mfenced><mrow><mn>102</mn><mo>,</mo><mo> </mo><msup><mn>8</mn><mn>2</mn></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo><</mo><mn>100</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>401</mn></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mi>w</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>444</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>w</mi><mo>=</mo><mn>103</mn><mo> </mo><mfenced><mtext>g</mtext></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"><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>100</mn><mo> </mo><menclose notation="left"><mo> </mo><mi>Χ</mi><mo>></mo><mn>105</mn></menclose></mrow></mfenced><mo>=</mo><mfrac><mrow><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>100</mn><mo>∩</mo><mi>Χ</mi><mo>></mo><mn>105</mn></mrow></mfenced></mrow><mrow><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>105</mn></mrow></mfenced></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>100</mn></mrow></mfenced></mrow><mrow><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>></mo><mn>105</mn></mrow></mfenced></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>15865</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>35383</mn><mo>…</mo></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>448</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mn>90</mn><mo><</mo><mi>Χ</mi><mo><</mo><mn>114</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mo>-</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo><</mo><mi>Z</mi><mo><</mo><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>866</mn><mo>…</mo><mo>×</mo><mn>500</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>433</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo><</mo><mn>95</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>19078</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>recognising <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>80</mn><mo>,</mo><mo> </mo><mi>p</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>now using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>80</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>19078</mn><mo>…</mo></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>Y</mi><mo>≥</mo><mn>20</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>116</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Hank sets up a bird table in his garden to provide the local birds with some food. Hank notices that a specific bird, a large magpie, visits several times per month and he names him Bill. Hank models the number of times per month that Bill visits his garden as a Poisson distribution with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Over the course of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> consecutive months, find the probability that Bill visits the garden:</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using Hank’s model, find the probability that Bill visits the garden on exactly four occasions during one particular month.</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>on exactly <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> occasions.</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>during the first and third month only.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that over a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math>-month period, there will be exactly <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> months when Bill does not visit the garden.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>After the first year, a number of baby magpies start to visit Hank’s garden. It may be assumed that each of these baby magpies visits the garden randomly and independently, and that the number of times each baby magpie visits the garden per month is modelled by a Poisson distribution with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>1</mn></math>.</p>
<p>Determine the least number of magpies required, including Bill, in order that the probability of Hank’s garden having at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> magpie visits per month is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[4]</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>X</mi><mn>1</mn></msub><mo>~</mo><mtext>Po</mtext><mfenced><mrow><mn>3</mn><mo>.</mo><mn>1</mn></mrow></mfenced></math></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>173</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>173349</mn><mo>…</mo></mrow></mfenced></math> <em>A1</em></strong></p>
<p><br><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>X</mi><mn>2</mn></msub><mo>~</mo><mtext>Po</mtext><mfenced><mrow><mn>3</mn><mo>×</mo><mn>3</mn><mo>.</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mtext>Po</mtext><mfenced><mrow><mn>9</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math><strong> <em>(M1)</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>2</mn></msub><mo>=</mo><mn>12</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>0799</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0798950</mn><mo>…</mo></mrow></mfenced></math> <em>A1</em></strong></p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>></mo><mn>0</mn></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>×</mo><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><msup><mn>95495</mn><mn>2</mn></msup><mo>×</mo><mn>0</mn><mo>.</mo><mn>04505</mn></math><strong> <em>(A1)</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0411</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0410817</mn><mo>…</mo></mrow></mfenced></math> <em>A1</em></strong></p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>04505</mn></math> <em>(A1)</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mn>1</mn></msub><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>12</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>04505</mn></mrow></mfenced></math> <em>(M1)(A1)</em></strong></p>
<p><strong><br>Note: </strong>Award <em><strong>M1</strong></em> for recognizing binomial probability, and <em><strong>A1</strong></em> for correct parameters.</p>
<p><strong><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0133</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>013283</mn><mo>…</mo></mrow></mfenced></math> <em>A1</em><br></strong></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD ONE</strong></p>
<p><img src="data:image/png;base64,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"><strong> <em>(M1)(A1)(A1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for evidence of a cumulative Poisson with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>.</mo><mn>1</mn><mi>n</mi></math>, <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>136705</mn></math> and <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>253384</mn></math>.</p>
<p><br>so require <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> magpies (including Bill)<strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>METHOD TWO</strong></p>
<p>evidence of a cumulative Poisson with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>.</mo><mn>1</mn><mi>n</mi></math><strong> <em>(M1)</em></strong></p>
<p>sketch of curve and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn></math><strong> <em>(A1)</em></strong></p>
<p>(intersect at) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>.</mo><mn>5810</mn><mo>…</mo></math><strong> <em>(A1)</em></strong></p>
<p>rounding up gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>11</mn></math></p>
<p>so require <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> magpies (including Bill)<strong> <em>A1</em></strong></p>
<p> </p>
<p><em><strong>[4 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.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">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>It is given that one in five cups of coffee contain more than 120 mg of caffeine.<br>It is also known that three in five cups contain more than 110 mg of caffeine.</p>
<p>Assume that the caffeine content of coffee is modelled by a normal distribution.<br>Find the mean and standard deviation of the caffeine content of coffee.</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>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> be the random variable “amount of caffeine content in coffee”</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X > 120) = 0.2,{\text{ P}}(X > 110) = 0.6">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>></mo>
<mn>120</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.2</mn>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>></mo>
<mn>110</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.6</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( \Rightarrow {\text{P}}(X < 120) = 0.8,{\text{ P}}(X < 110) = 0.4)">
<mo stretchy="false">(</mo>
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo><</mo>
<mn>120</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.8</mn>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo><</mo>
<mn>110</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.4</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for at least one correct probability statement.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{120 - \mu }}{\sigma } = 0.84162 \ldots ,{\text{ }}\frac{{110 - \mu }}{\sigma } = - 0.253347 \ldots ">
<mfrac>
<mrow>
<mn>120</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mn>0.84162</mn>
<mo>…</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mn>110</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>0.253347</mn>
<mo>…</mo>
</math></span> <strong><em>(M1)(A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for attempt to find at least one appropriate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span>-value.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="120 - \mu = 0.84162\sigma ,{\text{ }}110 - \mu = - 0.253347\sigma ">
<mn>120</mn>
<mo>−</mo>
<mi>μ</mi>
<mo>=</mo>
<mn>0.84162</mn>
<mi>σ</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>110</mn>
<mo>−</mo>
<mi>μ</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.253347</mn>
<mi>σ</mi>
</math></span></p>
<p>attempt to solve simultaneous equations <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 112,{\text{ }}\sigma = 9.13">
<mi>μ</mi>
<mo>=</mo>
<mn>112</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>σ</mi>
<mo>=</mo>
<mn>9.13</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>John likes to go sailing every day in July. To help him make a decision on whether it is safe to go sailing he classifies each day in July as windy or calm. Given that a day in July is calm, the probability that the next day is calm is 0.9. Given that a day in July is windy, the probability that the next day is calm is 0.3. The weather forecast for the 1st July predicts that the probability that it will be calm is 0.8.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Draw a tree diagram to represent this information for the first three days of July.</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 probability that the 3rd July is calm.</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 the 1st July was calm given that the 3rd July is windy.</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 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_2017-08-09_om_18.23.32.png" alt="M17/5/MATHL/HP2/ENG/TZ2/05.a/M"> <strong><em>M1A2</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for 3 stage tree-diagram, <strong><em>A2 </em></strong>for 0.8, 0.9, 0.3 probabilities correctly placed.</p>
<p> </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="0.2 \times 0.7 \times 0.3 + 0.2 \times 0.3 \times 0.9 + 0.8 \times 0.1 \times 0.3 + 0.8 \times 0.9 \times 0.9 = 0.768">
<mn>0.2</mn>
<mo>×</mo>
<mn>0.7</mn>
<mo>×</mo>
<mn>0.3</mn>
<mo>+</mo>
<mn>0.2</mn>
<mo>×</mo>
<mn>0.3</mn>
<mo>×</mo>
<mn>0.9</mn>
<mo>+</mo>
<mn>0.8</mn>
<mo>×</mo>
<mn>0.1</mn>
<mo>×</mo>
<mn>0.3</mn>
<mo>+</mo>
<mn>0.8</mn>
<mo>×</mo>
<mn>0.9</mn>
<mo>×</mo>
<mn>0.9</mn>
<mo>=</mo>
<mn>0.768</mn>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></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}}({\text{1st July is calm | 3rd July is windy)}} = \frac{{{\text{P}}({\text{1st July is calm and 3rd July is windy)}}}}{{{\text{P}}({\text{3rd July is windy)}}}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>1st July is calm | 3rd July is windy)</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>1st July is calm and 3rd July is windy)</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>3rd July is windy)</mtext>
</mrow>
</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{{0.8 \times 0.1 \times 0.7 + 0.8 \times 0.9 \times 0.1}}{{1 - 0.768}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.8</mn>
<mo>×</mo>
<mn>0.1</mn>
<mo>×</mo>
<mn>0.7</mn>
<mo>+</mo>
<mn>0.8</mn>
<mo>×</mo>
<mn>0.9</mn>
<mo>×</mo>
<mn>0.1</mn>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>0.768</mn>
</mrow>
</mfrac>
</math></span></p>
<p><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0.8 \times 0.1 \times 0.7 + 0.8 \times 0.9 \times 0.1}}{{0.2 \times 0.7 \times 0.7 + 0.2 \times 0.3 \times 0.1 + 0.8 \times 0.1 \times 0.7 + 0.8 \times 0.9 \times 0.1}}">
<mfrac>
<mrow>
<mn>0.8</mn>
<mo>×</mo>
<mn>0.1</mn>
<mo>×</mo>
<mn>0.7</mn>
<mo>+</mo>
<mn>0.8</mn>
<mo>×</mo>
<mn>0.9</mn>
<mo>×</mo>
<mn>0.1</mn>
</mrow>
<mrow>
<mn>0.2</mn>
<mo>×</mo>
<mn>0.7</mn>
<mo>×</mo>
<mn>0.7</mn>
<mo>+</mo>
<mn>0.2</mn>
<mo>×</mo>
<mn>0.3</mn>
<mo>×</mo>
<mn>0.1</mn>
<mo>+</mo>
<mn>0.8</mn>
<mo>×</mo>
<mn>0.1</mn>
<mo>×</mo>
<mn>0.7</mn>
<mo>+</mo>
<mn>0.8</mn>
<mo>×</mo>
<mn>0.9</mn>
<mo>×</mo>
<mn>0.1</mn>
</mrow>
</mfrac>
</math></span></p>
<p><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0.128}}{{0.232}}">
<mfrac>
<mrow>
<mn>0.128</mn>
</mrow>
<mrow>
<mn>0.232</mn>
</mrow>
</mfrac>
</math></span> <strong><em>(</em></strong><strong><em>A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for correct numerator, <strong><em>A1 </em></strong>for correct denominator.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.552">
<mo>=</mo>
<mn>0.552</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></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 curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is shown in the graph, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x \leqslant 10">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>10</mn>
</math></span>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> passes through the following points.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>It is required to find the area bounded by the curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span><em>-</em>axis, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span><em>-</em>axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 10">
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>One possible model for the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is a cubic function.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the trapezoidal rule to find an estimate for the area.</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>Use all the coordinates in the table to find the equation of the least squares cubic regression curve.</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>Write down the coefficient of determination.</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 an expression for the area enclosed by the cubic regression curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 10">
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
</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>Find the value of this area.</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>Area = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{2}\left( {2 + 2\left( {4.5 + 4.2 + 3.3 + 4.5} \right) + 8} \right)">
<mfrac>
<mn>2</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>4.5</mn>
<mo>+</mo>
<mn>4.2</mn>
<mo>+</mo>
<mn>3.3</mn>
<mo>+</mo>
<mn>4.5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>8</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>Area = 43 <em><strong>A1</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0.0389{x^3} - 0.534{x^2} + 2.06x + 2.06">
<mi>y</mi>
<mo>=</mo>
<mn>0.0389</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>0.534</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2.06</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2.06</mn>
</math></span> <em><strong>M1A2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></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="{R^2} = 0.991">
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0.991</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Area = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{10} {y\,dx} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>10</mn>
</mrow>
</munderover>
<mrow>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>42.5 <em><strong>A2</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.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">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 café serves sandwiches and cakes. Each customer will choose one of the following three options; buy only a sandwich, buy only a cake or buy both a sandwich and a cake.</p>
<p>The probability that a customer buys a sandwich is 0.72 and the probability that a customer buys a cake is 0.45.</p>
</div>
<div class="specification">
<p>Find the probability that a customer chosen at random will buy</p>
</div>
<div class="specification">
<p>On a typical day 200 customers come to the café.</p>
</div>
<div class="specification">
<p>It is known that 46 % of the customers who come to the café are male, and that 80 % of these buy a sandwich.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>both a sandwich and a cake.</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>only a sandwich.</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>Find the expected number of cakes sold on a typical day.</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>Find the probability that more than 100 cakes will be sold on a typical day.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A customer is selected at random. Find the probability that the customer is male and buys a sandwich.</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>A female customer is selected at random. Find the probability that she buys a sandwich.</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>use of formula or Venn diagram <em><strong> (M1)</strong></em></p>
<p>0.72 + 0.45 − 1 <em><strong>(A1)</strong></em></p>
<p>= 0.17 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.72 − 0.17 = 0.55 <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>200 × 0.45 = 90 <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>let <em>X</em> be the number of customers who order cake</p>
<p><em>X</em> ~ B(200,0.45) <em><strong>(M1)</strong></em></p>
<p>P(<em>X</em> > 100) = P(<em>X</em> ≥ 101)(= 1 − P(<em>X</em> ≤ 100)) <em><strong>(M1)</strong></em></p>
<p>= 0.0681 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.46 × 0.8 = 0.368 <em><strong>A1</strong></em></p>
<p><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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.368 + 0.54 \times {\text{P}}\left( {S\left| F \right.} \right) = 0.72">
<mn>0.368</mn>
<mo>+</mo>
<mn>0.54</mn>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>S</mi>
<mrow>
<mo>|</mo>
<mi>F</mi>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.72</mn>
</math></span> <em><strong>M1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for an appropriate tree diagram. Award <em><strong>M1</strong></em> for LHS, <em><strong>M1</strong></em> for RHS.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {S\left| F \right.} \right) = 0.652">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>S</mi>
<mrow>
<mo>|</mo>
<mi>F</mi>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.652</mn>
</math></span> <em><strong>A1</strong></em> </p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {S\left| F \right.} \right) = \frac{{{\text{P}}\left( {S \cap F} \right)}}{{{\text{P}}\left( F \right)}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>S</mi>
<mrow>
<mo>|</mo>
<mi>F</mi>
<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>S</mi>
<mo>∩</mo>
<mi>F</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>F</mi>
<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{{0.72 - 0.368}}{{0.54}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.72</mn>
<mo>−</mo>
<mn>0.368</mn>
</mrow>
<mrow>
<mn>0.54</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for numerator, <em><strong>A1</strong> </em>for denominator.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {S\left| F \right.} \right) = 0.652">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>S</mi>
<mrow>
<mo>|</mo>
<mi>F</mi>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.652</mn>
</math></span> <em><strong>A1</strong></em> </p>
<p> </p>
<p><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;">
[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.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">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>Dana has collected some data regarding the heights <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> (metres) of waves against a pier at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> randomly chosen times in a single day. This data is shown in the table below.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>She wishes to perform a <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>χ</mi><mn>2</mn></msup></math>-test at the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> significance level to see if the height of waves could be modelled by a normal distribution. Her null hypothesis is</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo>:</mo></math> The data can be modelled by a normal distribution.</p>
<p>From the table she calculates the mean of the heights in her sample to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>828</mn><mo> </mo><mtext>m</mtext></math> and the standard deviation of the heights <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mi>n</mi></msub></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>257</mn><mo> </mo><mtext>m</mtext></math>.</p>
</div>
<div class="specification">
<p>She calculates the expected values for each interval under this null hypothesis, and some of these values are shown in the table below.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the given value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mi>n</mi></msub></math> to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></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>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>, giving your answers correct to one decimal place.</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 the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>χ</mi><mn>2</mn></msup></math> test statistic <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msubsup><mi>χ</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow><mn>2</mn></msubsup></mfenced></math> for this 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>Determine the degrees of freedom for Dana’s test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is given that the critical value for this test is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>49</mn></math>.</p>
<p>State the conclusion of the test in context. Use your answer to part (c) to justify your conclusion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</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><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><msqrt><mfrac><mn>50</mn><mn>49</mn></mfrac></msqrt><mo>×</mo><mn>0</mn><mo>.</mo><mn>257</mn></math> <strong>(M1)</strong></p>
<p> </p>
<p><strong>Note: M1</strong> is for the use of the correct formula</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>260</mn></math> <strong>A</strong><strong>1</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>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¯</mo></mover><mo>=</mo><mn>0</mn><mo>.</mo><mn>828</mn></math> and <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>0</mn><mo>.</mo><mn>260</mn></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>7</mn><mo>.</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mn>7</mn><mo>.</mo><mn>6</mn></math> <strong>A</strong><strong>1A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>χ</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow><mn>2</mn></msubsup><mo>=</mo><mn>3</mn><mo>.</mo><mn>35</mn></math> <strong>(M1)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>Combining columns with expected values less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> leaves <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> cells <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>-</mo><mn>1</mn><mo>-</mo><mn>2</mn><mo>=</mo><mn>4</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>35</mn><mo><</mo><mn>9</mn><mo>.</mo><mn>49</mn></math> <strong>R1</strong></p>
<p>hence insufficient evidence to reject <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math> that the heights of the waves are normally distributed. <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> The <strong>A1</strong> can be awarded independently of the <strong>R1</strong>.</p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean 196 minutes and a standard deviation 24 minutes.</p>
</div>
<div class="specification">
<p>It is found that 5% of the male runners complete the marathon in less than <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> minutes.</p>
</div>
<div class="specification">
<p>The times taken for female runners to complete the marathon can be modelled by a normal distribution with a mean 210 minutes. It is found that 58% of female runners complete the marathon between 185 and 235 minutes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a runner selected at random will complete the marathon in less than 3 hours.</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>Calculate <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>.</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 standard deviation of the times taken by female runners.</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 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="T \sim N(196,{\text{ }}{24^2})">
<mi>T</mi>
<mo>∼</mo>
<mi>N</mi>
<mo stretchy="false">(</mo>
<mn>196</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mn>24</mn>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(T < 180) = 0.252">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo><</mo>
<mn>180</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.252</mn>
</math></span> <strong><em>(M1)A1</em></strong></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{P}}(T < {T_1}) = 0.05">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo><</mo>
<mrow>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.05</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_1} = 157">
<mrow>
<msub>
<mi>T</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>157</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></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="F \sim N(210,{\text{ }}{\sigma ^2})">
<mi>F</mi>
<mo>∼</mo>
<mi>N</mi>
<mo stretchy="false">(</mo>
<mn>210</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>σ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(F < 235) = 0.79">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mo><</mo>
<mn>235</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.79</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{235 - 210}}{\sigma } = 0.806421">
<mfrac>
<mrow>
<mn>235</mn>
<mo>−</mo>
<mn>210</mn>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mn>0.806421</mn>
</math></span> or equivalent <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma = 31.0">
<mi>σ</mi>
<mo>=</mo>
<mn>31.0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></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 Principal would like to compare the students in his school with a national standard. He decides to give a test to eight students made up of four boys and four girls. One of the teachers offers to find the volunteers from his class.</p>
</div>
<div class="specification">
<p>The marks out of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math>, for the students who took the test, are:</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mn>29</mn><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mn>38</mn><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mn>37</mn><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mn>12</mn><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mn>18</mn><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mn>27</mn><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mn>31</mn><mo>.</mo></math></p>
</div>
<div class="specification">
<p>For the eight students find</p>
</div>
<div class="specification">
<p>The national standard mark is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo>.</mo><mn>2</mn></math> out of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math>.</p>
</div>
<div class="specification">
<p>Two additional students take the test at a later date and the mean mark for all ten students is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn><mo>.</mo><mn>1</mn></math> and the standard deviation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>4</mn></math>.</p>
<p>For further analysis, a standardized score out of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> for the ten students is obtained by multiplying the scores by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> and adding <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math>.</p>
</div>
<div class="specification">
<p>For the ten students, find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Name the type of sampling that best describes the method used by the Principal.</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>the mean mark.</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>the standard deviation of the marks.</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>Perform an appropriate test at the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> significance level to see if the mean marks achieved by the students in the school are higher than the national standard. It can be assumed that the marks come from a normal population.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State one reason why the test might not be valid.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>their mean standardized score.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the standard deviation of their standardized score.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>quota <em><strong>A1</strong> </em></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"><mn>27</mn><mo>.</mo><mn>125</mn><mo>≈</mo><mn>27</mn><mo>.</mo><mn>1</mn></math> <strong> </strong><strong><em>(M1)A1</em></strong></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>29815</mn><mo>…</mo><mo>≈</mo><mn>8</mn><mo>.</mo><mn>30</mn></math> <strong> </strong><strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> be the national mean)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo>:</mo><mo> </mo><mi>μ</mi><mo>=</mo><mn>25</mn><mo>.</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub><mo>:</mo><mo> </mo><mi>μ</mi><mo>></mo><mn>25</mn><mo>.</mo><mn>2</mn></math> <strong> </strong><strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Accept hypotheses in words if they are clearly expressed and ‘<strong>population</strong> mean’ or ‘school mean’ is referred to. Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo>:</mo><mo> </mo><mi>μ</mi><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub></math> unless <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mn>0</mn></msub></math> is explicitly defined as “national standard mark” or given as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo>.</mo><mn>2</mn></math>.</p>
<p><br>recognizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>-test <strong> </strong><strong><em>(M1)</em></strong></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>0</mn><mo>.</mo><mn>279391</mn><mo>…</mo></math> <strong> </strong><strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>279391</mn><mo>…</mo><mo>></mo><mn>0</mn><mo>.</mo><mn>05</mn></math> <strong> </strong><strong><em>R1</em></strong></p>
<p><br><strong>Note:</strong> The <em><strong>R1</strong></em> mark is for the comparison of 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>.</p>
<p><br>insufficient evidence to reject the null hypothesis (that the mean for the school is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo>.</mo><mn>2</mn></math>) <strong> </strong><strong><em>A1</em></strong></p>
<p><strong><br>Note:</strong> Award the final <em><strong>A1</strong> </em>only if the <strong>null</strong> hypothesis is also correct (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mn>0</mn></msub><mo>=</mo><mn>25</mn><mo>.</mo><mn>2</mn></math> or (population) mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>25</mn><mo>.</mo><mn>2</mn></math>) and the conclusion is consistent with both the direction of the inequality and the alternative hypothesis.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>the sampling process is not random <em><strong>R1</strong></em></p>
<p><em>For example:</em></p>
<p>the school asked for volunteers</p>
<p>the students were selected from a single class</p>
<p><br><strong>OR</strong></p>
<p>the quota might not be representative of the student population <em><strong>R1</strong></em></p>
<p><em>For example:</em></p>
<p>the school may have only <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> boys and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>400</mn></math> girls.<br><br><br><strong>Note:</strong> Do not accept ‘the sample is too small’.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>28</mn><mo>.</mo><mn>1</mn><mo>×</mo><mn>2</mn><mo>+</mo><mn>20</mn><mo>=</mo><mo>)</mo><mo> </mo><mo> </mo><mn>76</mn><mo>.</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>4</mn><mo>×</mo><mn>2</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>16</mn><mo>.</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The most common answer to this question was ‘convenience sampling’. Though it is a convenience sample because four boys and four girls were required the most appropriate response was ‘quota sampling’.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Some candidates still try to calculate a mean and standard deviation by hand. This is not expected.</p>
<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;">
<p>This was surprisingly poorly answered given that statistical testing forms a large part of the course. Candidates need to give the null and alternative hypotheses, find a <em>p</em>-value, compare this to the significance level and write their conclusion, in context of the question; examination questions may ask for each element individually or the question may say “Perform the test” wherein it is expected that each individual element will be clearly stated (as the test is incomplete if any are omitted). Many candidates had the null hypothesis as an inequality. The easiest way to write the null hypothesis is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mtext>: μ=25.2</mtext></math>, but it could also be stated in words so long as it is clear that the population mean is being referred to rather than the sample mean. For example, H<sub>0</sub>: The mean score of the whole school is equal to 25.2.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The answer that the sample was self-selecting or unrepresentative was the expected response. The sample being small was also accepted if the additional reason of therefore ‘not able to assume a normal population’ was also given. In general, a small sample can be valid (though will probably not be reliable).</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Some candidates missed the point of this question, that it was concerned with transformations of the mean and standard deviation, and instead tried to work out the actual values for the extra two candidates.</p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of marathons that Audrey runs in any given year can be modelled by a Poisson distribution with mean 1.3 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the probability that Audrey will run at least two marathons in a particular year.</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 that she will run at least two marathons in exactly four out of the following five years.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \sim {\text{Po}}\left( {1.3} \right)"> <mi>X</mi> <mo>∼</mo> <mrow> <mtext>Po</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1.3</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X \geqslant 2} \right) = 0.373"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>⩾</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.373</mn> </math></span> <em><strong>(M1)</strong></em><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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V \sim {\text{B}}\left( {5{\text{, }}0.373} \right)"> <mi>V</mi> <mo>∼</mo> <mrow> <mtext>B</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mrow> <mtext>, </mtext> </mrow> <mn>0.373</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for recognition of binomial or equivalent, <em><strong>A1</strong></em> for correct parameters.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {V = 4} \right) = 0.0608"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo>=</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.0608</mn> </math></span> <em><strong>(M1)</strong></em><em><strong>A1</strong></em></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>Packets of biscuits are produced by a machine. The weights <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>, in grams, of packets of biscuits can be modelled by a normal distribution where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \sim {\text{N}}(\mu ,{\text{ }}{\sigma ^2})">
<mi>X</mi>
<mo>∼<!-- ∼ --></mo>
<mrow>
<mtext>N</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span>. A packet of biscuits is considered to be underweight if it weighs less than 250 grams.</p>
</div>
<div class="specification">
<p>The manufacturer makes the decision that the probability that a packet is underweight should be 0.002. To do this <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> is increased and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ<!-- σ --></mi>
</math></span> remains unchanged.</p>
</div>
<div class="specification">
<p>The manufacturer is happy with the decision that the probability that a packet is underweight should be 0.002, but is unhappy with the way in which this was achieved. The machine is now adjusted to reduce <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ<!-- σ --></mi>
</math></span> and return <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> to 253.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 253">
<mi>μ</mi>
<mo>=</mo>
<mn>253</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma = 1.5">
<mi>σ</mi>
<mo>=</mo>
<mn>1.5</mn>
</math></span> find the probability that a randomly chosen packet of biscuits is underweight.</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>Calculate the new value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span> giving your answer correct to two decimal places.</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>Calculate the new value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ</mi>
</math></span>.</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 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 < 250) = 0.0228">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo><</mo>
<mn>250</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.0228</mn>
</math></span> <strong><em>(M1)A1</em></strong></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="\frac{{250 - \mu }}{{1.5}} = - 2.878 \ldots ">
<mfrac>
<mrow>
<mn>250</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mrow>
<mn>1.5</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>2.878</mn>
<mo>…</mo>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \mu = 254.32">
<mo stretchy="false">⇒</mo>
<mi>μ</mi>
<mo>=</mo>
<mn>254.32</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Only award <strong><em>A1 </em></strong>here if the correct 2dp answer is seen. Award <strong><em>M0 </em></strong>for use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{1.5^2}">
<mrow>
<msup>
<mn>1.5</mn>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></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="\frac{{250 - 253}}{\sigma } = - 2.878 \ldots ">
<mfrac>
<mrow>
<mn>250</mn>
<mo>−</mo>
<mn>253</mn>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>2.878</mn>
<mo>…</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \sigma = 1.04">
<mo stretchy="false">⇒</mo>
<mi>σ</mi>
<mo>=</mo>
<mn>1.04</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></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>In a reforested area of pine trees, heights of trees planted in a specific year seem to follow a normal distribution. A sample of 100 such trees is selected to test the validity of this hypothesis. The results of measuring tree heights, to the nearest centimetre, are recorded in the first two columns of the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Describe what is meant by</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>a goodness of fit test (a complete explanation required);</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>the level of significance of a hypothesis test.</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>Find the mean and standard deviation of the sample data in the table above. Show how you arrived at your answers.</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>Most of the expected frequencies have been calculated in the third column. (Frequencies have been rounded to the nearest integer, and frequencies in the first and last classes have been extended to include the rest of the data beyond 15 and 225. Find the values of <span class="mjpage"><math alttext="a" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>a</mi> </math></span>, <span class="mjpage"><math alttext="b" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>b</mi> </math></span> and <span class="mjpage"><math alttext="c" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>c</mi> </math></span> and show how you arrived at your answers.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In order to test for the goodness of fit, the test statistic was calculated to be 1.0847. Show how this was done.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">State your hypotheses, critical number, decision rule and conclusion (using a 5% level of significance).</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>A goodness of fit test is a statistical test of the hypothesis that a set of observed counts of <span class="mjpage"><math alttext="k" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>k</mi> </math></span> cells of a certain large population is consistent with a set of theoretical counts. <em><strong>(R1)</strong></em></p>
<p>The test statistic has 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> distribution with <span class="mjpage"><math alttext="k - n" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>k</mi> <mo>−</mo> <mi>n</mi> </math></span> degrees of freedom. One degree of freedom is lost for every parameter that has to be estimated from the sample. <em><strong>(R1)</strong></em></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>The level of significance of a hypothesis test is the ma<em>x</em>imal probability that we reject a true null hypothesis. <em><strong>(R1)</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 class="indent1">We use the class midpoints in the calculation of the mean and standard deviation.</p>
<p><span class="mjpage"><math alttext="\bar x = \frac{{\sum {{x_i}f\left( {{x_i}} \right)} }}{{\sum {f\left( {{x_i}} \right)} }} = \frac{{30 \times 6 + 60 \times 11 + 90 \times 15 + \ldots }}{{100}} = \frac{{13350}}{{100}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>∑</mo> <mrow> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mrow> <mo>∑</mo> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>30</mn> <mo>×</mo> <mn>6</mn> <mo>+</mo> <mn>60</mn> <mo>×</mo> <mn>11</mn> <mo>+</mo> <mn>90</mn> <mo>×</mo> <mn>15</mn> <mo>+</mo> <mo>…</mo> </mrow> <mrow> <mn>100</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>13350</mn> </mrow> <mrow> <mn>100</mn> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p>= 133.5 <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math alttext="{\text{s}} = \sqrt {\frac{{\sum {x_i^2f\left( {{x_i}} \right)} }}{{\sum {f\left( {{x_i}} \right)} }} - {{\left( {\bar x} \right)}^2}} = \sqrt {\frac{{900 \times 6 + 3600 \times 11 + \ldots }}{{100}} - {{133.5}^2}} " xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext>s</mtext> </mrow> <mo>=</mo> <msqrt> <mfrac> <mrow> <mo>∑</mo> <mrow> <msubsup> <mi>x</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mrow> <mo>∑</mo> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </mfrac> <mo>−</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>=</mo> <msqrt> <mfrac> <mrow> <mn>900</mn> <mo>×</mo> <mn>6</mn> <mo>+</mo> <mn>3600</mn> <mo>×</mo> <mn>11</mn> <mo>+</mo> <mo>…</mo> </mrow> <mrow> <mn>100</mn> </mrow> </mfrac> <mo>−</mo> <mrow> <msup> <mrow> <mn>133.5</mn> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </math></span> <em><strong>(M1)</strong></em></p>
<p>= 56.345 (= 56.3 to 3 sf) <em><strong>(A1)</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 class="indent1">Every frequency is the product of the number of observations and the probability of a number in each class. Since by hypothesis we have a normal distribution, the probabilities can be read from a normal table with mean 133.5 and standard deviation 56.345 <em><strong>(M1)</strong></em></p>
<p class="indent1"><em>E</em><sub>1</sub> = 100 × <em>P</em>(45 ≤ <span class="mjpage"><math alttext="x" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> </math></span> ≤ 75) ≈ 9 so <span class="mjpage"><math alttext="a" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>a</mi> </math></span> = 9 <em><strong>(A1) </strong></em></p>
<p class="indent1"><em>E</em><sub>2</sub> = 100 × <em>P</em>(135 ≤ <span class="mjpage"><math alttext="x" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> </math></span> ≤ 165) ≈ 20 so <em><span class="mjpage">b</span></em> = 20 <em><strong>(A1) </strong></em></p>
<p class="indent1"><em>E</em><sub>3</sub> = 100 × <em>P</em>(195 ≤ <span class="mjpage"><math alttext="x" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> </math></span> ≤ 225) ≈ 9 so <span class="mjpage"><math alttext="c" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>c</mi> </math></span> = 9 <em><strong>(A1) </strong></em></p>
<p class="indent1"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="indent1">The test statistic is 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> variable. Hence <em><strong>(M1)</strong></em></p>
<p class="indent1"><span class="mjpage"><math alttext="{\chi ^2} = \sum {\frac{{{{\left( {{f_e} - {f_o}} \right)}^2}}}{{{f_e}}}} = \frac{{{{\left( {6 - 6} \right)}^2}}}{6} + \frac{{{{\left( {9 - 11} \right)}^2}}}{9} + \ldots \frac{{{{\left( {5 - 6} \right)}^2}}}{5}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mo>∑</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>f</mi> <mi>e</mi> </msub> </mrow> <mo>−</mo> <mrow> <msub> <mi>f</mi> <mi>o</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <msub> <mi>f</mi> <mi>e</mi> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>6</mn> <mo>−</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mn>6</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>9</mn> <mo>−</mo> <mn>11</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mn>9</mn> </mfrac> <mo>+</mo> <mo>…</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mo>−</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p class="indent1">= 1.0847 <em><strong>(A1)</strong></em></p>
<p class="indent1"><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="indent1"><em>H</em><sub>0</sub>: The distribution of tree heights is normally distributed</p>
<p class="indent1"><em>H</em><sub>1</sub>: The distribution is not normal <em><strong>(M1)</strong></em></p>
<p class="indent1">Since the mean and standard deviation were estimated from the sample, the number of degrees of freedom is 8 – 1 – 2 = 5 <em><strong>(A1)</strong></em></p>
<p class="indent1">The critical number is <span class="mjpage"><math alttext="\chi _{5{\text{,}}\,\,0.05}^2" xmlns="http://www.w3.org/1998/Math/MathML"> <msubsup> <mi>χ</mi> <mrow> <mn>5</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>0.05</mn> </mrow> <mn>2</mn> </msubsup> </math></span> = 110705</p>
<p class="indent1">If <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> > 11.0705 we reject <em>H</em><sub>0</sub> <em><strong>(A1)</strong></em></p>
<p class="indent1">Since <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> = 1.0847 < 11.0705, we fail to reject H<sub>0</sub> <em><strong>(R1) </strong></em></p>
<p class="indent1">Conclusion: we do not have enough evidence to claim that the distribution of tree heights is not normal <em><strong>(R1) </strong></em></p>
<p class="indent1"><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The random variable <em>X</em> has a normal distribution with mean <em>μ</em> = 50 and variance <em>σ </em><sup>2</sup> = 16 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the probability density function for<em> X</em>, and shade the region representing P(<em>μ</em> − 2σ < <em>X</em> < <em>μ</em> + σ).</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 P(<em>μ</em> − 2σ < <em>X</em> < <em>μ</em> + σ).</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 value of <em>k</em> for which P(<em>μ</em> − <em>k</em>σ < <em>X</em> < <em>μ</em> + <em>k</em>σ) = 0.5.</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 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 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"></p>
<p>normal curve centred on 50 <em><strong>A1</strong></em></p>
<p>vertical lines at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 42 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 54, with shading in between <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>P(42<em> </em>< <em>X</em> < 54) (= P(− 2<em> </em>< <em>Z</em> < 1)) <em><strong>(M1)</strong></em></p>
<p>= 0.819 <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>P(<em>μ</em> − <em>k</em>σ < <em>X</em> < <em>μ</em> + <em>k</em>σ) = 0.5 ⇒ P(<em>X</em> < <em>μ</em> + <em>k</em>σ) = 0.75 <em><strong>(M1)</strong></em></p>
<p><em>k</em> = 0.674 <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for <em>k</em> = −0.674<em>.</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="specification">
<p>Loreto is a manager at the Da Vinci health centre. If the mean rate of patients arriving at the health centre exceeds <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> per minute then Loreto will employ extra staff. It is assumed that the number of patients arriving in any given time period follows a Poisson distribution.</p>
<p>Loreto performs a hypothesis test to determine whether she should employ extra staff. She finds that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>320</mn></math> patients arrived during a randomly selected <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>-hour clinic.</p>
</div>
<div class="specification">
<p>Loreto is also concerned about the average waiting time for patients to see a nurse. The health centre aims for at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mo>%</mo></math> of patients to see a nurse in under <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> minutes.</p>
<p>Loreto assumes that the waiting times for patients are independent of each other and decides to perform a hypothesis test at a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> significance level to determine whether the health centre is meeting its target.</p>
<p>Loreto surveys <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>150</mn></math> patients and finds that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn></math> of them waited more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> minutes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down null and alternative hypotheses for Loreto’s test.</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>Using the data from Loreto’s sample, perform the hypothesis test at a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> significance level to determine if Loreto should employ extra staff.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down 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>Perform the test, clearly stating the conclusion in context.</p>
<div class="marks">[5]</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>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the random variable “number of patients arriving in a minute”, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>Po</mtext><mfenced><mi>m</mi></mfenced></math></p>
<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>m</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub><mo> </mo><mo>:</mo><mo> </mo><mi>m</mi><mo>></mo><mn>1</mn><mo>.</mo><mn>5</mn></math> <strong><em>A1</em></strong></p>
<p><strong>Note: </strong>Allow a value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>270</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>. Award at most <em><strong>A0A1</strong></em> if it is not clear that it is the population mean being referred to e.g<br> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo> </mo><mo>:</mo></math> <em>The number of patients is equal to 1.5 every minute</em><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub><mo> </mo><mo>:</mo></math> <em>The number of patients exceeds 1.5 every minute.<br></em>Referring to the “expected” number of patients or the use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> is sufficient for<em><strong> A1A1</strong>.</em></p>
<p> </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>under <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math> let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> be the number of patients in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> hours</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>~</mo><mtext>Po</mtext><mfenced><mn>270</mn></mfenced></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>Y</mi><mo>≥</mo><mn>320</mn></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>1</mn><mo>-</mo><mtext>P</mtext><mfenced><mrow><mi>Y</mi><mo>≤</mo><mn>319</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>00166</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00165874</mn></mrow></mfenced></math> <strong><em>(M1)A1</em></strong></p>
<p>since <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>00166</mn><mo><</mo><mn>0</mn><mo>.</mo><mn>05</mn></math> <strong><em>R1</em></strong></p>
<p>(reject <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math>)</p>
<p>Loreto should employ more staff <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.ii.</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></math> : The probability of a patient waiting less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> minutes is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>95</mn></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub></math> : The probability of a patient waiting less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> minutes is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>95</mn></math> <strong><em>A1</em></strong></p>
<p> </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>Under <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math> let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math> be the number of patients waiting more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> minutes</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>150</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>05</mn></mrow></mfenced></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>W</mi><mo>≥</mo><mn>11</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>132</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>132215</mn><mo>…</mo></mrow></mfenced></math> <strong><em>(M1)A1</em></strong></p>
<p>since <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>132</mn><mo>></mo><mn>0</mn><mo>.</mo><mn>1</mn></math> <strong><em>R1</em></strong></p>
<p>(fail to reject <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub></math>)</p>
<p>insufficient evidence to suggest they are not meeting their target <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Do not accept “they are meeting target” for the <em><strong>A1</strong></em>. Accept use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mo>(</mo><mn>150</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>95</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>W</mi><mo>≤</mo><mn>139</mn></mrow></mfenced></math> and any consistent use of a random variable, appropriate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value and significance level.<br><br></p>
<p><em><strong>[5 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;">
<p>In part (a) there was a general lack of consistency in how candidates wrote down their null and alternative hypotheses. It was surprising how many candidates solved a Poisson <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PDF</mi></math> rather than <math xmlns="http://www.w3.org/1998/Math/MathML" class="wrs_chemistry"><mi>CDF</mi></math> to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">p</mi></math>-value. This suggests a lack of understanding of the nature of distributions or more specifically the concepts of hypothesis testing. In part (b), which was challenging, there were issues for many candidates in interpreting the situation. This is understandable since it was difficult, but as previously mentioned interpretation is a general issue in the paper. When writing down the conclusion of the tests, there was often very loose use of the terms accept/reject and candidates seemed unclear of the significance and importance of the correct use of these terms.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (a) there was a general lack of consistency in how candidates wrote down their null and alternative hypotheses. It was surprising how many candidates solved a Poisson <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PDF</mi></math> rather than <math xmlns="http://www.w3.org/1998/Math/MathML" class="wrs_chemistry"><mi>CDF</mi></math> to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">p</mi></math>-value. This suggests a lack of understanding of the nature of distributions or more specifically the concepts of hypothesis testing. In part (b), which was challenging, there were issues for many candidates in interpreting the situation. This is understandable since it was difficult, but as previously mentioned interpretation is a general issue in the paper. When writing down the conclusion of the tests, there was often very loose use of the terms accept/reject and candidates seemed unclear of the significance and importance of the correct use of these terms.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (a) there was a general lack of consistency in how candidates wrote down their null and alternative hypotheses. It was surprising how many candidates solved a Poisson <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PDF</mi></math> rather than <math xmlns="http://www.w3.org/1998/Math/MathML" class="wrs_chemistry"><mi>CDF</mi></math> to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">p</mi></math>-value. This suggests a lack of understanding of the nature of distributions or more specifically the concepts of hypothesis testing. In part (b), which was challenging, there were issues for many candidates in interpreting the situation. This is understandable since it was difficult, but as previously mentioned interpretation is a general issue in the paper. When writing down the conclusion of the tests, there was often very loose use of the terms accept/reject and candidates seemed unclear of the significance and importance of the correct use of these terms.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (a) there was a general lack of consistency in how candidates wrote down their null and alternative hypotheses. It was surprising how many candidates solved a Poisson <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PDF</mi></math> rather than <math xmlns="http://www.w3.org/1998/Math/MathML" class="wrs_chemistry"><mi>CDF</mi></math> to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">p</mi></math>-value. This suggests a lack of understanding of the nature of distributions or more specifically the concepts of hypothesis testing. In part (b), which was challenging, there were issues for many candidates in interpreting the situation. This is understandable since it was difficult, but as previously mentioned interpretation is a general issue in the paper. When writing down the conclusion of the tests, there was often very loose use of the terms accept/reject and candidates seemed unclear of the significance and importance of the correct use of these terms.</p>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The marks achieved by eight students in a class test are given in the following list.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The teacher increases all the marks by 2. Write down the new value for</p>
</div>
<div class="question">
<p>the standard deviation.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>2.22 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</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 random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> is normally distributed with mean <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span> and standard deviation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ<!-- σ --></mi>
</math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X < 30.31) = 0.1180">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo><</mo>
<mn>30.31</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.1180</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X > 42.52) = 0.3060">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>></mo>
<mn>42.52</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.3060</mn>
</math></span>.</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="\mu ">
<mi>μ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ</mi>
</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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\left| {X - \mu } \right| < 1.2\sigma } \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>X</mi>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mo>|</mo>
</mrow>
<mo><</mo>
<mn>1.2</mn>
<mi>σ</mi>
</mrow>
<mo>)</mo>
</mrow>
</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X < 42.52) = 0.6940">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo><</mo>
<mn>42.52</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.6940</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p>either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {Z < \frac{{30.31 - \mu }}{\sigma }} \right) = 0.1180{\text{ or P}}\left( {Z < \frac{{42.52 - \mu }}{\sigma }} \right) = 0.6940">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Z</mi>
<mo><</mo>
<mfrac>
<mrow>
<mn>30.31</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.1180</mn>
<mrow>
<mtext> or P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Z</mi>
<mo><</mo>
<mfrac>
<mrow>
<mn>42.52</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.6940</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{30.31 - \mu }}{\sigma } = \underbrace {{\Phi ^{ - 1}}(0.1180)}_{ - 1.1850 \ldots }">
<mfrac>
<mrow>
<mn>30.31</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<munder>
<mrow>
<munder>
<mrow>
<mrow>
<msup>
<mi mathvariant="normal">Φ</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>0.1180</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>⏟</mo>
</munder>
</mrow>
<mrow>
<mo>−</mo>
<mn>1.1850</mn>
<mo>…</mo>
</mrow>
</munder>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{42.52 - \mu }}{\sigma } = \underbrace {{\Phi ^{ - 1}}(0.6940)}_{0.5072 \ldots }">
<mfrac>
<mrow>
<mn>42.52</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<munder>
<mrow>
<munder>
<mrow>
<mrow>
<msup>
<mi mathvariant="normal">Φ</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>0.6940</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>⏟</mo>
</munder>
</mrow>
<mrow>
<mn>0.5072</mn>
<mo>…</mo>
</mrow>
</munder>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempting to solve simultaneously <strong><em>(M1) </em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 38.9">
<mi>μ</mi>
<mo>=</mo>
<mn>38.9</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma = 7.22">
<mi>σ</mi>
<mo>=</mo>
<mn>7.22</mn>
</math></span> <strong><em>A1</em></strong></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="{\text{P}}(\mu - 1.2\sigma < X < \mu + 1.2\sigma )">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>μ</mi>
<mo>−</mo>
<mn>1.2</mn>
<mi>σ</mi>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>1.2</mn>
<mi>σ</mi>
<mo stretchy="false">)</mo>
</math></span> (or equivalent <em>eg</em>. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\text{P}}(\mu < X < \mu + 1.2\sigma )">
<mn>2</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>μ</mi>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>1.2</mn>
<mi>σ</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=" = 0.770">
<mo>=</mo>
<mn>0.770</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1)A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}( - 1.2 < Z < 1.2) = 0.770">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1.2</mn>
<mo><</mo>
<mi>Z</mi>
<mo><</mo>
<mn>1.2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.770</mn>
</math></span>.</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>A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.</p>
<p>The model is of the form</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><msub><mi>X</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>Z</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">M</mi><mfenced><mtable><mtr><mtd><msub><mi>X</mi><mi>n</mi></msub></mtd></mtr><mtr><mtd><msub><mi>Z</mi><mi>n</mi></msub></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>n</mi></msub></math> is the probability of the gene being in its normal state after dividing for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mtext>th</mtext></math> time, and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Z</mi><mi>n</mi></msub></math> is the probability of it being in another state after dividing for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mtext>th</mtext></math> time, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math>.</p>
<p>Matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math> is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo> </mo><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="specification">
<p>The gene is in its normal state when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn></math>. Calculate the probability of it being in its normal state</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</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>What does <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> represent in this context?</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>Find the eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></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>Find the eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>in the long term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.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"><mn>0</mn><mo>.</mo><mn>02</mn></math> <em><strong>A1</strong></em></p>
<p> </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>the probability of mutating from ‘not normal state’ to ‘normal state’ <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>A1</strong> </em>can only be awarded if it is clear that transformation is <strong>from</strong> the mutated state.</p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo>-</mo><mi>λ</mi><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>02</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for an attempt to find eigenvalues. Any indication that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mrow><mi mathvariant="bold-italic">M</mi><mo>-</mo><mi>λ</mi><mi mathvariant="bold-italic">I</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> has been used is sufficient for the <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>94</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mfenced><mrow><mn>0</mn><mo>.</mo><mn>98</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>-</mo><mn>0</mn><mo>.</mo><mn>0012</mn><mo>=</mo><mn>0</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>.</mo><mn>92</mn><mi>λ</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>92</mn><mo>=</mo><mn>0</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>92</mn><mo> </mo><mo> </mo><mfenced><mfrac><mn>23</mn><mn>25</mn></mfrac></mfenced></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>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>02</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</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> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>02</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</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><mn>0</mn><mo>.</mo><mn>92</mn><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><br><strong>Note:</strong> Award <em><strong>M1</strong></em> can be awarded for attempting to find either eigenvector.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>02</mn><mi>y</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>02</mn><mi>y</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>02</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Accept any multiple of the given eigenvectors.</p>
<p> </p>
<p><em><strong>[3 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"><msup><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>02</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</mn></mtd></mtr></mtable></mfenced><mn>5</mn></msup><mfenced><mtable><mtr><mtd><mn>1</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>744</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>0852</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>256</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>915</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Condone omission of the initial state vector for the <em><strong>M1</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>744</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>744311</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">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</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>25</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>25</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>75</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>75</mn></mtd></mtr></mtable></mfenced></math> seen.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>25</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>There was some difficulty in interpreting the meaning of the values in the transition matrix, but most candidates did well with the rest of the question. In part (d) there was frequently evidence of a correct method, but a failure to identify the correct probabilities. It was surprising to see a significant number of candidates diagonalizing the matrix in part (d) and this often led to errors. Clearly this was not necessary.</p>
<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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Scientists have developed a type of corn whose protein quality may help chickens gain weight faster than the present type used. To test this new type, 20 one-day-old chicks were fed a ration that contained the new corn while another control group of 20 chicks was fed the ordinary corn. The data below gives the weight gains in grams, for each group after three weeks.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The scientists wish to investigate the claim that Group B gain weight faster than Group A. Test this claim at the 5% level of significance, noting which hypothesis test you are using. You may assume that the weight gain for each group is normally distributed, with the same variance, and independent from each other.</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>The data from the two samples above are combined to form a single set of data. The following frequency table gives the observed frequencies for the combined sample. The data has been divided into five intervals.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>Test, at the 5% level, whether the combined data can be considered to be a sample from a normal population with a mean of 380.</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>This is a <em>t</em>-test of the difference of two means. Our assumptions are that the two populations are approximately normal, samples are random, and they are independent from each other. <em><strong>(R1)</strong></em></p>
<p>H<sub>0</sub>: <em>μ</em><sub>1</sub> − <em>μ</em><sub>2</sub> = 0</p>
<p>H<sub>1</sub>: <em>μ</em><sub>1</sub> − <em>μ</em><sub>2</sub> < 0 <em><strong>(A1)</strong></em> <em> </em></p>
<p><em>t </em>= −2.460, <em><strong>(A1)</strong></em></p>
<p>degrees of freedom = 38 <em><strong>(A1)</strong></em></p>
<p>Since the value of critical t = −1.686 we reject H<sub>0</sub>. <em><strong>(A1) </strong></em></p>
<p>Hence group B grows faster. <em><strong>(R1)</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This is 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.</p>
<p>To finish the table, the frequencies of the respective cells have to be calculated. Since the standard deviation is not given, it has to be estimated using the data itself. <em>s</em> = 49.59, <em>eg</em> the third expected frequency is 40 × 0.308 = 12.32, since P(350.5 < <em>W</em> < 390.5) = 0.3078...</p>
<p>The table of observed and expected frequencies is:</p>
<p><img src="data:image/png;base64,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"> <em><strong>(M1)(A2)</strong></em></p>
<p>Since the first expected frequency is 3.22, we combine the two cells, so that the first two rows become one row, that is,</p>
<p><img src="data:image/png;base64,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"> <em><strong>(M1)</strong></em></p>
<p>Number of degrees of freedom is 4 – 1 – 1 = 2 <em><strong> (C1)</strong></em> </p>
<p>H<sub>0</sub>: The distribution is normal with mean 380</p>
<p>H<sub>1</sub>: The distribution is not normal with mean 380 <em><strong>(A1)</strong></em></p>
<p>The test statistic is</p>
<p><span class="mjpage"><math alttext="\chi _{calc}^2 = \sum {\frac{{{{\left( {{f_e} - {f_0}} \right)}^2}}}{{{f_e}}}} = \frac{{{{\left( {11 - 11.04} \right)}^2}}}{{11.04}} + \frac{{{{\left( {8 - 12.32} \right)}^2}}}{{12.32}} + \frac{{{{\left( {15 - 10.48} \right)}^2}}}{{10.48}} + \frac{{{{\left( {6 - 6.17} \right)}^2}}}{{6.17}}" xmlns="http://www.w3.org/1998/Math/MathML"> <msubsup> <mi>χ</mi> <mrow> <mi>c</mi> <mi>a</mi> <mi>l</mi> <mi>c</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <mo>∑</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>f</mi> <mi>e</mi> </msub> </mrow> <mo>−</mo> <mrow> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <msub> <mi>f</mi> <mi>e</mi> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>11</mn> <mo>−</mo> <mn>11.04</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>11.04</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mo>−</mo> <mn>12.32</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>12.32</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>15</mn> <mo>−</mo> <mn>10.48</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>10.48</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>6</mn> <mo>−</mo> <mn>6.17</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>6.17</mn> </mrow> </mfrac> </math></span></p>
<p>= 3.469 <em><strong>(A1)</strong></em></p>
<p>With 2 degrees of freedom, the critical number is <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> = 5.99 <em><strong>(A2) </strong></em></p>
<p>So, we do not have enough evidence to reject the null hypothesis. Therefore, there is no evidence to say that the distribution is not normal with mean 380. <em><strong>(R1)</strong></em></p>
<p><em><strong>[10 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 random variable<em> X</em> has a binomial distribution with parameters <em>n</em> and <em>p</em>.<br>It is given that E(<em>X</em>) = 3.5.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least possible value of <em>n</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>It is further given that P(<em>X</em> ≤ 1) = 0.09478 correct to 4 significant figures.</p>
<p>Determine the value of <em>n</em> and the value of <em>p</em>.</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><em>np</em> = 3.5 <em><strong>(A1)</strong></em></p>
<p><em>p </em>≤ 1 ⇒ least <em>n</em> = 4 <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>(1 − <em>p</em>)<em><sup>n</sup></em> + <em>np</em>(1 − <em>p</em>)<sup><em>n</em>−1</sup> = 0.09478 <em><strong>M1A1</strong></em></p>
<p>attempt to solve above equation with <em>np</em> = 3.5 <em><strong>(M1)</strong></em></p>
<p><em>n</em> = 12, <em>p</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{7}{{24}}">
<mfrac>
<mn>7</mn>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span> (=0.292) <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Do not accept <em>n</em> as a decimal.</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="specification">
<p>A survey of British holidaymakers found that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo> </mo><mo>%</mo></math> of those surveyed took a holiday in the Lake District in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2019</mn></math>.</p>
</div>
<div class="specification">
<p>A random sample of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math> British holidaymakers was taken. The number of people in the sample who took a holiday in the Lake District in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2019</mn></math> can be modelled by a binomial distribution.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State two assumptions made in order for this model to be valid.</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 probability that at least three people from the sample took a holiday in the Lake District in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2019</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>From a random sample of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> holidaymakers, the probability that at least one of them took a holiday in the Lake District in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2019</mn></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>999</mn></math>.</p>
<p>Determine the least possible value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></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>people’s holidays are independent of each other <em><strong>R1</strong></em></p>
<p>the proportion is constant (at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>15</mn></math>) <em><strong>R1</strong></em></p>
<p><em><strong><br>[2 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>Χ</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>16</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>15</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>≥</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>439</mn></math> <em><strong>(M1)A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>probability of at least one <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo></math> probability of none</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>1</mn><mo>-</mo><mn>0</mn><mo>.</mo><msup><mn>85</mn><mi>n</mi></msup><mo>></mo><mn>0</mn><mo>.</mo><mn>999</mn></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><msup><mn>85</mn><mi>n</mi></msup><mo><</mo><mn>0</mn><mo>.</mo><mn>001</mn></math> <em><strong>(A1)</strong></em></p>
<p>attempt to solve inequality <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>42</mn><mo>.</mo><mn>503</mn><mo>…</mo></math></p>
<p>so least possible <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>43</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">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A discrete 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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Po}}(\mu )">
<mrow>
<mtext>Po</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">)</mo>
</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}}(X = x + 1) = \frac{\mu }{{x + 1}} \times {\text{P}}(X = x),{\text{ }}x \in \mathbb{N}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 2) = 0.241667">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.241667</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 3) = 0.112777">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.112777</mn>
</math></span>, use part (a) to find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span>.</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><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = x + 1) = \frac{{{\mu ^{x + 1}}}}{{(x + 1)!}}{{\text{e}}^{ - \mu }}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\mu }{{x + 1}} \times \frac{{{\mu ^x}}}{{x!}}{{\text{e}}^{ - \mu }}">
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mi>x</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\mu }{{x + 1}} \times {\text{P}}(X = x)">
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</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="\frac{\mu }{{x + 1}} \times {\text{P}}(X = x) = \frac{\mu }{{x + 1}} \times \frac{{{\mu ^x}}}{{x!}}{{\text{e}}^{ - \mu }}">
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mi>x</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\mu ^{x + 1}}}}{{(x + 1)!}}{{\text{e}}^{ - \mu }}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{P}}(X = x + 1)">
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}(X = x + 1)}}{{{\text{P}}(X = x)}} = \frac{{\frac{{{\mu ^{x + 1}}}}{{(x + 1)!}}{{\text{e}}^{ - \mu }}}}{{\frac{{{\mu ^x}}}{{x!}}{{\text{e}}^{ - \mu }}}}">
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mi>x</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo>!</mo>
</mrow>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>μ</mi>
</mrow>
</msup>
</mrow>
</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{{{\mu ^{x + 1}}}}{{{\mu ^x}}} \times \frac{{x!}}{{(x + 1)!}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>μ</mi>
<mi>x</mi>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mi>x</mi>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\mu }{{x + 1}}">
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p>and so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = x + 1) = \frac{\mu }{{x + 1}} \times {\text{P}}(X = x)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X = 3) = \frac{\mu }{3} \bullet {\text{P}}(X = 2){\text{ }}\left( {0.112777 = \frac{\mu }{3} \bullet 0.241667} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mn>3</mn>
</mfrac>
<mo>∙</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.112777</mn>
<mo>=</mo>
<mfrac>
<mi>μ</mi>
<mn>3</mn>
</mfrac>
<mo>∙</mo>
<mn>0.241667</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p>attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 1.40">
<mi>μ</mi>
<mo>=</mo>
<mn>1.40</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 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="question">
<p>Jenna is a keen book reader. The number of books she reads during one week can be modelled by a Poisson distribution with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>6</mn></math>.</p>
<p>Determine the expected number of weeks in one year, of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>52</mn></math> weeks, during which Jenna reads at least four books.</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>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Χ</mi></math> be the random variable “number of books Jenna reads per week.”</p>
<p>then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Χ</mi><mo>~</mo><mtext>Po </mtext><mfenced><mrow><mn>2</mn><mo>.</mo><mn>6</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>Χ</mi><mo>≥</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.264 </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>263998</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>263998</mn><mo>…</mo><mo>×</mo><mn>52</mn></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>7</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn></math> weeks.</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>It is known that 56 % of Infiglow batteries have a life of less than 16 hours, and 94 % have a life less than 17 hours. It can be assumed that battery life is modelled by the normal distribution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{N}}\left( {\mu ,\,\,{\sigma ^2}} \right)">
<mrow>
<mtext>N</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>σ<!-- σ --></mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</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="\mu ">
<mi>μ</mi>
</math></span> and the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ</mi>
</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>Find the probability that a randomly selected Infiglow battery will have a life of at least 15 hours.</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>use of inverse normal (implied by ±0.1509… or ±1.554…) <em><strong>(M1)</strong></em></p>
<p>P(<em>X</em> < 16) = 0.56</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{16 - \mu }}{\sigma } = 0.1509 \ldots ">
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mn>16</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mn>0.1509</mn>
<mo>…</mo>
</math></span> <em><strong>(A1)</strong></em></p>
<p>P(<em>X</em> < 17) = 0.94</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{17 - \mu }}{\sigma } = 1.554 \ldots ">
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mn>17</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mi>σ</mi>
</mfrac>
<mo>=</mo>
<mn>1.554</mn>
<mo>…</mo>
</math></span> <em><strong>(A1)</strong></em></p>
<p>attempt to solve a pair of simultaneous equations <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span> = 15.9, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sigma ">
<mi>σ</mi>
</math></span> = 0.712 <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correctly shaded diagram or intent to find P(<em>X</em> ≥ 15) <em><strong> (M1)</strong></em></p>
<p>= 0.895 <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Accept answers rounding to 0.89 or 0.90. Award <em><strong>M1A0</strong></em> for the answer 0.9.</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;">
[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 continuous random variable <em>X</em> has probability density function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> given by</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3ax}&,&{0 \leqslant x < 0.5} \\ {a\left( {2 - x} \right)}&,&{0.5 \leqslant x < 2} \\ 0&,&{{\text{otherwise}}} \end{array}} \right.">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mi>a</mi>
<mi>x</mi>
</mrow>
</mtd>
<mtd>
<mo>,</mo>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mn>0.5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mo>,</mo>
</mtd>
<mtd>
<mrow>
<mn>0.5</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mo>,</mo>
</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>
<p> </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="a = \frac{2}{3}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < 1} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {s < X < 0.8} \right) = 2 \times {\text{P}}\left( {2s < X < 0.8} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>s</mi>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mn>0.8</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>s</mi>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mn>0.8</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, and that 0.25 < <em>s</em> < 0.4 , find the value of <em>s</em>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</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> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left[ {\int_0^{0.5} {3x\,{\text{d}}x} + \int_{0.5}^2 {\left( {2 - x} \right)} \,{\text{d}}x} \right] = 1">
<mi>a</mi>
<mrow>
<mo>[</mo>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>0.5</mn>
</mrow>
</msubsup>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>+</mo>
<msubsup>
<mo>∫</mo>
<mrow>
<mn>0.5</mn>
</mrow>
<mn>2</mn>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note</strong>: Award the <em><strong>M1</strong></em> for the total integral equalling 1, or equivalent.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {\frac{3}{2}} \right) = 1">
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{2}{3}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>AG</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><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{0.5} {2x\,{\text{d}}x} + \frac{2}{3}\int_{0.5}^1 {\left( {2 - x} \right)} \,{\text{d}}x">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>0.5</mn>
</mrow>
</msubsup>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>+</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<msubsup>
<mo>∫</mo>
<mrow>
<mn>0.5</mn>
</mrow>
<mn>1</mn>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong> (M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{2}{3}">
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</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="\frac{2}{3}\int_1^2 {\left( {2 - x} \right)} \,{\text{d}}x = \frac{1}{3}">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<msubsup>
<mo>∫</mo>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < 1} \right) = \frac{2}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>(M1)A1</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="{\text{P}}\left( {s < X < 0.8} \right) = \int_s^{0.5} {2x\,{\text{d}}x} + \frac{2}{3}\int_{0.5}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>s</mi>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mn>0.8</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mo>∫</mo>
<mi>s</mi>
<mrow>
<mn>0.5</mn>
</mrow>
</msubsup>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>+</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<msubsup>
<mo>∫</mo>
<mrow>
<mn>0.5</mn>
</mrow>
<mrow>
<mn>0.8</mn>
</mrow>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {{x^2}} \right]_s^{0.5} + 0.27">
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mi>s</mi>
<mrow>
<mn>0.5</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mn>0.27</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.25 - {s^2} + 0.27">
<mn>0.25</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>s</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>0.27</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}}\left( {2s < X < 0.8} \right) = \frac{2}{3}\int_{2s}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>s</mi>
<mo><</mo>
<mi>X</mi>
<mo><</mo>
<mn>0.8</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<msubsup>
<mo>∫</mo>
<mrow>
<mn>2</mn>
<mi>s</mi>
</mrow>
<mrow>
<mn>0.8</mn>
</mrow>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{2}{3}\left[ {2x - \frac{{{x^2}}}{2}} \right]_{2s}^{0.8}">
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mrow>
<mn>2</mn>
<mi>s</mi>
</mrow>
<mrow>
<mn>0.8</mn>
</mrow>
</msubsup>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.28</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>s</mi>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>s</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>equating</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.25 - {s^2} + 0.27 = \frac{4}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)">
<mn>0.25</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>s</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>0.27</mn>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.28</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>s</mi>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>s</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p>attempt to solve for <em>s</em> <em><strong>(M1)</strong></em></p>
<p><em>s</em> = 0.274 <em><strong>A1</strong></em></p>
<p><em><strong>[7 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>Steffi the stray cat often visits Will’s house in search of food. 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 “the number of times per day that Steffi visits Will’s house”.</p>
<p>The random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> can be modelled by a Poisson distribution with mean 2.1.</p>
</div>
<div class="specification">
<p>Let<em> Y</em> be the discrete random variable “the number of times per day that Steffi is fed at Will’s house”. Steffi is only fed on the first four occasions that she visits each day.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that on a randomly selected day, Steffi does not visit Will’s house.</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>Copy and complete the probability distribution table for <em>Y</em>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaIAAABdCAYAAADnuGIXAAAPdElEQVR4Ae3dD1CUZR4H8O+unnkGyHB6x642crKBzklakF7FeQix5Fw2Dk5SFkFwNWSZyWQUaTWnohlDWmNj14GinRcWOzXTnKbJcTd4XgxbejQhDHiQwtLocYibGcn73ry7QC/7B5Y/77v/vswwvLzv8z6/5/m8z/v+eP/wrkYURRH8ogAFKEABCnhJQOuluAxLAQpQgAIUsAkwEXEgUIACFKCAVwUmD0TXaDQDk/xJAQpQgAIUUFRAfldoMBFJEeULFG1BkFUuJflgt6UBQAMaSIc+jgO7gTwN8NKcXIPTFKAABSigugATkerkDEgBClCAAnIBJiK5BqcpQAEKUEB1ASYi1ckZkAIUoAAF5AJMRHINTlOAAhSggOoCTESqkzMgBShAAQrIBZiI5BqcpgAFKEAB1QWYiFQnZ0AKUIACFJALMBHJNThNAQpQgAKqCzARqU7OgBSgAAUoIBdgIpJrcJoCFKAABVQXYCJSnZwBKUABClBALsBEJNfgNAUoQAEKqC7ARKQ6OQNSgAIUoIBcgIlIrhFo04IF5vICLNNooNFn4fVaC4RA6+NI/bE2ocp0CMVZaSioujRS6cBbbm3C8eIs6KUxoNFjWcFBmC29gdfPYXvUC0vtHmTpBwxMaLIG3Z4gE+pFu+kp6NPK0OQjDExEss0TWJPdMJesxbONd+NQn4g+88O4WLAWJebuwOrmcL0RzqLsgU0or9yOjQf+O1zJwFwmtOGjd6qA+95Ehyiir+N93Ne5Ewlr9sAcNAdiAdYvjuAY0rGvY8DgZSRtqUZPYG71EXsltH+Ml57aA8uIJdUrwESknrWqkYQmEwpL5mPz8ynQaQGtLgXPb56PkkKTz/wVpDiIdh5yPj6MfS9vgFHxYL4XQPjPBUxfnYPUmDBb47S6u7D+xQ0wVv8Zh2u7fK/BSrRIaEPd5duQuVgH6WCn1d2B3KwVwMFPUdfjI6cDSvTbXZ3WWpQU/gMzUhe4K+GV+UxEXmFXOug1NNccxbE4A2aHDGxiLcIS7kZm/VHUNF9TugGs3wcEtNF3IWnWlCEt0UZGYZFuyKzA/kX7SyQl3WRLQvaO9qKztQ2x+SuxOGxg3whsgh971w3z3kPA02uRFnnDj7N9YMpPtsR5mLIMto/Y1WjuRfHg5aVLqCpIgD7XhPYg/OPG7fgRWlFTUQPdoihEOm3hGlTUtAbfvSK3WEG4YNqvsSTWfpYUVL2X7heW/QHbmrNxKH8xQoKq8wKs5gN4A2uQFx/hcz13Okz5XAttDboJ6eXN6GsshRGf4/iZzv4DaRjmp61A7IVufOubDfdOq6wdaKwH4mL1QbazeYfbf6IK6Kmrxum8R2B0OFPynz6MpaUCeqoKoQ+NRUrudhw49Tecag6yO0TWOux9A3g6L8Enjwl+kojsg087dyFSoy1o6fq2PxFNwS9mR2Fu6gLo/aonY9mZuA4FxilgrcMfy2egyEcPRuPs3TCraxGWXIQO8Xt01B3Ac9iPVUmFMLUHy9OD3TC/U4W5RXmIH7xUPwyXFxb51+FbeyMionVoqW/DRQlLaMOHJV/hdw8u8sks74XtaQ8ZokdsHFDf2AGr1xrBwL4lIB2MjiDihWyfPRgp7zUFuvhMbH97K4yWf+GzxmA4K5Iuyb2HyjkPYqUPnwX7XSIKj5zWP17tj2XWGdf6NLDyO5eLCNooJGYkOi0QOltx2pKIjMQo2c1bp2KcEXACvWj/6D18uXwDcuYF4b0hh+2pNdyJDKNv3ax3aOIE/tqF2sN/wvZVUZhk+18y6X+pZiJlpxk4lovYSXqklZ31+j1j/0pE+CnCI8OBk+dw/vwJ7KzU4cmVc3hQdRq2U2FIvAdxQx5RFWC90Ix64z1INEx1WoMzAlWgF5aqMhwOXYHMwSTUg7PlB1EdjI8vS5tZuofacmuQPLAxA8mv1kEURdn3RZx4Lh4wlqKxrwOf5Mzz+jHUzxLRVEyfGQpcPYk9u84gad1yzPKzHqh1uNPGpKMovwFbdpyARQAEywns2NKA/KJ0xNBMrc3g5Tg9aDK9gjUpTyA/ZbbsL+LpmH/oB+h99H7BRKIJ7Sbk6tNQUF5r2w8gtKNqx150FuYF2QMbE6k68XX52SFpMkIjZgKIwG8fexzJuqH/IzHxPP5cYzji89/CqzPfRfwkDSat+RSxxW8hPz7cnzs1yrbbH++fFJuLYzBjZ8pMaHzotSaj7Mwoi0uvcSlE0qrtqHZaUwdjxp0w+Nne79QND2Zop8fgrtQO7MxeAv0kDfSP/gXdq3ZjX84C3lf2wE+tIhpROmcDbP+j0z+pVuwxxLmGprIXsT8qH1uTZ3n9dNLTDmg0GttpsaflA7EcDfxlH1N29HEccBxII8xxHPjR30TS0x/7UYpHUehHSUjZ3Zq1U4ACFPB/gcm+3QUp+ezGihVtyNx3Mxq/uh2bN/CU2re3GVtHAQpQYHQCPn9GJFy+iEZLAxo7b8cz64PttRyj25gsTQEKUMAfBfzsHpE/EjtfD/XPXoyv1Y7XhMdXm3+uTQPuC9LI5ThwNvD5MyL/POSw1RSgAAUo4KkAE5GnUixHAQpQgAKKCDARKcLKSilAAQpQwFMBJiJPpViOAhSgAAUUEWAiUoSVlVKAAhSggKcCTESeSrEcBShAAQooIsBEpAgrK6UABShAAU8FmIg8lWI5ClCAAhRQRICJSBFWVkoBClCAAp4KMBF5KsVyFKAABSigiMCQV/woEoGVUoACFKAABRwE5B87NOTt2/IFDuvw13EI8N1Szu+WGgen367KccBxIA1ejgO7gXxH5qU5uQanKUABClBAdQEmItXJGZACFKAABeQCTERyDU5TgAIUoIDqAkxEqpMzIAUoQAEKyAWYiOQanKYABShAAdUFmIhUJ2dAClCAAhSQCzARyTU4TQEKUIACqgswEalOzoAUoAAFKCAXYCKSa3CaAhSgAAVUF2AiUp2cASlAAQpQQC7ARCTX4DQFKEABCqguwESkOjkDUoACFKCAXICJSK7BaQpQgAIUUF2AiUh1cgakAAUoQAG5gNcSkWCpRXnhG6jqEeTt8ZHpS6gq3gFTU4+PtIfNoAAFKBC4AuNMRNfQVLba9vka0mdsDPnWZ6H4g2o0WZ0TjWA5jk3Pf4zQ7Gwkh0lNuA6LKW/o+oZimK9bYS5eJptvQJbpvApbYwaSHvoN2rb9HoVV7XDugQpNYAgKUIACwSIg9n8BGJgc/c++BrHUGC0aSxvEPtval8XGyhfEJEDUPbJfbLhin2tbdOUz8TXjerHywvcOcfrEK3UltnWiX6sTfxhc+p3YWHq/iKQXxMrGy4NzVZm4Ui+W5qwXSxvGF3dctqp0VPkgNBBFGtBA2tM4DpwNxnlG5C5dhyEm/SW8XXo/LAfexL7arv6C19B0eDeOZ+Rh5awpDitrEXLrcmQadWipb8NF21IB1rMV2Fb0c1S++wrSY8Ic1lH415AFyC64BRXrP0ATT4sUxmb1FKBAsAoolIgkzimIjDJAhw6cbr1kv7zV80+UbvoBGYlRcBlYOxsLU2OBk+fQcR2AtQ57n/gQcRVbke6UuNTZZFrDnchAKUqrL6kTkFEoQAEKBJmAy3wwMQa96GxthgV6LIqaAS0E9NR9ioNx9yDRMNVNiGm4+bY7oGtpRts3bagqKkZ95hbkxYe7KT/G2RYTsgbuaS17HeaB+1g9VSjQxyHX1PbjfSFtFBIzZuHgJ/8GH10YozdXowAFKDCMwAQmoquo//tn/Q8n9MJSW4ptm94Hkh7C6sURALpQ98kxYFEUIt1G1SJsfgJScRKVhc/g8W8eQFH2AoS47MB5mLIMsgcZHB6W6E80hmIzpJOrIV+6dJSL36Gx9H6g+hTOdPTaF4fEIi17Di50fzekuPSL5XQrOnl5zsmFMyhAAQqMV8BtShhLxZYD2ZgfOgkazQ3QL9kL5Fei7tCTiA/RAsIltJ7uGLnamXMQF/0lDn29FBW7V2KW2xbehPTyZukJi2G/m5+Nx2SXUadi7sLFiMZFdF3pT1Xan2G24VdIXRgpu3TYf4mxvhkXBs6cXNbHmRSgAAUoMBYB18fosdSEaTCWNuBIzjzZQXz0FQnnzuB4iw7GwuW4VUpgCn5pQyMQjfOob/sfEB8Cof2vKKldgs3ZLi4FWprR2tkLhLm7rKhgQ1k1BShAgQAWUPZIP2q4a2iuOYpjSHT/QMNgneO4NNdfhzY0HJGD9XXji8NNMG6+1/VZmM6AqEjHJ/0GV+YEBShAAQqMUWACz4hGaIF2BqIW6YcvJLSipqIGMG4d5oGGgSr6L82VD/w+hp83hiNS14LKc1/jfNUpVM55EFvdPZ0XZ8Bshc/QxtADrkIBClDA7wUm5ozoag+6eoHerh5cdUsSgYQ0I+D2pn8vLNUVOHjMgujUhZg7MS1z2xrbgmnTMXMacPWLMuyqXoh1K+e4uKxof/pv+Icshg/DpRSgAAUo4F5gnIf7/lf8hC7BxuoWVG9cglDNapQ1XXMRUYuwhLuRWX8UNc0Oy22PU98AfcorqAbQsjEBP7G94sdFNRM5S3sjIqJ1wOSleOy5FOhcadjO0tqRmXYLVP532onsKeuiAAUo4LMCmv5XTtgeg7a/fULJtkqJKxfrsHncDzVMSCuFsyh7rBJRWzciWef6/o/QVIbl64A3j+QgxlWi8qAh0jv4lLf1oCFeLEIDqLSPeXEjexCa44DjQBomjuNgjIdWD0acyyJTEbN6PVIrdmH/WW//e2g3zLsrgI3r3CYhCG348NUTSC1KH3MScsnAmRSgAAUoMCig8hmRPa7t7dvZlTDs3omceWpe8OqGufhhrLCkY9/sVny1NB8b3L21wXoWpi27UJe2GVuTZ7m4dzRoOOKEY/YfcYUALEAD578CA3Azj9gljgOOA2mQOI4Dlc+I7ONUq0tFUeUzCD+wX+XPI7qOyxc7Yfm8EZ1Ln8B6d0kIl1C1txJXVr887iQ04p7JAhSgAAWCXMArZ0TBZu6Y/YOt/1J/aUADjgP7ns99wXlf8MoZUTAeiNlnClCAAhRwLcBE5NqFcylAAQpQQCUBJiKVoBmGAhSgAAVcCzARuXbhXApQgAIUUEmAiUglaIahAAUoQAHXAkxErl04lwIUoAAFVBJgIlIJmmEoQAEKUMC1ABORaxfOpQAFKEABlQSYiFSCZhgKUIACFHAtwETk2oVzKUABClBAJYEhr/hRKSbDUIACFKBAkAvIPxpn8KPC5TOD3IfdpwAFKEABFQV4aU5FbIaiAAUoQAFngf8DryRfl5vMKHwAAAAASUVORK5CYII="></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>Hence find the expected number of times per day that Steffi is fed at Will’s house.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In any given year of 365 days, the probability that Steffi does not visit Will for at most <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> days in total is 0.5 (to one decimal place). Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</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="X \sim {\text{Po}}\left( {2.1} \right)">
<mi>X</mi>
<mo>∼</mo>
<mrow>
<mtext>Po</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2.1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X = 0} \right) = 0.122\left( { = {{\text{e}}^{ - 2.1}}} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.122</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2.1</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</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><img 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"> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct probability for <em>Y</em> = 1, 2, 3, 4. Accept 0.162 for P(<em>Y</em> = 4).</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( Y \right) = \sum {y{\text{P}}\left( {Y = y} \right)} ">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>Y</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>∑</mo>
<mrow>
<mi>y</mi>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>Y</mi>
<mo>=</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 \times 0.257 \ldots + 2 \times 0.270 \ldots + 3 \times 0.189 + 4 \times 0.161 \ldots ">
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mn>0.257</mn>
<mo>…</mo>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mn>0.270</mn>
<mo>…</mo>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<mn>0.189</mn>
<mo>+</mo>
<mn>4</mn>
<mo>×</mo>
<mn>0.161</mn>
<mo>…</mo>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2.01">
<mo>=</mo>
<mn>2.01</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> be the no of days per year that Steffi does not visit</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T \sim B\left( {365{\text{,}}\,\,0.122 \ldots } \right)">
<mi>T</mi>
<mo>∼</mo>
<mi>B</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>365</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0.122</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>require <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.45 \leqslant {\text{P}}\left( {T \leqslant n} \right) < 0.55">
<mn>0.45</mn>
<mo>⩽</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>⩽</mo>
<mi>n</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo><</mo>
<mn>0.55</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {T \leqslant 44} \right) = 0.51">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>⩽</mo>
<mn>44</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.51</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 44">
<mi>n</mi>
<mo>=</mo>
<mn>44</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V">
<mi>V</mi>
</math></span> be the discrete random variable “number of times Steffi is not fed per day”</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( V \right) = 1 \times {\text{P}}\left( {X = 5} \right) + 2 \times {\text{P}}\left( {X = 6} \right) + 3 \times {\text{P}}\left( {X = 7} \right) + \ldots ">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>V</mi>
<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>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>7</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>…</mo>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 \times 0.0416 \ldots + 2 \times 0.0145 \ldots + 3 \times 0.00437 \ldots + \ldots ">
<mo>=</mo>
<mn>1</mn>
<mo>×</mo>
<mn>0.0416</mn>
<mo>…</mo>
<mo>+</mo>
<mn>2</mn>
<mo>×</mo>
<mn>0.0145</mn>
<mo>…</mo>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<mn>0.00437</mn>
<mo>…</mo>
<mo>+</mo>
<mo>…</mo>
</math></span> <em><strong>A1</strong></em></p>
<p>= 0.083979... <em><strong>A1</strong></em></p>
<p>expected no of occasions per year > 0.083979... × 365 = 30.7 <em><strong>A1</strong></em></p>
<p>hence Steffi can expect not to be fed on at least 30 occasions <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Candidates may consider summing more than three terms in their calculation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( V \right)">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>V</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right) - {\text{E}}\left( Y \right) = 0.0903 \ldots ">
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>X</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>Y</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.0903</mn>
<mo>…</mo>
</math></span> <em><strong>M1A</strong></em><em><strong>1</strong></em></p>
<p>0.0903… × 365 <em><strong>M1</strong></em></p>
<p>= 33.0 > 30 <em><strong> A1AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Timmy owns a shop. His daily income from selling his goods can be modelled as a normal distribution, with a mean daily income of $820, and a standard deviation of $230. To make a profit, Timmy’s daily income needs to be greater than $1000.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the probability that, on a randomly selected day, Timmy makes a profit.</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 shop is open for 24 days every month.</p>
<p>Calculate the probability that, in a randomly selected month, Timmy makes a profit on between 5 and 10 days (inclusive).</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><em>X</em> ~ N(820, 230<sup>2</sup>) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> M1</strong></em> for an attempt to use normal distribution. Accept labelled normal graph.</p>
<p>⇒P(<em>X</em> > 1000) = 0.217 <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>Y</em> ~ B(24,0.217...) <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for recognition of binomial distribution with parameters.</p>
<p>P(<em>Y</em> ≤ 10) − P(<em>Y</em> ≤ 4) <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for an attempt to find P(5 ≤ <em>Y</em> ≤ 10) or P(<em>Y</em> ≤ 10) − P(<em>Y</em> ≤ 4).</p>
<p>= 0.613 <em><strong>A1</strong></em></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>The number of bananas that Lucca eats during any particular day follows a Poisson distribution with mean 0.2.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Lucca eats at least one banana in a particular day.</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 number of weeks in the year in which Lucca eats no bananas.</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>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> be the number of bananas eaten in one day</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \sim {\text{Po}}(0.2)">
<mi>X</mi>
<mo>∼</mo>
<mrow>
<mtext>Po</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>0.2</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X \geqslant 1) = 1 - {\text{P}}(X = 0)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</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>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=" = 0.181{\text{ }}( = 1 - {{\text{e}}^{ - 0.2}})">
<mo>=</mo>
<mn>0.181</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>0.2</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></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><strong>EITHER</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y">
<mi>Y</mi>
</math></span> be the number of bananas eaten in one week</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Y}} \sim {\text{Po}}(1.4)">
<mrow>
<mtext>Y</mtext>
</mrow>
<mo>∼</mo>
<mrow>
<mtext>Po</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>1.4</mn>
<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}}(Y = 0) = 0.246596 \ldots {\text{ }}( = {{\text{e}}^{ - 1.4}})">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>Y</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.246596</mn>
<mo>…</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1.4</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><strong>OR</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Z">
<mi>Z</mi>
</math></span> be the number of days in one week at least one banana is eaten</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Z \sim {\text{B}}(7,{\text{ }}0.181 \ldots )">
<mi>Z</mi>
<mo>∼</mo>
<mrow>
<mtext>B</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>7</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0.181</mn>
<mo>…</mo>
<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}}(Z = 0) = 0.246596 \ldots ">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>Z</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.246596</mn>
<mo>…</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="52 \times 0.246596 \ldots ">
<mn>52</mn>
<mo>×</mo>
<mn>0.246596</mn>
<mo>…</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 12.8{\text{ }}( = 52{{\text{e}}^{ - 1.4}})">
<mo>=</mo>
<mn>12.8</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>52</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1.4</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</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>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 such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = 0.95,{\text{ P}}(A \cap B) = 0.6">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.95</mn>
<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>
<mn>0.6</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A|B) = 0.75">
<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>
<mn>0.75</mn>
</math></span>.</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="{\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">[2]</div>
<div class="question_part_label">a.</div>
</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="{\text{P}}(A)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
</math></span>.</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 show that events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A’">
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are independent.</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 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}}(A|B) = \frac{{{\text{P}}(A \cap B)}}{{{\text{P}}(B)}}">
<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>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 0.75 = \frac{{0.6}}{{{\text{P}}(B)}}">
<mo stretchy="false">⇒</mo>
<mn>0.75</mn>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.6</mn>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{P}}(B){\text{ }}\left( { = \frac{{0.6}}{{0.75}}} \right) = 0.8">
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.6</mn>
</mrow>
<mrow>
<mn>0.75</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.8</mn>
</math></span> <strong><em>A1</em></strong></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{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></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 0.95 = {\text{P}}(A) + 0.8 - 0.6">
<mo stretchy="false">⇒</mo>
<mn>0.95</mn>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>0.8</mn>
<mo>−</mo>
<mn>0.6</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{P}}(A) = 0.75">
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.75</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</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}}(A'|B) = \frac{{{\text{P}}(A' \cap B)}}{{{\text{P}}(B)}} = \frac{{0.2}}{{0.8}} = 0.25">
<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>
<mfrac>
<mrow>
<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>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.2</mn>
</mrow>
<mrow>
<mn>0.8</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.25</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}}(A'|B) = {\text{P}}(A’)">
<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>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>R1</em></strong></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A’">
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are independent <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> If there is evidence that the student has calculated <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B) = 0.2">
<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>
<mn>0.2</mn>
</math></span> by assuming independence in the first place, award <strong><em>A0R0</em></strong>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = {\text{P}}(A|B)">
<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>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) \times {\text{P}}(B) = 0.75 \times 0.80 = 0.6 = {\text{P}}(A \cap B)">
<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>
<mn>0.75</mn>
<mo>×</mo>
<mn>0.80</mn>
<mo>=</mo>
<mn>0.6</mn>
<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>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p><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 independent <strong><em>R1</em></strong></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A’">
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are independent <strong><em>AG</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A') \times {\text{P}}(B) = 0.25 \times 0.80 = 0.2">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.25</mn>
<mo>×</mo>
<mn>0.80</mn>
<mo>=</mo>
<mn>0.2</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}}(A') \times {\text{P}}(B) = {\text{P}}(A' \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<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>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>R1</em></strong></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A’">
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are independent <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></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" style="padding-left: 20px; padding-right: 20px;">
<p>A horse breeder records the number of births for each of 100 horses during the past eight years. The results are summarized in the following table:</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p class="indent1" style="margin-top:12.0pt;">Stating null and alternative hypotheses carry out an appropriate test at the 5% significance level to decide whether the results can be modelled by B (6, 0.5).</p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Without doing any further calculations, explain briefly how you would carry out a test, at the 5% significance level, to decide if the data can be modelled by B(6, <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span>), where <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span> is unspecified.</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>A different horse breeder collected data on the time and outcome of births. The data are summarized in the following table:</p>
<p><img style="display: block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>Carry out an appropriate test at the 5% significance level to decide whether there is an association between time and outcome.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="indent1"><strong>METHOD 1</strong></p>
<p class="indent1">H<sub>0</sub>: distribution is B(6, 0.5); H<sub>1</sub>: distribution is not B(6, 0.5) <em><strong>A1</strong></em></p>
<p class="indent1"><img src="data:image/png;base64,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"></p>
<p class="indent1"><span class="mjpage"><math alttext="\left( {{E_0} = 100{{\left( {0.5} \right)}^6} = \frac{{25}}{{16}} = 0.015625} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> <mo>=</mo> <mn>100</mn> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.5</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>6</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>25</mn> </mrow> <mrow> <mn>16</mn> </mrow> </mfrac> <mo>=</mo> <mn>0.015625</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A3</strong></em></p>
<p class="indent1">Combining the first two columns and the last two columns: <em><strong>A1 </strong></em></p>
<p class="indent1"><span class="mjpage"><math alttext="{\chi ^2} = \sum {\frac{{{O^2}}}{E}} - \sum E " xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mo>∑</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mi>O</mi> <mn>2</mn> </msup> </mrow> </mrow> <mi>E</mi> </mfrac> </mrow> <mo>−</mo> <mo>∑</mo> <mi>E</mi> </math></span></p>
<p class="indent1"><span class="mjpage"><math alttext=" = \frac{{{6^2}}}{{\left( {\frac{{175}}{{16}}} \right)}} + \frac{{{{26}^2}}}{{\left( {\frac{{375}}{{16}}} \right)}} + \frac{{{{37}^2}}}{{\left( {\frac{{500}}{{16}}} \right)}} + \frac{{{{18}^2}}}{{\left( {\frac{{375}}{{16}}} \right)}} + \frac{{{{13}^2}}}{{\left( {\frac{{175}}{{16}}} \right)}} - 100" xmlns="http://www.w3.org/1998/Math/MathML"> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>175</mn> </mrow> <mrow> <mn>16</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>26</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>375</mn> </mrow> <mrow> <mn>16</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>37</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>500</mn> </mrow> <mrow> <mn>16</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>18</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>375</mn> </mrow> <mrow> <mn>16</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>13</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>175</mn> </mrow> <mrow> <mn>16</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>−</mo> <mn>100</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p class="indent1">= 5.22 <em><strong>A1</strong></em></p>
<p class="indent1"><span class="mjpage"><math alttext="v" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>v</mi> </math></span> = 4, so critical value of <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 class="indent1">Since 5.22 < 9.488 the result is not significant and we accept H<sub>0</sub> <em><strong>R1</strong></em></p>
<p class="indent2"> </p>
<p class="indent2"><strong>METHOD 2</strong></p>
<p class="indent1">H<sub>0</sub>: distribution is B(6, 0.5); H<sub>1</sub>: distribution is not B(6, 0.5) <em><strong>A1</strong></em></p>
<p class="indent1">By GDC, <span class="mjpage"><math alttext="p = 0.266" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> <mo>=</mo> <mn>0.266</mn> </math></span> <em><strong>A8</strong></em></p>
<p class="indent1">Since 0.266 > 0.05 the result is not significant and we accept H<sub>0</sub> <em><strong>R1</strong></em></p>
<p class="indent1"> </p>
<p class="indent1"><em><strong>[10 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="indent1">Estimate <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span> from the data which would entail the loss of one degree of freedom <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p class="indent1"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="indent1">H<sub>0</sub>: there is no association H<sub>1</sub>: there is an association <em><strong> </strong></em><em><strong>A1</strong></em></p>
<p class="indent1"><img 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"> <em><strong> </strong></em><em><strong>A2</strong></em></p>
<p class="indent1"><span class="mjpage"><math alttext="{\chi ^2} = \frac{{{{68}^2}}}{{64.8}} + \frac{{{{42}^2}}}{{45.2}} + \frac{{{{103}^2}}}{{94.3}} + \ldots + \frac{{{6^2}}}{{10.6}} + \frac{{{{12}^2}}}{{7.4}} - 314" 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> <mn>68</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>64.8</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>42</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>45.2</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>103</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>94.3</mn> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>10.6</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mn>12</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>7.4</mn> </mrow> </mfrac> <mo>−</mo> <mn>314</mn> </math></span> <em><strong> </strong></em><em><strong>(M1)</strong></em></p>
<p class="indent1"><strong> = </strong>15.7</p>
<p class="indent1"><span class="mjpage"><math alttext="v = 3{\text{,}}\,\,\chi _{5{\text{% }}}^2\left( 3 \right) = 7.815" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>v</mi> <mo>=</mo> <mn>3</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <msubsup> <mi>χ</mi> <mrow> <mn>5</mn> <mrow> <mtext>% </mtext> </mrow> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>7.815</mn> </math></span> <em><strong> </strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p class="indent1">Since 15.7 > 7.815 we reject H<sub>0</sub> <em><strong>R1</strong></em></p>
<p class="indent1"> </p>
<p class="indent1"><strong>METHOD 2</strong></p>
<p class="indent1">H<sub>0</sub>: there is no association H<sub>1</sub>: there is an association <em><strong> </strong></em><em><strong>A1</strong></em></p>
<p class="indent1">By GDC, <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span> = 0.00129 <em><strong>A6 </strong></em></p>
<p class="indent1">Since 0.00129 < 0.05 we reject H<sub>0</sub>. <em><strong>R1 </strong></em></p>
<p class="indent1"> </p>
<p class="indent1"><em><strong>[8 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>Each of the 25 students in a class are asked how many pets they own. Two students own three pets and no students own more than three pets. The mean and standard deviation of the number of pets owned by students in the class are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{18}}{{25}}">
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{24}}{{25}}">
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</math></span> respectively.</p>
<p>Find the number of students in the class who do not own a pet.</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>let <em>p</em> have no pets, <em>q</em> have one pet and <em>r</em> have two pets <em><strong> (M1)</strong></em></p>
<p><em>p</em> + <em>q</em> + <em>r</em> + 2 = 25 <em><strong>(A1)</strong></em></p>
<p>0<em>p</em> + 1<em>q</em> + 2<em>r</em> + 6 = 18 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept a statement that there are a total of 12 pets.</p>
<p>attempt to use variance equation, or evidence of trial and error <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0p + 1q + 4r + 18}}{{25}} - {\left( {\frac{{18}}{{25}}} \right)^2} = {\left( {\frac{{24}}{{25}}} \right)^2}">
<mfrac>
<mrow>
<mn>0</mn>
<mi>p</mi>
<mo>+</mo>
<mn>1</mn>
<mi>q</mi>
<mo>+</mo>
<mn>4</mn>
<mi>r</mi>
<mo>+</mo>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p>attempt to solve a system of linear equations <em><strong>(M1)</strong></em></p>
<p><em>p</em> = 14 <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><img src="data:image/png;base64,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"> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p + q + r + \frac{2}{{25}} = 1">
<mi>p</mi>
<mo>+</mo>
<mi>q</mi>
<mo>+</mo>
<mi>r</mi>
<mo>+</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q + 2r + \frac{6}{{25}} = \frac{{18}}{{25}}\left( { \Rightarrow q + 2r = \frac{{12}}{{25}}} \right)">
<mi>q</mi>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
<mo>+</mo>
<mfrac>
<mn>6</mn>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mi>q</mi>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>12</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</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="q + 4r + \frac{{18}}{{25}} - {\left( {\frac{{18}}{{25}}} \right)^2} = \frac{{576}}{{625}}\left( { \Rightarrow q + 4r = \frac{{18}}{{25}}} \right)">
<mi>q</mi>
<mo>+</mo>
<mn>4</mn>
<mi>r</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>576</mn>
</mrow>
<mrow>
<mn>625</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mi>q</mi>
<mo>+</mo>
<mn>4</mn>
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</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="q = \frac{6}{{25}},\,\,r = \frac{3}{{25}}">
<mi>q</mi>
<mo>=</mo>
<mfrac>
<mn>6</mn>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{14}}{{25}}">
<mi>p</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>14</mn>
</mrow>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>so 14 have no pets</p>
<p><em><strong>[7 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>Iqbal attempts three practice papers in mathematics. The probability that he passes the first paper is 0.6. Whenever he gains a pass in a paper, his confidence increases so that the probability of him passing the next paper increases by 0.1. Whenever he fails a paper the probability of him passing the next paper is 0.6.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the given probability tree diagram for Iqbal’s three attempts, labelling each branch with the correct probability.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></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>Calculate the probability that Iqbal passes at least two of the papers he attempts.</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 Iqbal passes his third paper, given that he passed only one previous paper.</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 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><em><img 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"> <strong>A1A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct column of probabilities.</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>probability (at least twice) =</p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0.6 \times 0.7 \times 0.8} \right) + \left( {0.6 \times 0.7 \times 0.2} \right) + \left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.7</mn>
<mo>×</mo>
<mn>0.8</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.7</mn>
<mo>×</mo>
<mn>0.2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.3</mn>
<mo>×</mo>
<mn>0.6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.4</mn>
<mo>×</mo>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.7</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0.6 \times 0.7} \right) + \left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.7</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.3</mn>
<mo>×</mo>
<mn>0.6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.4</mn>
<mo>×</mo>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.7</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> M1</strong></em> for summing all required probabilities.</p>
<p><strong>THEN</strong></p>
<p>= 0.696<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>P(passes third paper given only one paper passed before)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{P}}\,\left( {{\text{passes third AND only one paper passed before}}} \right)}}{{{\text{P}}\,\left( {{\text{passes once in first two papers}}} \right)}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>passes third AND only one paper passed before</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>passes once in first two papers</mtext>
</mrow>
</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{{\left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)}}{{\left( {0.6 \times 0.3} \right) + \left( {0.4 \times 0.6} \right)}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.3</mn>
<mo>×</mo>
<mn>0.6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.4</mn>
<mo>×</mo>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.7</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.6</mn>
<mo>×</mo>
<mn>0.3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.4</mn>
<mo>×</mo>
<mn>0.6</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>= 0.657<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="specification">
<p>There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_16.43.55.png" alt="M17/5/MATHL/HP2/ENG/TZ2/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>One of the players is chosen at random. Find the probability that this player’s score was 5 or more.</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>Calculate the mean score.</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="{\text{P}}(5{\text{ or more}}) = \frac{{29}}{{75}}\,\,\,( = 0.387)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>5</mn>
<mrow>
<mtext> or more</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mn>29</mn>
</mrow>
<mrow>
<mn>75</mn>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>0.387</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">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{mean score}} = \frac{{2 \times 3 + 3 \times 15 + 4 \times 28 + 5 \times 17 + 6 \times 9 + 7 \times 3}}{{75}}">
<mrow>
<mtext>mean score</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>3</mn>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<mn>15</mn>
<mo>+</mo>
<mn>4</mn>
<mo>×</mo>
<mn>28</mn>
<mo>+</mo>
<mn>5</mn>
<mo>×</mo>
<mn>17</mn>
<mo>+</mo>
<mn>6</mn>
<mo>×</mo>
<mn>9</mn>
<mo>+</mo>
<mn>7</mn>
<mo>×</mo>
<mn>3</mn>
</mrow>
<mrow>
<mn>75</mn>
</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{{323}}{{75}}\,\,\,( = 4.31)">
<mo>=</mo>
<mfrac>
<mrow>
<mn>323</mn>
</mrow>
<mrow>
<mn>75</mn>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>4.31</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></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>Jorge is carefully observing the rise in sales of a new app he has created.</p>
<p>The number of sales in the first four months is shown in the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Jorge believes that the increase is exponential and proposes to model the number of sales <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> in month <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> with the equation</p>
<p style="text-align: left; padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mi>r</mi><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mi>A</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math></p>
</div>
<div class="specification">
<p>Jorge plans to adapt Euler’s method to find an approximate value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<p>With a step length of one month the solution to the differential equation can be approximated using Euler’s method where</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≈</mo><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>×</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math></p>
</div>
<div class="specification">
<p>Jorge decides to take the mean of these values as the approximation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> for his model. He also decides the graph of the model should pass through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>52</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The sum of the square residuals for these points for the least squares regression model is approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>555</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Jorge’s model satisfies the differential equation</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mi>N</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow><mrow><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow></mfrac></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>Hence find three approximations for the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation for Jorge’s model.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum of the square residuals for Jorge’s model using the values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>4</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment how well Jorge’s model fits the data.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give two possible sources of error in the construction of his model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mi>A</mi><msup><mtext>e</mtext><mrow><mi>r</mi><mi>t</mi></mrow></msup></math> <strong>(M1)A1</strong></p>
<p> </p>
<p><strong>Note: M1</strong> is for an attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>r</mi><mi>N</mi></math> <strong>AG</strong></p>
<p> </p>
<p><strong>Note:</strong> Accept solution of the differential equation by separating variables</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>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≈</mo><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>×</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>⇒</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>≈</mo><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>≈</mo><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></math> <strong>M1A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mo>≈</mo><mfrac><mrow><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow><mrow><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow></mfrac></math> <strong>AG</strong></p>
<p> </p>
<p><strong>Note:</strong> Do not penalize the use of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> sign.</p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Correct method <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mn>52</mn><mo>-</mo><mn>40</mn></mrow><mn>40</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mn>70</mn><mo>-</mo><mn>52</mn></mrow><mn>52</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>346</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mn>98</mn><mo>-</mo><mn>70</mn></mrow><mn>70</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> <strong>A2</strong></p>
<p> </p>
<p><strong>Note: A1</strong> for a single error <strong>A0</strong> for two or more errors.</p>
<p> </p>
<p><strong>[3 marks]</strong></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"><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>349</mn><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>34871</mn><mo>…</mo></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>68</mn><mn>195</mn></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>52</mn><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>34871</mn><mo>…</mo><mo>×</mo><mn>2</mn></mrow></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>25</mn><mo>.</mo><mn>8887</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>25</mn><mo>.</mo><mn>9</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>349</mn><mi>t</mi></mrow></msup></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>36</mn><mo>.</mo><mn>6904</mn><mo>…</mo><mo>-</mo><mn>40</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>0</mn><mo>+</mo><msup><mfenced><mrow><mn>73</mn><mo>.</mo><mn>6951</mn><mo>…</mo><mo>-</mo><mn>70</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>104</mn><mo>.</mo><mn>4435</mn><mo>…</mo><mo>-</mo><mn>98</mn></mrow></mfenced><mn>2</mn></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>66</mn><mo>.</mo><mn>1</mn><mo> </mo><mfenced><mrow><mn>66</mn><mo>.</mo><mn>126</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The sum of the square residuals is approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> times as large as the minimum possible, so Jorge’s model is unlikely to fit the data exactly <strong>R1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>For example</p>
<p>Selecting a single point for the curve to pass through</p>
<p>Approximating the gradient of the curve by the gradient of a chord <strong>R1R1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">f.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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> has a probability distribution given in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-02-28_om_17.11.47.png" alt="N16/5/MATHL/HP2/ENG/TZ0/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}({X^2})">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</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 value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</math></span>.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}({X^2}) = \Sigma {x^2} \bullet {\text{P}}(X = x) = 10.37{\text{ }}( = 10.4{\text{ 3 sf)}}">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">Σ</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>∙</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>10.37</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>10.4</mn>
<mrow>
<mtext> 3 sf)</mtext>
</mrow>
</math></span> <strong><em>(M1)A1</em></strong></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><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sd}}(X) = 1.44069 \ldots ">
<mrow>
<mtext>sd</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1.44069</mn>
<mo>…</mo>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = 2.08{\text{ }}( = 2.0756)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.08</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>2.0756</mn>
<mo stretchy="false">)</mo>
</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{E}}(X) = 2.88{\text{ }}( = 0.06 + 0.27 + 0.5 + 0.98 + 0.63 + 0.44)">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.88</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>0.06</mn>
<mo>+</mo>
<mn>0.27</mn>
<mo>+</mo>
<mn>0.5</mn>
<mo>+</mo>
<mn>0.98</mn>
<mo>+</mo>
<mn>0.63</mn>
<mo>+</mo>
<mn>0.44</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = {\text{E}}({X^2}) - {\left( {{\text{E}}(X)} \right)^2}">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>only if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{E}}(X)} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> is used correctly.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{Var}}(X) = 10.37 - 8.29} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>10.37</mn>
<mo>−</mo>
<mn>8.29</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = 2.08{\text{ }}( = 2.0756)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.08</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>2.0756</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept 2.11.</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}(X) = 2.88{\text{ }}( = 0.06 + 0.27 + 0.5 + 0.98 + 0.63 + 0.44)">
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.88</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>0.06</mn>
<mo>+</mo>
<mn>0.27</mn>
<mo>+</mo>
<mn>0.5</mn>
<mo>+</mo>
<mn>0.98</mn>
<mo>+</mo>
<mn>0.63</mn>
<mo>+</mo>
<mn>0.44</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(X) = {\text{E}}\left( {{{\left( {X - {\text{E}}(X)} \right)}^2}} \right)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>−</mo>
<mrow>
<mtext>E</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</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="(0.679728 + \ldots + 0.549152)">
<mo stretchy="false">(</mo>
<mn>0.679728</mn>
<mo>+</mo>
<mo>…</mo>
<mo>+</mo>
<mn>0.549152</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Var}}(E) = 2.08{\text{ }}( = 2.0756)">
<mrow>
<mtext>Var</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>E</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2.08</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>2.0756</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 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>When carpet is manufactured, small faults occur at random. The number of faults in Premium carpets can be modelled by a Poisson distribution with mean 0.5 faults per 20<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,">
<mspace width="thinmathspace"></mspace>
</math></span>m<sup>2</sup>. Mr Jones chooses Premium carpets to replace the carpets in his office building. The office building has 10 rooms, each with the area of 80<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,">
<mspace width="thinmathspace"></mspace>
</math></span>m<sup>2</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the carpet laid in the first room has fewer than three faults.</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 probability that exactly seven rooms will have fewer than three faults in the carpet.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 4 \times 0.5"> <mi>λ</mi> <mo>=</mo> <mn>4</mn> <mo>×</mo> <mn>0.5</mn> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 2"> <mi>λ</mi> <mo>=</mo> <mn>2</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}}(X \leqslant 2) = 0.677"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>⩽</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.677</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="Y \sim B(10,{\text{ }}0,677)"> <mi>Y</mi> <mo>∼</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mn>10</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0</mn> <mo>,</mo> <mn>677</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y = 7) = 0.263"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>=</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.263</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for clear recognition of binomial distribution.</p>
<p> </p>
<p><strong><em>[3 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 class="question" style="tab-stops: 1.0cm 36.0pt 72.0pt 108.0pt 144.0pt 180.0pt 216.0pt 252.0pt 288.0pt 324.0pt 360.0pt 396.0pt 432.0pt 468.0pt;">The hens on a farm lay either white or brown eggs. The eggs are put into boxes of six. The farmer claims that the number of brown eggs in a box can be modelled by the binomial distribution, B(6, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>)<em>.</em> By inspecting the contents of 150 boxes of eggs she obtains the following data.</p>
<p class="question" style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXgAAABcCAYAAABk164yAAATGElEQVR4Ae2dv44VRxPFl088BUIIef0MBAgTELB+AAKbiAhpiS2TOHSyiNhIREReB34Am4CAtQh4Bq+FEOI19tNv4IyL2rl378z0zO07Pi3d7f/V1ad7q3uqq2cunZ2dne3ZGQEjYASMwOIQ+N/ieuQOGQEjYASMQIOABbwnghEwAkZgoQhYwC90YN0tI2AEjMDlDMGlS5dykuNGwAgYASOwAwjkI9VzAp4+5EI70K/Fs8jCW+O4mK9+U69GvGrkCVTNV/+5lWtYRZMRcdwIGAEjsBAELOAXMpDuhhEwAkYgI2ABnxFx3AgYASOwEAQs4BcykO6GETACRiAjYAGfEXHcCBgBI7AQBCzgFzKQ7oYRMAJGICNgAZ8RcdwIGAEjsBAEJhPw2LDq9/XXX7dw/fPPP206+b/99lubVyIQ6X/77bclSA6mQfvCYNu8DO6EKxoBI7CzCEwm4LmUc3h42ABzenq6JwH31VdfNRd2Dg4OGv+7774rCh70j4+Pi9IcQuzJkyd7f/75597R0dEeff3777+HkHEdI2AEjMBgBCYT8OIIAYdD2CH05O7cuaPgIv1Xr161/frjjz8s4Fs0HDACRmAuBCYX8D/++GOzg6VDjx8/PqeSQehLjYG6hp/iWhBQ8Sgt5v/111/NoqE84tlFNQnqG7moyokqJLWFrzIxX/XxI+/wIPfo0aNmQSNOn9UP5ds3AkbACMyBwOQCnk6wg93f32/68/333zeCU52LCwBpqGxQaUT34sWLNvr69etWBfPNN9806VLJ/Pzzz205Ajw1/PTTT3uoiHDiQWGpiRDgCGVcbOvu3buNiqXJSH9YaBDetI06CtpaCH755ZdWPcUTDH20MwJGwAjMjcAsAp5OvXz5su1bFLRt4oYBhOfVq1eb0tBZJzwR4Ldu3dpDL69FA8GsHbXUROQ/e/bsCw5YFNCbQ79Lf87CgRMvCHfqQN/OCBgBI1ADArMJeIToyclJ2+fnz5+34VKBLkG8jjY7cFQrEu5RhTN0EXr//v26Jp1nBIyAEZgNgdkEPD1iNy11ymw9TA1px00yVj6oV/RjERrrrl27NpaE6xsBI2AEiiAwq4CHY3TsMp9UD6JgZReN7ryE045eNKWyuXfvXkOenbt27ejgFSZTevtVfEjf/+HDh6aI2ipt9rmq/aHp8WC461B6KN0S9eJZSAl6JWjoAB+/JrwiX3HelujzWBqoKXUeNZZWifqMW8SrNjWqDEFkSl6izy2Ns+Q+fe8jJQ6IQke/4+PjcxQODw/btNPT07bs/v7+2cHBQRsnjzTRop7C+LEs8aOjozPaw+enspSLjjLKwxePua1YJ4cjfWjIZR4pN9ZF+kNp0Uf6JwdN8B3jSvAVxz/Oi23zFbHSWNeAV+Yrz+2+uJUYQ7WpsYw8Kq+vX4qvUnNK/Jfi6+TkpJFBpfjr4utfqfSZ+65C6pj97SFQYlz4p9NCRk8QWmMnVwm+hCqCaiw/ojWWL/75sjCHZsRPbfXxx/JFW5Ev+KxJwMML86oWAQ9WJTZYcYxLjSF0GL9Srouv2VU07aODA7MiwGM8aqcbN2607XJeEK2b2gwHWuurCAUH7/EMJ+bNGY4qTUyFsSyrwaHmrIUX4fH06dPGnBkVTU2qLPjChJrxg7epVFoW8JoJC/c/fvzY9DAKB4TVRWcNC4dl4+4hHPgnxFCgBocwRTDU4tBr3759uzFJroUn+Pjhhx8aIwqMO1igdd9l2zxy/sdtdxl48H84BW8W8Nseabe/Ewiw46ppdwovCAcMBxBc29yd0jYXEGs0MNCGBt4QogjWbR+Wq30ugMqxm5e5ttJK+BbwJVDcARpXrlxpuIyCAAugofb+O9DlYizWujulgxISekIr1ukehFj8EE6yVOF+CcK0NrUIwh4LPlm+9eji5EVv3rw5SRsW8JPAWh9RJjfCPAoCLmXxOga71Qiw26p1dyqut71I62lC6gZ2o/BEXDto8VqDv+1zFKn5tJMXJrptr3gJ3wK+BIo7QgPbfdnvs5Nnp4WO0q4bAf4BwSuqZqbQk3a3vlmqbM4lNDar9d8sJYFaA1Y8STx48KAdCMIx3maMDWQTnS5Tm1xmXRwzN2iUNgFa1+Z/IW/suAgj2XOXGp9SfEFHvxpM7CJO4gt/rMkdNMY42U6LpxJmpWN5yv0BoxrGEL6EE/5Yc1L1sxReUVaOnVfqq3iUf+lzRrtOoDf7hEub1DuAtQE6ON49U8Nq2bsDFVYoMS5TdMt89UO1Rrxq5AlUzdf4uWUVTT8MXdoIGAEjsDMITCrg37x5056sR90lKzM/3r2AXkxhUFNcZaQ30/saSIdWjBNGFyk6PEGovtKJi1bX6MR3tOh1wpSL9UUz0oppCsMfOm7F8SPNrvadZgSMgBEojoB0NfJL6JfQv0Wdl3RN0jNx3Zt8yul6Or7eYyG9ourparbo6ro4daCjuOiLDnnQUHurdHDKx1dddJ3Sd0qfqPZ0vVj8EBcN8U5ZtUeeeBPOfX36UqMzX/1GpUa8auQJVM3X+Lk16Q5eH8W4f/9+szB1vQMeO16sFPCxp8Vdv379C1/psgDBbC3acyve9areaAHREO34Iz65xi+zrl9//bW1l9U1YuXxZILjnAGHjblMr3T1n7K8FZPdPJcs1n2YpCHiP0bACBiBwghMKuAv4lWC86JyEuZ6jwpC9O3bt635GpcsiCt/FT290ndVPra7qFNwtCmhrXriQwuJ7I+xLdflCfWJhQVTKF0AUd1VbTvdCBgBI1AagVkFvIRf305o54zPZQB2zuzaiUvIKt6XdiyP1Q9Phvx4osACiDTdymNHzjstdCUbwU/7vDCIb83Cm24WQhchr12+LxRFpB02AkZgDgQmFfDa1aLuwEkVoo5pZ6y4Lt28e/euSZIvFQ+J+o4qLzbCPXz4sPGl1mkiPf+IhlRAVOfgFkcaQlqCX8KdPA5gqau8KNypz6EuixA3++yMgBEwArMjkNX4JQ42dCgJLf10MKnDE6Xnw0cdbCo/1qOuDkF18JrjlNHhJzR0MCp6uT31P5cjPfMiGjpIje0oDx9HmUgz90PtbuqL7qbl5ypnvvohXSNeNfIEquZr/Nya5KLT7KvUhA2yE+/6hCCqG55QUM1kh+59k8PdXG9dnLOBT3N+Xan588xXP8xrxKtGnkDVfI2fW5OqaPqxV19pVDCokaSCkQ+nWM4g3KPennyE+xh1UX0omCMjYAR2FQEL+DUjp3dIs5OIP3Ty6NYR5voii/IR7jaJXAOqs4yAEZgNAatoZoN6XEN+XO2Hn/HaHC9jtTlWlNwlvLyD7ze2Lm0EjIAR2BkELOB3ZqjMqBEwAkagHwKdKpp+JFzaCBgBI2AEakAgW9pd7mIqF+oq47R5Edglvd+8yHS3Zry6celKNVZdqKxOqxmvzLVVNBkRx42AETACC0HAAn4hA+luGAEjYAQyAhbwGRHHjYARMAILQcACfiED6W4YASNgBDICFvAZEceNgBEwAgtBwAJ+IQPpbhgBI2AEMgIW8BkRx42AETACC0FgMgGPrah+8UtOfLpO6fi8sbGki/T10Y6S9PvQon31ddu89OHbZY2AEVgGApMJeL06F5h4+6IEHG9hJI/P2+HHLySVgBT6fFZv2+7JkyfNe+T5mhN9zV+v2jZ/bt8IGIHlIzCZgBd0+lwdH81A6Mnp03uKL81/9epV2yU+5WcB38LhgBEwAjMhMLmA593o7GBxjx8/PqeSQehLjYG6hp/iWhBQ8Sgt5vPN01ifeHZRTYL6Ri6qcqIKSW3hq0zMV3382Db8yT169Kj9ChR9Vj+Ub98IGAEjMAcCkwt4OsEOdn9/v+kPX0GKgjYuABRAZaMFQQC8ePFCwb3Xr1+3Khg+toGTSiZ+NJt0nhr40DcqIpx4UFhqIgQ4QhkX27p79+7KD2az0CC8aRtVE7S1EPC5Pj4GguMJxh8AaaDwHyNgBGZGYBYBT59evnzZdi0K2jZxwwDC8+rVq01p6KwTngjwW7duNV9f0qKBYNaOWmoi9PbPnj37ggMWBdQq0O9Sr7Bw4MQLwp060LczAkbACNSAwGwCHiHK90vlnj9/rmAxv0sQryPODhzVioR7fLIYugi9f/9+XZPOMwJGwAjMhsBsAp4esZuWOmW2HqaGtOMmGTUK6hX9WITGumvXro0l4fpGwAgYgSIIzCrg4Rgdu/TT6kEUrOyi0Z2XcNrRi6ZUNvfu3WvIs3PXrh0dvMJkSm+/ig/p+z98+NAUUVulzT5XtT8kPR8K6+B6CK2xdcBb7UtlJprxIF3nGsqb2o+H8l3qtsh3PFifmi/oCy98nRmpXXAifU68IlY1jaEwyT7jOSc+uf0cj/+PXQYiufyg+Flye3tsaMc76Oh3fHx8juDh4WGbdnp62pbd398/Ozg4aOPkkSZa1FMYP5YlfnR0dEZ7+PxUlnLRUUZ5+OIxtxXr5HCkDw25zCPlxrpIfyitzAd9juMwhO4QvuBDeGvs8XEnJydf8KT50Je3IXyBBe2Ljy4a28ALfsBLvGUsmNsaW8rluZ7L53hXP3OZHO/CSmO6zTHMfCquecZ8GuuG4JXbBKvICzT1P5DLbhrv4utfqfSZSlehTRtwuekQGDsuXZMHQbBKaGzakyF8ZV6Y6EqTkFD7xIe0MaSOeFDbmQZCNJdR2U39THPTeqsWFsYv0wTPPuOa62/CUx4n5pJ4zHlzjuEq3rUIRqG6quxF6UPwyjThI+LE3BJ+ueym8S6+ZlfRDHrMcKXRCEQ1GMRQR6FW4lxkbhd5QdXw8OHDxtIJPrKKizOToQfeffsV+UL9kM+LOJSHF/LmdKgWUCeigslqozdv3pwzK0YNQfqULo8T2F2/fr1pMufNOYZdfWaOYX1Xi+N/DxXwjRs3WpY4u4uWhm3GyIAF/EgAd7X627dv97Dz35ZD54jAkgXTKj4QVDrvWFWmZDoCFL7yORD/lDqMpz3KTKY3TR1CENA2QoF7JFGP/O7du3ZxVDWELelzOoTTKpPluccw9pvxvH379jmMYpm5wx8/fmyajBsKFsGLzv2G8GkBPwS1BdThQtf9+/e31hOeHBBa7JLX3fbllQ95Rzgl07QFX1xQQ5hqxxz/Gbm4R/6DBw+mZKWlrbbx4Q2XDzXbwlsIgNG6RXjuMRQELMpcjJxz/qjtWnwL+FpGYkY+tqmeyd3knw+rqq4dJ6oQhOk2HLtRrK5W3Wsgf4od1yZ9RZgKL9QijGd0xKUuielThC8Sotscw6dPn7aqLZ642EgwZoQzZlNgs4rmlStXmqzIA9Z4U6giLeBXjcKC03///fdG711LF7uEEXpT3RbeFp/aOa9qf4p/yFVt5XRhdvPmzXM3rTlbIX0Oh5pvlX5722MIXzzx6MdTF2NG/KKxnRI72oYPqWpoi43EJCrTfELbdRKbyzg+PwIlxyVarYztSQm+oBGtU7AmiBYG5MkMcFN+x/JFm+towGMfSxXxvY6mylzkZ0sQLESEFzgR7+OG8pTrRSujGsYwYwB/GbtcZpN47vcmdXIZxkvjpLkW/wdy+U3iXXyxmn3hugp9UWCmCIMBL/w0eWdquspmSo0LQkkTq0RHh/DFP5nGNtfPeSrXd/Jnupv0VW3hZ0HAHIz5Q+fkEL4Yr9h2V1+EW+a7q2xO68uTTDMjT4Q1r8RLzp9jDHPfYrwmAQ9fUcYN2SzEvhHuGsdLnzPaJxL0U5/KtklbC6C/w5qBg7j/8kEJA1DTuMQJYb4iGheHa8SrRp485y+eS7lE1zhaB59RctwIGAEjsBAEdkLAcwDB6sQvXzLhIEd50T6YsNKpwy/XVz6+nOyzlSczOeXbNwJGwAjsCgI7IeB5tTBqI07BUdkg1HHYAnNRhtcQk48JlBYALAn0DnhOrfXhD5ndsQDoRB0zPdXDtllvmaQ9OyNgBIzAziKwiaI+l5krrsMlHWjptFmHCTrE0eGNDnd0YBHL6wAI3nVgpnc/6LCD8mqTtJqc+lwTT/BivvqNSI141ciT51a/ebUKr53YwfddPfUKX3buejVxl92r3u/BBQgcdqns8Nn5kxZVN315cHkjYASMwLYR2EkBf9EFk/hRD10IQZjHm2MAjyBnE6qfXryFkNfXp6S62fZAuX0jYASMQF8EdkLA67o4NzBxeu+FdOS6EYYOHuEvQa0DUpXTTTGZXKLP1wuj0OcTRjfPQgAN7f77guryRsAIGIEqEMianpr0cejD0YtLtw5v0seLb/TopPOLlzxUBxoKqxx182UN6dyhF8tLv6/2tuXDe43OfPUblRrxqpEnUDVf4+dW1RedqlgBK2GC84BPc74Shj6zYb76jUeNeNXIE6iar/FzaydUNP266dJGwAgYASMAAhbwngdGwAgYgYUiYAG/0IF1t4yAETACFvCeA0bACBiBhSJgAb/QgXW3jIARMAKdVjSGxQgYASNgBHYPgWxpdzl3IRfI+Y4bASNgBIzAbiBgFc1ujJO5NAJGwAj0RsACvjdkrmAEjIAR2A0E/g/fWYLNevBqPgAAAABJRU5ErkJggg=="></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that this data leads to an estimated value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 0.4">
<mi>p</mi>
<mo>=</mo>
<mn>0.4</mn>
</math></span>.</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 class="indent1">Stating null and alternative hypotheses, carry out an appropriate test at the 5 % level to decide whether the farmer’s claim can be justified.</p>
<div class="marks">[11]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>from the sample, the probability of a brown egg is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0 \times 7 + 1 \times 32 + \ldots }}{{6 \times 150}} = \frac{{360}}{{900}} = 0.4">
<mfrac>
<mrow>
<mn>0</mn>
<mo>×</mo>
<mn>7</mn>
<mo>+</mo>
<mn>1</mn>
<mo>×</mo>
<mn>32</mn>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mrow>
<mn>6</mn>
<mo>×</mo>
<mn>150</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>360</mn>
</mrow>
<mrow>
<mn>900</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.4</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 0.4">
<mi>p</mi>
<mo>=</mo>
<mn>0.4</mn>
</math></span> <em><strong>AG</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>if the data can be modelled by a binomial distribution with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 0.4">
<mi>p</mi>
<mo>=</mo>
<mn>0.4</mn>
</math></span>, the expected frequencies of boxes are given in the table</p>
<p><img src="data:image/png;base64,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"> <em><strong>A3</strong></em></p>
<p><strong>Notes: </strong>Deduct one mark for each error or omission.<br>Accept any rounding to at least one decimal place.</p>
<p>null hypothesis: the distribution is binomial <em><strong>A1 </strong></em></p>
<p>alternative hypothesis: the distribution is not binomial <em><strong>A1 </strong></em></p>
<p>for a chi-squared test the last two columns should be combined <em><strong>R1 </strong></em></p>
<p><img src="data:image/png;base64,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"></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\chi _{{\text{calc}}}^2 = \frac{{{{\left( {7 - 7} \right)}^2}}}{7} + \frac{{{{\left( {32 - 28} \right)}^2}}}{{28}} + \ldots = 6.05">
<msubsup>
<mi>χ</mi>
<mrow>
<mrow>
<mtext>calc</mtext>
</mrow>
</mrow>
<mn>2</mn>
</msubsup>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>7</mn>
<mo>−</mo>
<mn>7</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>7</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>32</mn>
<mo>−</mo>
<mn>28</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>28</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mo>…</mo>
<mo>=</mo>
<mn>6.05</mn>
</math></span> (Accept 6.06) <em><strong>(M1)A1</strong></em></p>
<p>degrees of freedom = 4 <em><strong>A1 </strong></em></p>
<p>critical value = 9.488 <em><strong>A1 </strong></em></p>
<p>Or use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>-value</p>
<p>we conclude that the farmer’s claim can be justified <em><strong>R1</strong></em></p>
<p><em><strong>[11 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 number of taxis arriving at Cardiff Central railway station can be modelled by a Poisson distribution. During busy periods of the day, taxis arrive at a mean rate of 5.3 taxis every 10 minutes. Let T represent a random 10 minute busy period.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that exactly 4 taxis arrive during T.</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 most likely number of taxis that would arrive during T.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.</p>
<p>Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \sim {\text{Po}}\left( {5.3} \right)">
<mi>X</mi>
<mo>∼</mo>
<mrow>
<mtext>Po</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>5.3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X = 4} \right) = {{\text{e}}^{ - 5.3}}\frac{{{{5.3}^4}}}{{4{\text{!}}}}">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>5.3</mn>
</mrow>
</msup>
</mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mn>5.3</mn>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 0.164 <em><strong>A1</strong></em></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><strong>METHOD 1</strong></p>
<p>listing probabilities (table or graph) <em><strong>M1</strong></em></p>
<p>mode <em>X</em> = 5 (with probability 0.174) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M0A0</strong></em> for 5 (taxis) or mode = 5 with no justification.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>mode is the integer part of mean <em><strong>R1</strong></em></p>
<p>E(<em>X</em>) = 5.3 ⇒ mode = 5 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not allow <em><strong>R0A1</strong></em>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt at conditional probability <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{P}}\left( {X = 7} \right)}}{{{\text{P}}\left( {X \geqslant 6} \right)}}">
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>7</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>⩾</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> or equivalent <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \frac{{0.1163 \ldots }}{{0.4365 \ldots }}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.1163</mn>
<mo>…</mo>
</mrow>
<mrow>
<mn>0.4365</mn>
<mo>…</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>= 0.267 <em><strong> A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p>the possible arrivals are (2,0), (1,1), (0,2) <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y \sim {\text{Po}}\left( {0.65} \right)">
<mi>Y</mi>
<mo>∼</mo>
<mrow>
<mtext>Po</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.65</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt to compute, using sum and product rule, <em><strong>(M1)</strong></em></p>
<p>0.070106… × 0.52204… + 0.026455… × 0.33932… + 0.0049916… × 0.11028… <em><strong>(A1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for one correct product and <em><strong>A1</strong> </em>for two other correct products.</p>
<p>= 0.0461 <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>recognising a sum of 2 independent Poisson variables <em>eg Z</em> = <em>X</em> + <em>Y</em> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 5.3 + \frac{{1.3}}{2}">
<mi>λ</mi>
<mo>=</mo>
<mn>5.3</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mn>1.3</mn>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p>P(<em>Z</em> = 2) = 0.0461 <em><strong>(M1)A3</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<p> </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 age, <em>L</em>, in years, of a wolf can be modelled by the normal distribution <em>L</em> ~ N(8, 5).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a wolf selected at random is at least 5 years old.</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>Eight wolves are independently selected at random and their ages recorded.</p>
<p>Find the probability that more than six of these wolves are at least 5 years old.</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>P(<em>L</em> ≥ 5) = 0.910 <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><em>X</em> is the number of wolves found to be at least 5 years old recognising binomial distribution <em><strong>M1</strong></em></p>
<p><em>X</em> ~ B(8, 0.910…)</p>
<p>P(<em>X</em> > 6) = 1 − P(<em>X</em> ≤ 6) <em><strong>(M1)</strong></em></p>
<p>= 0.843 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for finding P(<em>X</em> ≥ 6).</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>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}}(A) = k,{\text{ P}}(B) = 3k,{\text{ P}}(A \cap B) = {k^2}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>k</mi>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mi>k</mi>
<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>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = 0.5">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0.5</mn>
</math></span>.</p>
</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="k">
<mi>k</mi>
</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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B)">
<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">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 <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="0.5 = k + 3k - {k^2}">
<mn>0.5</mn>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
<mi>k</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - 4k + 0.5 = 0">
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>0.5</mn>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.129">
<mi>k</mi>
<mo>=</mo>
<mn>0.129</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award the final <strong><em>A1 </em></strong>if two solutions are given.</p>
<p> </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>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B) = {\text{P}}(B) - {\text{P}}(A \cap B)">
<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>
<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> or alternative <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A' \cap B) = 3k - {k^2}">
<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>
<mn>3</mn>
<mi>k</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.371">
<mo>=</mo>
<mn>0.371</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 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="question">
<p>The mean number of squirrels in a certain area is known to be 3.2 squirrels per hectare of woodland. Within this area, there is a 56 hectare woodland nature reserve. It is known that there are currently at least 168 squirrels in this reserve.</p>
<p>Assuming the population of squirrels follow a Poisson distribution, calculate the probability that there are more than 190 squirrels in the reserve.</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><em>X</em> is number of squirrels in reserve<br><em>X</em> ∼ Po(179.2) <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> A1</strong></em> if 179.2 or 56 × 3.2 seen or implicit in future calculations.</p>
<p>recognising conditional probability <em><strong>M1</strong></em></p>
<p>P(<em>X</em> > 190 | <em>X</em> ≥ 168)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{P}}\left( {X > 190} \right)}}{{{\text{P}}\left( {X \geqslant 168} \right)}} = \left( {\frac{{0.19827 \ldots }}{{0.80817 \ldots }}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mn>190</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>⩾</mo>
<mn>168</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>0.19827</mn>
<mo>…</mo>
</mrow>
<mrow>
<mn>0.80817</mn>
<mo>…</mo>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)(A1)</strong></em></p>
<p>= 0.245 <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>