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<h2>HL Paper 1</h2><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right)"> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right) = {x^3} + 3{x^2} + 8x - 24"><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>24</mn></math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}"> <mi>x</mi> <mo>∈<!-- ∈ --></mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the remainder when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is divided by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 2} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the remainder when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is divided by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 3} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( x \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> has only one real zero.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the transformation that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 8{x^3} + 12{x^2} + 16x - 24"><mi>y</mi><mo>=</mo><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mo>−</mo><mn>24</mn></math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> follows a Poisson distribution with a mean of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6{\text{P}}\left( {X = 3} \right) = 3{\text{P}}\left( {X = 2} \right) - 2{\text{P}}\left( {X = 1} \right) + 3{\text{P}}\left( {X = 0} \right)">
<mn>6</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the time, in days, from December <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mtext>st</mtext></math> and the percentage of Christmas trees in stock at a shop on the beginning of that day.</p>
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"></p>
<p>The following table shows the natural logarithm of both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> on these days to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> decimal places.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the data in the second table to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> for the regression line, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>m</mi><mo>(</mo><mi>ln</mi><mo> </mo><mi>d</mi><mo>)</mo><mo>+</mo><mi>b</mi></math>.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming that the model found in part (a) remains valid, estimate the percentage of trees in stock when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>25</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The weights of apples from Tony’s farm follow a normal distribution with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>158</mn><mtext> g</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn><mtext> g</mtext></math>. The apples are sold in bags that contain six apples.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the mean weight of a bag of apples.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the standard deviation of the weights of these bags of apples.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a bag selected at random weighs more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A manufacturer of chocolates produces them in individual packets, claiming to have an average of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>85</mn></math> chocolates per packet.</p>
<p>Talha bought <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> of these packets in order to check the manufacturer’s claim.</p>
<p>Given that the number of individual chocolates is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, Talha found that, from his <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> packets, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Σ</mtext><mi>x</mi><mo>=</mo><mn>2506</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Σ</mtext><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>209</mn><mo> </mo><mn>738</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an unbiased estimate for the mean number <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>μ</mi><mo>)</mo></math> of chocolates per packet.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the formula <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>s</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><mfrac><mrow><mtext>Σ</mtext><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mfenced><mrow><mtext>Σ</mtext><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mi>n</mi></mfrac></mstyle></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> to determine an unbiased estimate for the variance of the number of chocolates per packet.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>95</mn><mo>%</mo></math> confidence interval for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math>. You may assume that all conditions for a confidence interval have been met.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Suggest, with justification, a valid conclusion that Talha could make.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A factory, producing plastic gifts for a fast food restaurant’s Jolly meals, claims that just <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>%</mo></math> of the toys produced are faulty.</p>
<p>A restaurant manager wants to test this claim. A box of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn></math> toys is delivered to the restaurant. The manager checks all the toys in this box and four toys are found to be faulty.</p>
</div>
<div class="specification">
<p>The restaurant manager performs a one-tailed hypothesis test, at the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> significance level, to determine whether the factory’s claim is reasonable. It is known that faults in the toys occur independently.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Identify the type of sampling used by the restaurant manager.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the null and alternative hypotheses.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value for the test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the conclusion of the test. Give a reason for your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Saloni wants to find a model for the temperature of a bottle of water after she removes it from the fridge. She uses a temperature probe to record the temperature of the water, every 5 minutes.</p>
<p><img src="data:image/png;base64,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"></p>
<p>After graphing the data, Saloni believes a suitable model will be</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T = 28 - a{b^t}">
<mi>T</mi>
<mo>=</mo>
<mn>28</mn>
<mo>−<!-- − --></mo>
<mi>a</mi>
<mrow>
<msup>
<mi>b</mi>
<mi>t</mi>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{,}}\,\,b \in {\mathbb{R}^ + }">
<mi>a</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="28 - T">
<mn>28</mn>
<mo>−</mo>
<mi>T</mi>
</math></span> can be modeled by an exponential function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the least squares exponential regression curve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="28 - T">
<mn>28</mn>
<mo>−</mo>
<mi>T</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coefficient of determination, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{R^2}">
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Interpret what the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{R^2}">
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> tells you about the model.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence predict the temperature of the water after 3 minutes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A newspaper vendor in Singapore is trying to predict how many copies of <em>The Straits Times</em> they will sell. The vendor forms a model to predict the number of copies sold each weekday. According to this model, they expect the same number of copies will be sold each day.</p>
<p>To test the model, they record the number of copies sold each weekday during a particular week. This data is shown in the table.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>A goodness of fit test at the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> significance level is used on this data to determine whether the vendor’s model is suitable. The critical value for the test is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>49</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an estimate for how many copies the vendor expects to sell each day.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the null and alternative hypotheses for this test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the degrees of freedom for this test.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the conclusion to the test. Give a reason for your answer.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>Two unbiased tetrahedral (four-sided) dice with faces labelled 1, 2, 3, 4 are thrown and the scores recorded. Let the random variable <em>T</em> be the maximum of these two scores.</p>
<p>The probability distribution of <em>T</em> is given in the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>a</em> and the value of <em>b</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the expected value of <em>T</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The cars for a fairground ride hold four people. They arrive at the platform for loading and unloading every <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> seconds.</p>
<p>During the hour from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> am the arrival of people at the ride in any interval of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> minutes can be modelled by a Poisson distribution with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mi>t</mi><mo> </mo><mfenced><mrow><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mn>60</mn></mrow></mfenced></math>.</p>
<p>When the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> am car leaves there is no one in the queue to get on the ride.</p>
<p>Shunsuke arrives at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>01</mn></math> am.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> people arrive at the ride before Shunsuke.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability there will be space for him on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>01</mn></math> car.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Chloe and Selena play a game where each have four cards showing capital letters A, B, C and D.<br>Chloe lays her cards face up on the table in order A, B, C, D as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_14.39.35.png" alt="N17/5/MATHL/HP1/ENG/TZ0/10"></p>
<p>Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Chloe’s card directly above.<br>Chloe wins if <strong>no</strong> matches occur; otherwise Selena wins.</p>
</div>
<div class="specification">
<p>Chloe and Selena repeat their game so that they play a total of 50 times.<br>Suppose the discrete random variable <em>X </em>represents the number of times Chloe wins.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability that Chloe wins the game is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{8}"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the mean of <em>X</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the variance of <em>X</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A company produces bags of sugar with a labelled weight of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mtext>kg</mtext></math>. The weights of the bags are normally distributed with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mtext>kg</mtext></math> and a standard deviation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mtext>g</mtext></math>. In an inspection, if the weight of a randomly chosen bag is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>950</mn><mo> </mo><mtext>g</mtext></math> then the company fails the inspection.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the company fails the inspection.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A statistician in the company suggests it would be fairer if the company passes the inspection when the mean weight of five randomly chosen bags is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>950</mn><mo> </mo><mtext>g</mtext></math>.</p>
<p>Find the probability of passing the inspection if the statistician’s suggestion is followed.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The discrete random variable <em>X</em> has the following probability distribution, where<em> p</em> is a constant.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>p</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>μ</em>, the expected value of <em>X</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find P(<em>X</em> > <em>μ</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The sex of cuttlefish is difficult to determine visually, so it is often found by weighing the cuttlefish.</p>
<p>The weights of adult male cuttlefish are known to be normally distributed with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mtext>kg</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<p>The weights of adult female cuttlefish are known to be normally distributed with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo> </mo><mtext>kg</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<p>A zoologist uses the null hypothesis that in the absence of information a cuttlefish is male.</p>
<p>If the weight is found to be above <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>kg</mtext></math> the cuttlefish is classified as female.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>%</mo></math> of adult cuttlefish are male.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of making a Type I error when weighing a male cuttlefish.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of making a Type II error when weighing a female cuttlefish.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of making an error using the zoologist’s method.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider two events, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( A \right) = {\text{P}}\left( {A' \cap B} \right) = 0.4">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩<!-- ∩ --></mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.4</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cap B} \right) = 0.1">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∩<!-- ∩ --></mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By drawing a Venn diagram, or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {A \cup B} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are not independent.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>It is believed that the power <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> of a signal at a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> km from an antenna is inversely proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>d</mi><mi>n</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<p>The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is recorded at distances of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi mathvariant="normal">m</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><mi mathvariant="normal">m</mi></math> and the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>d</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>P</mi></math> are plotted on the graph below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The values of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>d</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>P</mi></math> are shown in the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why this graph indicates that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is inversely proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>d</mi><mi>n</mi></msup></math>.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the least squares regression line of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>P</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>d</mi></math>.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your answer to part (b) to write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> to the nearest integer.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of fish that can be caught in one hour from a particular lake can be modelled by a Poisson distribution.</p>
<p>The owner of the lake, Emily, states in her advertising that the average number of fish caught in an hour is three.</p>
<p>Tom, a keen fisherman, is not convinced and thinks it is less than three. He decides to set up the following test. Tom will fish for one hour and if he catches fewer than two fish he will reject Emily’s claim.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State a suitable null and alternative hypotheses for Tom’s test.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of a Type I error.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The average number of fish caught in an hour is actually 2.5.</p>
<p>Find the probability of a Type II error.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="indent2" style="margin-top:12.0pt;">The number of cars passing a certain point in a road was recorded during 80 equal time intervals and summarized in the table below.</p>
<p class="indent2" style="margin-top:12.0pt;"><img style="display: block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p class="indent2" style="margin-top:12.0pt;">Carry out a <span class="mjpage"><math alttext="{\chi ^2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> </math></span> goodness of fit test at the 5% significance level to decide if the above data can be modelled by a Poisson distribution.</p>
</div>
<br><hr><br><div class="specification">
<p>A zoologist believes that the number of eggs laid in the Spring by female birds of a certain breed follows a Poisson law. She observes 100 birds during this period and she produces the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The zoologist wishes to determine whether or not a Poisson law provides a suitable model.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Calculate the mean number of eggs laid by these birds.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Write down appropriate hypotheses.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Carry out a test at the 1% significance level, and state your conclusion.</p>
<div class="marks">[14]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Product research leads a company to believe that the revenue (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span>) made by selling its goods at a price (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>) can be modelled by the equation.</p>
<p style="text-align: center;"><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R\left( p \right) = cp{{\text{e}}^{dp}}">
<mi>R</mi>
<mrow>
<mo>(</mo>
<mi>p</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>p</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>d</mi>
<mi>p</mi>
</mrow>
</msup>
</mrow>
</math></span></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}">
<mi>d</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span></p>
<p>There are two competing models, A and B with different values for the parameters <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>. </p>
<p>Model A has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> = 3, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> = −0.5 and model B has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> = 2.5, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> = −0.6.</p>
<p>The company experiments by selling the goods at three different prices in three similar areas and the results are shown in the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The company will choose the model with the smallest value for the sum of square residuals.</p>
<p>Determine which model the company chose.</p>
</div>
<br><hr><br><div class="specification">
<p>Mr Burke teaches a mathematics class with 15 students. In this class there are 6 female students and 9 male students.</p>
<p>Each day Mr Burke randomly chooses one student to answer a homework question.</p>
<p>In the first month, Mr Burke will teach his class 20 times.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability he will choose a female student 8 times.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The Head of Year, Mrs Smith, decides to select a student at random from the year group to read the notices in assembly. There are 80 students in total in the year group. Mrs Smith calculates the probability of picking a male student 8 times in the first 20 assemblies is 0.153357 correct to 6 decimal places.</p>
<p>Find the number of male students in the year group.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A company sends a group of employees on a training course. Afterwards, they survey these employees to gather data on the effectiveness of the training. In order to test the reliability of the survey, they design two sets of similar questions, which are given to the employees one week apart.</p>
</div>
<div class="specification">
<p>The questions in the survey were grouped in different sections. The mean scores of the employees on the first section of each survey are given in the table.</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT8AAABICAYAAACA9vJmAAAVVklEQVR4Ae2dzYsdxdfHOw/PWonuXIgYN2IgIqNughBBR6JLYeLLTjFGRBDBaDS4UKMxoiDiG4IiSBSzEhSNEMG4SQwSQXARRkRcjvHtD8iPT+v3cqamuvt03+7bM3NPQVPdVafOy/ecqq7q29V3y4ULFy4UkQKBQCAQmDME/m/O7A1zA4FAIBAoEYjBLwIhEAgE5hKBGPzm0u1hdCAQCMTgFzEQCAQCc4lADH5z6fYwOhAIBP4/hWDLli1pUVwHAoFAILAhEah7mWXN4IeFdQ02JAIjKM1NZEwc510+Lp93DML++olcLHtHGBhDZCAQCIyPQAx+4/sgNAgEAoEREIjBbwTQQ2QgEAiMj8CGG/yuuuqq8lnOt99+Oz56oUEgEAhsWAQ6DX4agHigmh5DD0qnTp3asGDXKf7222+XWP788891ZL3W5fx4ySWXFHv27Cl+//33XmVVMUPOQw89VCCXWCLnelbycxjcdtttxUcffVSlcq/lOfnqU0P3JQwZWz46VMXA4H2BDxvY9O+PvbZk7fnJkyf5GMIFcqWVlZULCwsLq8pU13eeyu6bfx/8PDhaOdu2bbuwdevWC2+99ZYt7nzukZ/z42effVb6dmlpqbNsGnrkEzPYTdwsLy+X8rC/L/96dEgxQI/Dhw+XOuzbt29wDFL5COyrL3WxH/mKAdu/uwDhkS9bbQzkMBlCfvZVly63tksvvbT47rvvujSd+zaff/55ceutt5Y4vPvuu8XevXtHw2T37t3FwsLCTOQfPHiwOH/+fMFsnvghYTsYjJWuvPLK4vHHHy/F79+/v7jpppvKmfAs9Rm7L80yBl555ZVieXm5OHfu3CQGdu7cWezbt29wyDste1Otfvjhh4Klgk2aTtOxWcYwlacMWspU/9RTT02aXX/99SUdSw7KtQxiSViX4MlSTcsF+FBG0nKSOvGxZaJ76aWXJjrZ9vBIp+V9Lws/+OCD4u677y7uuOOO4syZM8Xg0/0KMLETbAjGJ598soKqv+Ljx4+Xg74GPnHmJkoHGDPdd999pfhvvvlmpmoQj2lfmqkC/wmblQ8+/vjjbAy88cYbw8dAOp30TFU1LYVWx+Li4ipWoqGcpYRd4mhpp+UFdUrwo42m3FoGnT17ViRrlkUsnViiwIeDc5aQ4ssSjmm1TciQHkePHi3pJePAgQPlckz08KO9+NO2aUnkwRH+YIP+Sn0tfT3y5SP5kBz5LHumTR750BADQyWPDsJA8WZ1oT2+7prayIdWxzQyra5t5Mt++oCnnZVTde7hA81QMdAkn10Iq1JTA4jTgAGw1GEpDe2gsXQ5mhwYdEgLEDRylp5PaKBDDufQMKhZffVcKR1wGNgY8JSopz12iZcdEDRYij6X096TkKtBGHoNtJ62dTQe+Sn+2KqbjbW3Tk5VnUc+NNavVby6lnt0SDGwsmhv49XWec67yM/1JY+sHE0b+dDaI8evbZlHPjRpDDAZoJx+P01qkt/LsnfHjh3FF198Mdjs/IYbbqjk/eOPP5Z1dumk819//bWsYwnFc6yXX365vCZ/7rnnJjxZah46dGiybN62bVtZ988//xQ//fRTeX777bdP6u+6667ijz/+mLSf5uTNN98sHnzwwQlvrsda+oIbz9wWFxeL1157bRqz3G1/+eUXN+0sCfVr88033zxLscXQfanKmJMnT5bbMc+ePVtFMlj5999/v4q33uh44YUXVpX3fdHL4CeleLY3xM/zp0+fLi6++GKJWZVffvnl5bWClQudb9++fUL72GOPla8v8EyFZ008t1NisDt8+HDpfH6o1MGgedlll5VkCg7V/fuDpjh0y3m2yQ8dlufKykrJ7KuvvurGdAO1Wlpayr5SwjPXIeKoDTTHjh0rybkRjJGG6ktNtjD49hHbTXJUTwzQH9VnKdfk5ZprrhHZIHlvgx+DSl/vRp04cWICBj9EkG655ZYyF0i//fbbpHzr1q0FvxxSx8E5A9qNN95Y0tj2u3btWjXro+6BBx4oXnzxxUmHwxYNjvz6Rwd49NFHJz9E8ION/aFmIqTlydNPP11cd911q1rheOQdOXJkVfksLsCOHzy+/PLLYhYznocffricQdv3+pDPzHesxI9NxByz8QMHDpQzsVnr0mdfmrXubeUpBtR/ad/XONKoS7qmblonQ681ObTpoWdxotG6nedkotUzLuoosw/8ueYZnOo4t8+fLF+ej5DIoRN/fuBQnbUv/SFDdTzr0o8k8OA5j23POTzFH1ra1CVo65K1Q88moYe35Fhc6njl6prk00Y6SB45ZekzmBz/pjKPfHjgW+s75OOnJnyb5FPv0SGHAf63MeeRlaPpKp92HOpLOd6esjby6W825j38m2g88uGRxgB9gBgY2v4tCLcjJK+EJEW2evBz5LME1btWfQpkhjGr97bWA45j+3FM+cRN+GD8vjxmDDT5v7eXnPscpPrkxRKG54IXXXRRuWzl/aFIgUAgEAisq8FPz9l4s54BS9fTuolfZ3mO9uGHH07LKtoHAoHAJkEgu+zdJLaFGYFAIDDnCNQtu7Mzv7oGc46l2/ym5w1uRh0J510+sM07BmF/fMa+4/ARzQKBQGAzI9Dbe36bGaSwLRAIBDYfAjH4bT6fhkWBQCDgQCAGPwdIQTI7BHi7f+ytbetBh9khPr+SOg1+bIPihWF9epxv82kb2kaFki1F7CnFtlklvtvGQ+n0AE8ltlulWPe1/Qdb8Zu+rYg/Uz+muukanaZNDHLip5zXkqoSdkPXZ/LokPMBWxz7TmztIwbxg01Dys/FQLp1U75J8z5ioGsf6AX/dIuKZ0sK23/Y7qUtSGxF8bRLZa2Xa7ZzoX+fNnh4gaM+swUW4Mk2I7vdDT7aDki9ttnZdjkcvfLtVj1trbPbith6Jj8jhzp49yFfvHL6p2VsvdKWx7Su6tqDgUeH1Adggi4Wl5wOHvm0gw888a3FXjyHlE8M2hhQX7AxOGQMdOkDfeG/ZgOqx2HQpHsfCYaNmBi4scXTCdrY58Ex5ccg14QjwdlEA98u8umEtKvb20sH5WhKHvlezDU46FuDTbJV36cO4klOvMA7N1BZOo982cYA5E19yk9lyid1+3z7jIFUvqcP9GV/p8GPzeDp3UBG6AMGjOhK2jyuYLE0gKz69957b3J3VzDYO77uRhiPfIKLtipX56Ccc5Itq3OonC6dp809gZ/KwJaqgYeZFnaCa50d4tlFPjJoB765pHr5MUejMo98L+bEAn700g+hg3iSK34ZuOqSBwPs8swirZw+5YsvtuB3Bjb1J9XZvO8YsLw5r+sDou3L/k6DnwYsnJb7AgdldvADWALBdhrqMVQzL3hRr1EdkJWg0aAg8OWglB7nMTDahCwNhrbcnrftWLZt7twT+LYd9lR1AmyHHwcDgcXG8rDnbeXTtumOjmzrVysvPffI141NNzLsl5/FTx2S67Y+6ksH6UJOLBO3TfEErUc+toMp2GI/bbiu8nHf8i2ukm37qbWd875jwPKv6wOi69P+ToMfiqAEgUogaOCSgpSnnQRgLajUp4OU2tvgkrHkJAbCtJ2lVwdR8JBT35TUronOW+8JfMsLHbGtLjFYYHvVIGnbtpWPz+AtnC0vzsERngxGntRWPjwZUGinGxsyrU7WR2l85XTqQ4eUL3IZADzJIx8abNRsXjZzI8qlvuVbGcjGNnSyfVU0Q8eApw/0aX/nwU+AkOuupTLv4FcVwBpU4cc5hxJtcE56WBqCSQFKrs4kHrncdqxcfdsy9PMm9IOe4GpKmunmgtO2bSsfnKoGPvhS77mJSIc28tWGHN+p45PDp+qw7XLnfehg+YKBZ8anNh750NjYpS3XubZDyJeuyomBnE7UDxkDnj7Qt/2tX3Xh1QB+jreJv13s6z8t4HvnnXeWf5+IrE8++aTQ3whSpy8rs6qwh/3+X90n663e6+X8/fffL/+nFNuaEp/m6jPx+gjy+dQXX5HmlZ/0dRdeh4DO/u9JnzpYXjaOkGl9zF8JkFRm2/V5bnUQX17ruOKKKyb/qcxrKcTntIkvjuf+x4Svk9s0lHwrg3N9Qj4tHzoGmvrAIPZrxFeeu+OojlwzJN0BuVOkdwTduZjKU89yDr52BsYMrmrmhxzqmGmkd8VUvqbqdtaETJaGHFamtSM914zK8klp2lw34SheskfLHpWTq46chF3MhjwzMI98LaPhqwTeKeZcg2Wb5JGPj+Et+Vr2yt5UXlsf9aUDehHjNqF7lZ6i88gXtuJF/KXLvyHlo6P1N+eUpf1Aeso2T+6xHz7YDm2uD1A/lP1r1mZNChOoDGY4CFoOOqRVHBqCgzpygOScDgSdpriUpUElUKGBXh1D5eQ4QvLJc8+hpKNtlztHjnSVjlxbe3Ltmsrg5Ulgh7yqhONZCsKPA/o0MHNtPfItX/Ent50B3vghLcvJtGUe+fC0OnCe8yV88Qd6wBefe1IfOoC15MLPHhqwqnTxyKctOEgGtlmsh5YP3sSU7MIHObuGigHsr+sDQ9qf/Z4fy4qNnpgmz+qT9TmseBt+TBznXT4+mXcMwv76Ppj9nl+uM2+EMp5VxSfrN4KnQsdAYHwENtXgB5zxyfrxgyo0CAQ2AgKbdtk7Nvix5KhfcszCP+GDcX2w3vHPDn6zCMyQEQgEAoHA0AjUPXfPLnvrGgyt7Gbhv97vekPjPLb92De2DiF//JlnXZy3fsm5jlnUBQKBQCCwURCIwW+jeCr0DAQCgV4RiMGvVziDWSAQCGwUBGLw2yieWkd68gL5mGls+dg+pA7sae5j3/CYPhpadi/4p1tzPFty2HLGtrSqLTkpz/V8zfaZ1JaqLVZt7OiCI3iiS25LH9u72HqUq8vp5ZXPViptFcxtYaJeW59snpNpyzzyLb228aGDTV3lw6OtDtpCl27v6qqDRz6yLK46tzqgl92Clm6Bs3jZc4984imNgfTTark+4tkz75Fvt5bKdnK7hXEo/DvN/O65557i/Pnzxblz58otXEtLS8X+/fuHHuwH4c9XNXbs2FHas7KyUvA1jXvvvXcmf2QEjvw5zZkzZ0ocP/300+L48ePFwYMHV9nKzpVrr722pKv66saqBs4L5PNFkVOnTpXy9+zZU/oxnXUsLi6W9YwnOpwiGsn4WghfCfr666+LV199tfRD2mhI+ZKFHvfff3/l14mG1kG4Kt+5c2epGnrt2rWrIE6JT+r5ug79jRnitCmNgSeeeKI4dOjQKt5pH1lYWCg3ExC7faTl5eVJXKkPpl8QGgR/e6fw3i0ZmdPZUXrHTvlulGvswj575+2iOzyaUg5H7nL2QwfchdFJM4Qmnqr3yBetcmYBtEMHpVQflTflHvnIYzZb9XELZHSVT1uPDrIDHZh90ib1fVcdPPKb/Kr6dMZPjFg/yQ6be+Rbes4lj9lmVUKup793kY8PUt5D4d9p5sed4Jlnnlk1O2ImSOJuxPtN/CWdkv4aUTMKS8P341TPN730d5ha01OvMt3p+Ns6ZgvIoa3K+cYaZRyck2wZvJrS33//XZJcffXVTaRT13M3A0fdQcnfeeed4pFHHpnwfv7554vdu3dProc8+euvv0r227dvH1LMhPexY8fK7zY+++yzk7IxTogf4nfv3r1jiK+VSRyyGmE1wCyQRD86ffp0+d3L2sYtKuFNv3r99deLo0ePlquhtDnxCVYnTpwoZ+ppfR/XR44cKZh9ziSlo7tntOauwJqcEZqZSXpXoszOXjSjsHdU6uGhmQ28qNfMi+cMStDoLkc5OuqZQ0rPsxFmEzYhiztKU0JPdPLQNvHy4AgP+yzH2pXy1x05La+69sq37dGFwyZ9Wgxc4EkO5k3JIx8/4Rv7zJVr6/uu8tHPowOy7LNU2tg4hU9XHTzy9ZwRHaCnHyjWhbHiknrR1M3M1M4jH1rFFvTgn9oPDTpJPv6yPpK8NPfKVzvhnI4nKu87BteszbwKoyCAoJAGLhlBOSDaBF8LKvXpICV6eGoAkuMFCANh2s7Sy5FyDjn1noROdUswDw/ReHDERmyRruieC354yi7xb8o98i0PfIYuwtnW6Zw68IG3dFZdmnvkQ4NMdWR4cp0OwOLdRj5tPDpY+WoD1uBBnqY2Onjkp/yJCdrR4ZXAA9zlG9Hk9FMb8rbywV/+reKNr8CMOJU+VqY9byuffkr/rkt94t958LMKAhhgKHkHv3SAtO01YMGLQ4k2gJoelgbnoBOJ3AaS+KQ5dBpw07ou1x7HQ5MGGc63WEo2dB6eom9DCz7Y3xTM8IYG3k1YeeRDY/0Gf67r2nrlw6uOD/UaRKDLHalvaEPy6tAk/z92azJ7A6jyu6VZw+C/gi7yZVvqFytDq60qfETbRj4xCH3TTRXe0nHaGGz9zI/nDZvtPzx4vjjE/zN4nlvoGaOlzf2HhK3v85xnOE3/4WHl6bmgLet6zrNjz/9XWP59yucZH2OkPZDFf4VQpl9crXzO+9Qh5c11zv963mfp//zzT3vZy7nnbYK+/0cGxZv+w8Ma1xv+GqWVN43WuhNp1GUUZtagmRp8dPdmikw9sxn42hkYM7iqmR88qINnegdK5Wuqbu8YyGT2xGFlykabYwf624Tspruapc+dN+FIG+RwB5fuyETn3NRfd1vR5mTaMo98LWHASwm8Lebw0TM++RodbRu1tblHPnLgJayxLV36dJWPLh4drM7YRJs0Zrrq4JFPDICD8NRsVJhQDkZ2Zi4a+cXaYM898qGx/uacMsUZenBt9WEZbvu7lWnPPfKhlww9/rA8OIePbAUPsOgjBlsvexFO58R4lOIADKs4NDiVOnKA5ByFodMUl7J04JHh0FQZiIMkn1zAqC25dLRl6Tl6IQM90kPOTtt4r+HXlKwjoccWG4i0BwdhCQ36cm3xzsnxyGfghS49rA4KdNGkvs7JpswjHzpkyQc5+7vKb6ODbFBMoY/Ft6sOHgyw3/qB8zSe0QUd5IMcjWywuUc+slLeaewz2FododfgaOWl5x75tIEfMV2VqJdv4Mm19U9Vuyb52e/5/Ru7dqK58c5ZysZ/eNBfxkm8bjR2HI2tQ8gfNwaa8M9+z2+c7jK91PgPj+kxDA6BwLwgsKkGP5wW/+ExL6EbdgYC0yGQXfZOxzJaBwKBQCCwPhCoe/SyZuZXR7w+zAktAoFAIBCYHoHW7/lNLzI4BAKBQCAwPgIx+I3vg9AgEAgERkAgBr8RQA+RgUAgMD4CMfiN74PQIBAIBEZAIAa/EUAPkYFAIDA+AjH4je+D0CAQCARGQOB/HQ1rqo9smFgAAAAASUVORK5CYII="></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the name of this test for reliability.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State a possible disadvantage of using this test for reliability.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate Pearson’s product moment correlation coefficient for this data.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine, with a reason, if the survey is reliable.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>On Paul’s farm, potatoes are packed in sacks labelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo> </mo><mtext>kg</mtext></math>. The weights of the sacks of potatoes can be modelled by a normal distribution with mean weight <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>49</mn><mo>.</mo><mn>8</mn><mo> </mo><mtext>kg</mtext></math> and standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>9</mn><mo> </mo><mtext>kg</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a sack is under its labelled weight.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the lower quartile of the weights of the sacks of potatoes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The sacks of potatoes are transported in crates. There are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> sacks in each crate and the weights of the sacks of potatoes are independent of each other.</p>
<p>Find the probability that the total weight of the sacks of potatoes in a crate exceeds <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>500</mn><mo> </mo><mtext>kg</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Observations on <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> pairs of values of the random variables <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>,</mo><mo> </mo><mi>Y</mi></math> yielded the following results.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∑</mo><mi>x</mi><mo>=</mo><mn>76</mn><mo>.</mo><mn>3</mn><mo>,</mo><mo> </mo><mo>∑</mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>563</mn><mo>.</mo><mn>7</mn><mo>,</mo><mo> </mo><mo>∑</mo><mi>y</mi><mo>=</mo><mn>72</mn><mo>.</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>∑</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>460</mn><mo>.</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>∑</mo><mi>x</mi><mi>y</mi><mo>=</mo><mn>495</mn><mo>.</mo><mn>4</mn></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, the product moment correlation coefficient of the sample.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming that the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>,</mo><mo> </mo><mi>Y</mi></math> is bivariate normal with product moment correlation coefficient <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math>, calculate the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value of your result when testing the hypotheses <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>0</mn></msub><mo> </mo><mo>:</mo><mo> </mo><mo> </mo><mi>ρ</mi><mo>=</mo><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><msub><mi>H</mi><mn>1</mn></msub><mo> </mo><mo>:</mo><mo> </mo><mo> </mo><mi>ρ</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State whether your <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>-value suggests that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> are independent.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given a further value <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>2</mn></math> from the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math>, predict the corresponding value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>. Give your answer to one decimal place.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable <em>X</em> has a probability density function given by</p>
<p style="padding-left: 120px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \left\{ {\begin{array}{*{20}{l}} {k\sin \left( {\frac{{\pi x}}{6}} \right),}&{0 \leqslant x \leqslant \,6} \\ {0,}&{{\text{otherwise}}} \end{array}} \right.">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π<!-- π --></mi>
<mi>x</mi>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0</mn>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<mtext>otherwise</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <em>f </em>write down the mean of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <em>f </em>write down the median of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <em>f </em>write down the mode of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(0 \leqslant X \leqslant 2) = \frac{1}{4}">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>⩽</mo>
<mi>X</mi>
<mo>⩽</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state the interquartile range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P(X \leqslant 4|X \geqslant 3)">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>⩽</mo>
<mn>4</mn>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>X</mi>
<mo>⩾</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of coffees sold per hour at an independent coffee shop is modelled by a Poisson distribution with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>22</mn></math> coffees per hour.</p>
<p>Sheila, the shop’s owner wants to increase the number of coffees sold in the shop. She decides to offer a discount to customers who buy more than one coffee.</p>
<p>To test how successful this strategy is, Sheila records the number of coffees sold over a single <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>-hour period. Sheila decides to use a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> level of significance in her test.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the null and alternative hypotheses for the test.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Sheila will make a type I error in her test conclusion.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sheila finds <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>126</mn></math> coffees were sold during the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>-hour period.</p>
<p>State Sheila’s conclusion to the test. Justify your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The heights, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> metres, of the 241 new entrants to a men’s college were measured and the following statistics calculated.</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\sum {x = 412.11,\,\,\sum {{x^2} = 705.5721} } ">
<mo>∑<!-- ∑ --></mo>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mn>412.11</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>∑<!-- ∑ --></mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>705.5721</mn>
</mrow>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>The Head of Mathematics decided to use a <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\chi ^2}">
<mrow>
<msup>
<mi>χ<!-- χ --></mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> test to determine whether or not these heights could be modelled by a normal distribution. He therefore divided the data into classes as follows.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Calculate unbiased estimates of the population mean and the population variance.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">State suitable hypotheses.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Calculate the value of the <span class="mjpage"><math alttext="{\chi ^2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> </math></span> statistic and state your conclusion using a 10% level of significance.</p>
<div class="marks">[11]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The graph below shows a small maze, in the form of a network of directed routes. The vertices <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math> show junctions in the maze and the edges show the possible paths available from one vertex to another.</p>
<p>A mouse is placed at vertex <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and left to wander the maze freely. The routes shown by dashed lines indicate paths sprinkled with sugar.</p>
<p>When the mouse reaches any junction, she rests for a constant time before continuing.</p>
<p>At any junction, it may also be assumed that</p>
<ul>
<li> the mouse chooses any available normal path with equal probability</li>
<li> if the junction includes a path sprinkled with sugar, the probability of choosing this path is twice that of a normal path.</li>
</ul>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the transition matrix for this graph.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>If the mouse was left to wander indefinitely, use your graphic display calculator to estimate the percentage of time that the mouse would spend at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment on your answer to part (b), referring to at least one limitation of the model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Eggs at a farm are sold in boxes of six. Each egg is either brown or white. The owner believes that the number of brown eggs in a box can be modelled by a binomial distribution. He examines 100 boxes and obtains the following data.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVcAAADwCAYAAAC0Xyf6AAAaiElEQVR4Ae2dsU7kuvfHs3/dp7haIcTwDBQroKBgeIBbwK2oVoL6CpotaVhRg0RFtVDsA8xSUADagmeAFVqteI3965v7O3M9HodJIGFsz8fSkMQ5sc/5HOeMx3Hwu9+/f/8uSBCAAAQg0CqB/2u1NAqDAAQgAIGSAMGVhgABCECgAwIE1w6gUiQEIACBP3wE796987M4hgAEIACBCQT8x1djwVXX+0ITyuQ0BCDQIQF1eLgnOwTcQtGhTinDAi2ApQgIQAACPgGCq0+EYwhAAAItECC4tgCRIiAAAQj4BAiuPhGOIQABCLRAgODaAkSKgAAEIOATILj6RDiGAAQg0AIBgmsLECkCAhCAgE/gVcH19va20Pwu+1xcXIyU//nz5+E5yUi+i6R6FhcXuyi6cZmuzbu7uyPXG68fP36M5HNQj4B4vpWfc/SV2Nm9GtrSLuu1w9pS+q9Ybvr3HQI3Z/L+w8OD/rNW+dG+n/r9vp/V2vHh4WFZb6/Xa63MlxZ0fn7+22zVvpgYj5ubm2cZvbROrmufQGy+esk9WUXFb5cmp/vI2qrlsa1PIOSjV/Vc3Qi+s7NTHq6vr7vZne/v7e0Vh4eHnddTp4Kzs7NibW2tFN3c3CzfqllYWCiPl5eXi5ubmzrFIDNlArPoK91H1lanjD+b6lsLriKi4PHw8FD4P4ezoTXBkPv7+wkSnIZAfAQ0lEVqn0CrwVXf+OpFnpycFP74q1RXno31aHxHHzs2ecvTmNfGxkZ53sbZFLRNPjR+65bvNxh3LNTKk042tmbn3XM+btNHOrhyVoa+WPb390sdzR6/DB0/PT0N7VCZlsx200X1WKqq282XHu619iVneVaXypV+Lk/V/VySvcZe5bnJPWcyVpfkXB3tvPEzm5Vv+rtlu/sqxy1X+uvY+KsMXzf3ercuyZoOroy/X+UryfnlWd1+vo5dRjqOJUmXx8fHEXXEVGxNZ2vL1o5C7FybdZ350s23cqxca59Wucq1j8nqnOmjvNB5k7Fz1ka0tTyry20rVm9nW39UITR24Mv4xxqr2dnZGWZr3NEdb7RxSAnYeJY7viNZjQUpT/v2kaybJxklleeWaeOupoONK5m8zutjya43XVSfXWsy/lbXuGVojNcf59WxK+OXYfXZdWab6rZ96eLapjIm1a063WusHpVpyewzttoaH+lj503e3eq8lWV62rWq163bZ6lyzV6VqX1XXvtWts5V8ZOcy0blmi1Wnq5VXlWSnNnp2+FfYwxNd5P3rzfdTd7VX9fa9W7dfl2Tjp+zadK1/nm7N4ydtmaTZI2z8s025csu1zbJ6aNkbMRAKcRC5VmbkYzPQ+ct2fXauvqYntoaV78s08Xqks6urMlbXW1tXf2tzP8s+l9OSMiEq7YyyAw3GZVjRpkTdM7AuY6TrMEwOJKzpHJcx6out8wqgCaj61WH/6nSx+q1relsx9qanqa38nw9XXntWzmu7dJdeilZma7tdo1blslZ3SZj5dqxMdOxPpZc3soTJ99/Jmtl+exM3rfZ941ftu8rnTc/WZ1VW1/Wr0s8jGVVGcp3bTKGvrzJGFOdd32luo2BXeuetzzjZr6w/CbbOjbVLc8YmV3a+nb4PlPZz91DIRbS2bVZxy5rtw7TyVjZ1q53ZaWL8qWPkvmpPKj449Yte63cCvEXZasOP7U6LOB2r6c9/uoOzuvnuj76t23ux9X3uf1fv36Nnbbyf/78OXauScbc3Nyz4nXq1nBMr9crvn79Wpb1/fv3cnjm9PS0PP7y5UshmZck1d/v90e4ieHx8XFZ3MePHwurRxmXl5fDh3o63t7eLoeJrO6rq6vCfej56dOn4tu3b8Ofb13+ZLafhAcHB6U9plPdreurkJ523j13fn5eFi+7Y0xqx/Pz8xNVe+4ecu2dWFBAQPeQHoi796b29ZBtUgrdH/41GqpUO1PSPfLXX3/5Ip0cdxZc3fHXTjSfUKgcbk/uFXju7u4mXDH5tG5OP9kN5efXPVbDUvCalCbVbUHOGroakG4IXVfn5qmq//3792XwqzpvN4CNbWk8zfJ0jWZNyD47r4d+Fph1Xu1EN5J0VZKvukorKyuFgt1gMHhRFb6v9EUSSvbFK1/oppZ9+gKxcb/QNdPMc/1VpUdb91CofN1DVSxD8m6e2qeStXv3nO3LPrUvjdlqfNn8Y+e72nYWXKWwjKoKHHpQoGQPAba2toIPweoaLngGWBDVmK3RKPCofDuvMps0dAUINS7dnJZUh/J0rmmyQC999ADMvlVD5dSt24Lp0dFRabcakNir5/iaLwDrFdtDAtPR+GmrKWjW6/ADl/wrHey8P6NC5YqD9LUAa3V0sbVfGqEvq1B9Vb6Sz6SvcdC14uBOC1QP3eyVrB70uvKh+mLNe+4eUidGtrlMQ1+SLnvdn8ZDbdxnqTZhseE5JtY+3V9Dkvc5yy+KAaurq88V1+45f5wgNHbgy7jHGm+xMRJtbSzHldGYiZt0bNdo/EP7Gnex8RM754/FqC73Wht3UdlWjq51861eX0+V7Zev+p9L7riTW4evt3QIcVDZvqzJ+fnSzU1VdfsystOS2WfHqsvYamtjV5bn+8mu09ZkbGvnXH/YOW1Nf5+7ydi4l69DlQ/cerTvHouN63/VYVxNT21dGV1vTE0XV1b7vk/8Mv3zbjm+na6+2m+SVFYbyew13dy2YuW7Mr6evi/Nx7rWP6c6XB4+e5Xt1u+3Temh5Ovj1xPSu4qXlWnXtLkN1TnmtZBQm0pQVl4E3BvItczybeue035Vvi/H8b9fbKlx8IPrtPVX8O6yzYXiZqfDAu32sSktNgIaGnEfZpl+ytdQhH4matjDT+7PR/8cxxDogoCGy97qQdZQf/8bJRSBfRmOIWAE/J9paj/uzz31FpTnfvyfm1YW2zCB1O5J19dd9hbDtP7LtWEx6eMOYfwn0d5eyEfvVPww0hZF+VTXy3JPsw8BCLwxAc204J58Y+gNqwv5iGGBhhARhwAEIFCHAMG1DiVkIAABCDQkQHBtCAxxCEAAAnUIEFzrUEIGAhCAQEMCBNeGwBCHAAQgUIfAHyEhPfkiQQAC8RDgnozHF3U1CQZXpn3UxYccBLonEJrm032t1NCEQOjLj2GBJgSRhQAEIFCTAMG1JijEIAABCDQhQHBtQgtZCEAAAjUJEFxrgkIMAhCAQBMCBNcmtJCFAAQgUJMAwbUmKMQgAAEINCFAcG1CC1kIQAACNQkQXGuCSklM6wdp3p0+7rphKdmArnkQsLYYskZt09qpv+ZVSD61PIJrah6boK81Ur0IokXftFAcAXYCNE53QkCBU4sQhpLapNqm2qi9tGRtNySfYh7/LDtFr1Xo7DZYWz7YGqy7nHXF5WRHSkBBygJQpCpWqqUVXLXUj6+/3y5Dbbey0AhPhHxEzzVCR71UJS0Drd6ABVaVo6WEX7om/Ev14DoITCKgNukuc602q7ZrS5lPuj6F8wTXFLxUU8fr6+ticXFxTFo/vUgQiIWAeqlqk+/fvx9T6efPn2N5qWYQXFP1HHpDAAJREyC4Ru0elIMABFIlQHBN1XMBvTWGdX9/P3JGP7M0lkWCQCwEbHz1169fIyppqGBubm4kL+UDgmvK3vN0X1paKseyNKZl6fHxsfj48aMdsoVAFATW19cLPSOwZG12c3PTspLfMhUreReOGuBOcbm9vS1WVlbGpsGMXsFR7ARC03xi19n0q5qK5U+92tjYKNbW1oq9vT27NKltyEf0XJNy4WRlNZ9VDVfOVmBlpsBkZkh0Q0AzVzTHVUnt8eLiYliRhgZubm7KISud03GqgXVolLdDz9UDwiEEYiMQ6hXFpuOs6xPyET3XWW8V2A8BCHRCgODaCVYKhQAEZp0AwXXWWwD2QwACnRAguHaClUIhAIFZJ0BwnfUWgP0QgEAnBAiunWClUAhAYNYJEFxnvQVgPwQg0AmBP0Klas4WCQIQiIcA92Q8vqirSTC4+v81vG5hyEEAAu0TCE1Qb78WSnwNgdCXH8MCryHKtRCAAAQqCBBcK8CQDQEIQOA1BAiur6HHtRCAAAQqCBBcK8CQDQEIQOA1BAiur6HHtRCAAAQqCBBcK8CQDQEIQOA1BAiur6HHtRCAAAQqCBBcK8Cknq15d7bkS+q2oH+aBLQSgdqhPlruxU/Ks/Na5iW3RHDNzKNaSkMNlgSBaRJQsLy8vCzXbzs/Py+Xe3EDrPZPT0/L8/bSUm4BlmVeptkCO6xbvQatsKk1tUhpE9CXpQWgFCzRGm5PT0/F8vLyUF0LnIPBoMxT+zw7OxvK2IKFWlfLvW5YQOQ7IR/Rc43caagHgdQIaLFBP0Aqz5ICqb9wps73er3i+/fvJpb8luCavAsxAAJpENDS2W7KKZC6dtk+wdVIsIUABDojoPFXWzpbvdR+vz9cdtsq9Xuzlp/qNvhfsVI1Br0hAIH4CGjWisZX3aSxV5tN4OZ/+PDBPUx6n+CatPtQHgJxE9DsldXV1bExWGl9f38/VF4BWGOx/ljtUCDBHYJrgk5DZQikQOD29ra4vr6eOGNFcicnJ2MPuVKw8TkdGXN9jg7nIACBFxFQwDw4OBgJrOrF6uMm9VhXVlYKTcFyZxS4MqnuM881Vc9V6K1GrcbqppTmSLp6s/8vgdAcypjZKIBubW2NqaipVjYUoPFWPcDSgy2b+zp2QUIZIR8RXBNyIKrOJoHQjTubJOK1OuQjhgXi9ReaQQACCRMguCbsPFSHAATiJUBwjdc3aAYBCCRMgOCasPNQHQIQiJcAwTVe36AZBCCQMAGCa8LOQ3UIQCBeAgTXeH2DZhCAQMIECK4JOw/VIQCBeAkE/7eAJsSSIACBeAhwT8bji7qaBIMrr0vWxYccBLonEHr7p/taqaEJgdCXH8MCTQgiCwEIQKAmAYJrTVCIQQACEGhCgODahBayEIAABGoSILjWBIUYBCAAgSYECK5NaCELAQhAoCYBgmtNUIhBAAIQaEKA4NqEFrIQgAAEahIguNYElYqYlnnRnDv7aEVNEgSmQcDaoLb+2lmmz+fPn8u2mmM7JbialzPYqoFqUTi9BKKP1ifSukU5NtwM3JW1CVojy9rh+fl5uaaW3w4ls7+/ny0HgmtGrr27uxtZ7O34+Li0TvkkCLwlgcvLy2F1m5ub5f7T09MwTztarFCBN9dEcM3Is9aIzaTclio2u9jGT8Bte+qx6lfU8vJy/Iq3qGHwfwu0WD5FRUBgaWkpAi1QYVYJrK+vD5fUniUG9Fwz9rYeIhweHhZuLyJjczEtMgJqf3qY9fDwkO1Dq+eQE1yfo5P4uU+fPhV7e3uJW4H6qRLQMJUeaukLXuno6ChVU16kN8MCL8IW/0UbGxsz+VMsfs/Mnob6gn98fJw5wwmuGbp8d3e3UK+VBIFYCMzPz8eiypvpQXB9M9RvU5EC6+rq6siTWfViB4PB2yhALRDwCGi2gOazaohglhJjrhl5W0H05OSknLDtvh2ztraWkZWYEjsB/y1BzRYIBVa1162trdIcveyit7VySu9+e1brpvSycrIXWyCQHAHuyfhdFvIRPdf4/YaGEIBAggQIrgk6DZUhAIH4CRBc4/cRGkIAAgkSILgm6DRUhgAE4idAcI3fR2gIAQgkSIDgmqDTUBkCEIifAME1fh+hIQQgkCCB4BtamrNFggAE4iHAPRmPL+pqEgyuvERQFx9yEOieQGiCeve1UkMTAqEvP4YFmhBEFgIQgEBNAgTXmqAQgwAEINCEAMG1CS1kIQABCNQkQHCtCQoxCEAAAk0IEFyb0EIWAhCAQE0CBNeaoBCDAAQg0IQAwbUJLWQhAAEI1CRAcK0JKhUxW85Y8+5Cc+9SsQM98yCgZYeq2qFWHrB2WiWTMgWCa8re83TX8hrX19flShJ6EaTf7xdaSoMEgWkQUMDUskOhpE6A1tV6eHgo2+v5+XmxuLgYEk02j+CarOvGFf/zzz+L4+Pj4Ynt7W2W1x7SYOetCegL/vDwMFitOgH68l9YWCjPLy0tlYE2KJxoJsE1UceF1LaGaud+/vxZHBwc2CFbCERDQEttf/v2bUQfLVKYUyK45uRNxxb97FLa3Nx0ctmFQBwE9vb2SkVsKEArxN7f38ehXEtaBP9xS0tlU8yUCGic1XoFV1dXxWAwmJImVAuBagIab1VvVWOz2s8t0XPNzaNFUQZTjXep4SrI6kEXCQKxEfj69WuhB1kae1Vbza2dElxja3Et6pPbz6wW0VDUlAlo2Eq/qjRspV9WevC1srJS/PjxY8qatVc9wbU9llGWpB6BZhGQIBATgbOzs+FMAemlMVi11bu7u5jUfJUuBNdX4Yv7Yk3S1oMCfxZB3Fqj3SwQWFtbK+fAWk9VW427akpWLongmosni6Kwt2HsrReZ5s57zchUTEmAgGYC6EUBJbVJm8GiY/VUd3Z2hg+01GtVcM2pI/Dut7emiyB4WQm4ERUhkC8B7sn4fRvyET3X+P2GhhCAQIIECK4JOg2VIQCB+AkQXOP3ERpCAAIJEiC4Jug0VIYABOInQHCN30doCAEIJEiA4Jqg01AZAhCInwDBNX4foSEEIJAgAYJrgk5DZQhAIH4CwX85qAmxJAhAIB4C3JPx+KKuJsHgyhtadfEhB4HuCYTe/um+VmpoQiD05cewQBOCyEIAAhCoSYDgWhMUYhCAAASaECC4NqGFLAQgAIGaBAiuNUEhBgEIQKAJAYJrE1rIQgACEKhJgOBaExRiEIAABJoQILg2oYUsBCAAgZoECK41QaUopmVf9CFBIAYC7jJEWgIm90RwzdTDWgP+5OQkU+swKzUCWixT7VEvKOnz8ePHYmNjIzUzGulLcG2EKx3hL1++FP1+Px2F0TRrAldXV+WChGbkhw8fivv7ezvMcktwzdCt6iX8888/GVqGSakS0Kqul5eXI+rnPjRAcB1xd/oHGg5QymmJ4vS9ggX6stfS2TYUsL29XQwGg6zBBP9xS9YWZ26chgOOj48ztxLzUiOgL/ubm5tiZWWlmJV/REPPNbVW+oy+DAc8A4dTUyegL34F2F6vVwbYHz9+TF2nLhV499v7/4Kz8q3SJdRpla0xLP30CiXlM1QQIhN/Xg73pL74Hx8fh7+qNC3LZg/E74HJGoZ8RM91MrdkJPT01aa6aKvZAjs7O2UegTUZN2ap6OnpaTE/Pz+0zYau7BnB8ERGOwTXjJyJKRCIlcD6+nqxv78/VM+C6vLy8jAvtx0eaOXmUeyBQIQE1FPVGKt+PlvyRiQtO5stY67ZuBJDciUQGs/L1dZU7Qr5iGGBVL2J3hCAQNQECK5RuwflIACBVAkQXFP1HHpDAAJREyC4Ru0elIMABFIlQHBN1XPoDQEIRE2A4Bq1e1AOAhBIlQDBNVXPoTcEIBA1geBLBO5E36i1RzkIzAgB7sn0HB0Mrrm/OZGem9B4lgmEJqjPMo8YbQ99+TEsEKOn0AkCEEieAME1eRdiAAQgECMBgmuMXkEnCEAgeQIE1+RdiAEQgECMBAiuMXoFnSAAgeQJEFyTdyEGQAACMRIguMboFXSCAASSJ0BwTd6F4wZcXFyU//Fdc+/00eJwJAhMm4C1R23VRnNPwZcIcjc6d/uur6/LRQlztxP70iGglYnt5SQF1q2trWJpaSnrFYnpuabTPmtpqob7999/15JFCAJvReDy8nJY1ebmZrn/9PQ0zMtxhzW0MvOqeggPDw+lVdZTyMzEmTNHP6Nz8qUWKtzd3S0Gg0E2vgz5iJ5rNu791xD1EHQj7uzsMN6amW9zMUfLbOcUWKv8Qs+1ikwG+Ta2pZ7swsJCBhbNpgmhXlGKJKw9mu45tcuQj+i5mqcz3Gpsq9/vF3d3dxlah0mpEVB71K+qw8PDUvWjo6PUTGikL7MFGuFKT3htbS09pdE4awJ7e3vF4+Nj1jbKOHqumbv46uqqnPKSuZmYlxiB+fn5Qp+cE2OuGXvXXh5QT4GULoHQeF661hSFZgv0er2sZkCEfETPNeVW6umuBwZysn10msDqQeLwzQnc3t4O26TapmYL5DS1rAooPdcqMuRDIBICoV5RJKqhxv8IhHxEz5XmAQEIQKADAgTXDqBSJAQgAAGCK20AAhCAQAcECK4dQKVICEAAAgRX2gAEIACBDggQXDuASpEQgAAECK60AQhAAAIdECC4dgCVIiEAAQgE/3GLJsSSIACBeAhwT8bji7qaBIPrLLyaVhcQchCYNoHQ2z/T1on6RwmEvvwYFhhlxBEEIACBVggQXFvBSCEQgAAERgkQXEd5cAQBCECgFQIE11YwUggEIACBUQIE11EeHEEAAhBohQDBtRWMFAIBCEBglADBdZQHRxCAAARaIUBwbQVjnIVsbGyUy2ssLi7GqSBazRyB3d3dsk3OguHBlwhmwfCcbdSaRSsrK0W/35+JtYpy9mVOtoUm2udkn28LPVefSOLHFlgPDw+LwWCQuDWonxMBvfmpdjkrieCamae3t7eLnZ0dVn3NzK+Ykx4Bgmt6PqvUWL3Wh4eHYn5+friUMeOtlbg4AYFOCRBcO8X7toV///59WKF+ginQ6qOHCCQIQOBtCRBc35Z3p7U9Pj6WD7H29vbKehYWFsoxrpOTk07rpXAIQGCcAMF1nElWOXNzc1nZgzEQSIUAwTUVT9XQc3V1tfj27duYZK/XG8sjAwIQ6JYAwbVbvm9a+ubmZjks4I6xfvr0qTg4OHhTPagMAhAoine/vWUH+K/n6TcLzRDQgyyl8/PzQkGXlC6BXO5Jt13m1jZDPiK4pnvPofmMEAjduDNiejJmhnzEsEAy7kNRCEAgJQIE15S8ha4QgEAyBAiuybgKRSEAgZQIEFxT8ha6QgACyRAguCbjKhSFAARSIkBwTclb6AoBCCRDgOCajKtQFAIQSIkAwTUlb6ErBCCQDIHgMi+aEEuCAATiIcA9GY8v6moyFly9t2HrloMcBCAAAQg4BBgWcGCwCwEIQKAtAgTXtkhSDgQgAAGHAMHVgcEuBCAAgbYIEFzbIkk5EIAABBwCBFcHBrsQgAAE2iLw/5QSPb5ORFDhAAAAAElFTkSuQmCC"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the mean number of brown eggs in a box.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Hence estimate <span class="mjpage"><math alttext="p" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>p</mi> </math></span>, the probability that a randomly chosen egg is brown.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">By calculating an appropriate <span class="mjpage"><math alttext="{\chi ^2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> </mrow> </math></span> statistic, test, at the 5% significance level, whether or not the binomial distribution gives a good fit to these data.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Adesh wants to model the cooling of a metal rod. He heats the rod and records its temperature as it cools.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>He believes the temperature can be modeled by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T\left( t \right) = a{{\text{e}}^{bt}} + 25">
<mi>T</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>b</mi>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>25</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{,}}\,\,b \in \mathbb{R}">
<mi>a</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Hence</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {T - 25} \right) = bt + {\text{ln}}\,a">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>b</mi>
<mi>t</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>a</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the regression line of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {T - 25} \right)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mo>−</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>predict the temperature of the metal rod after 3 minutes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Katie likes to cycle to work as much as possible. If Katie cycles to work one day then she has a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> of not cycling to work on the next work day. If she does not cycle to work one day then she has a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn></math> of not cycling to work on the next work day.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the following transition diagram to represent this information.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Katie works for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn></math> days in a year.</p>
<p>Find the probability that Katie cycles to work on her final working day of the year.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A psychologist records the number of digits (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math> that a sample of IB Mathematics higher level candidates could recall.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The psychologist has read that in the general population people can remember an average of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>4</mn></math> digits of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math>. The psychologist wants to perform a statistical test to see if IB Mathematics higher level candidates can remember more digits than the general population.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo> </mo><mo>:</mo><mo> </mo><mi>μ</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>4</mn></math> is the null hypothesis for this test.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an unbiased estimate of the population mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an unbiased estimate of the population variance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the alternative hypothesis.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that all assumptions for this test are satisfied, carry out an appropriate hypothesis test. State and justify your conclusion. Use a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> significance level.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A manager wishes to check the mean mass of flour put into bags in his factory. He randomly samples 10 bags and finds the mean mass is 1.478 kg and the standard deviation of the sample is 0.0196 kg.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{n - 1}}">
<mrow>
<msub>
<mi>s</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</math></span> for this sample.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a 95 % confidence interval for the population mean, giving your answer to 4 significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The bags are labelled as being 1.5 kg mass. Comment on this claim with reference to your answer in part (b).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>As part of the selection process for an engineering course at a particular university, applicants are given an exam in mathematics. This year the university has produced a new exam and they want to test if it is a valid indicator of future performance, before giving it to applicants. They randomly select 8 students in their first year of the engineering course and give them the exam. They compare the exam scores with their results in the engineering course.</p>
</div>
<div class="specification">
<p>The results of the 8 students are shown in the table.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the name of this test for validity.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate Pearson’s product moment correlation coefficient for this data.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine, with a reason, if the new exam is a valid indicator of future performance.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>In a coffee shop, the time it takes to serve a customer can be modelled by a normal distribution with a mean of 1.5 minutes and a standard deviation of 0.4 minutes.</p>
<p>Two customers enter the shop together. They are served one at a time.</p>
<p>Find the probability that the total time taken to serve both customers will be less than 4 minutes.</p>
<p>Clearly state any assumptions you have made.</p>
</div>
<br><hr><br><div class="specification">
<p>A <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 \times 2">
<mn>2</mn>
<mo>×<!-- × --></mo>
<mn>2</mn>
</math></span> transition matrix for a Markov chain will have the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf{M}} = \left( {\begin{array}{*{20}{c}} a&{1 - b} \\ {1 - a}&b \end{array}} \right){\text{,}}\,\,0 < a < 1{\text{,}}\,\,0 < b < 1">
<mrow>
<mrow>
<mi mathvariant="bold">M</mi>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo><</mo>
<mi>a</mi>
<mo><</mo>
<mn>1</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo><</mo>
<mi>b</mi>
<mo><</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 1">
<mi>λ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> is always an eigenvalue for <strong>M</strong> and find the other eigenvalue in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the steady state probability vector for <strong>M</strong> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span> has the Poisson distribution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Po}}(m)"> <mrow> <mtext>Po</mtext> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X > 0) = \frac{3}{4}"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m"> <mi>m</mi> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\ln a"> <mi>ln</mi> <mo></mo> <mi>a</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> is an integer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y"> <mi>Y</mi> </math></span> has the Poisson distribution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Po}}(2m)"> <mrow> <mtext>Po</mtext> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> </math></span>. Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y > 1)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{b - \ln c}}{c}"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>ln</mi> <mo></mo> <mi>c</mi> </mrow> <mi>c</mi> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> are integers.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A robot moves around the maze shown below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Whenever it leaves a room it is equally likely to take any of the exits.</p>
<p style="text-align: left;">The time interval between the robot entering and leaving a room is the same for all transitions.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the transition matrix for the maze.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A scientist sets up the robot and then leaves it moving around the maze for a long period of time.</p>
<p>Find the probability that the robot is in room <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> when the scientist returns.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of cars arriving at a junction in a particular town in any given minute between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>:</mo><mn>00</mn><mo> </mo><mtext>am</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>:</mo><mn>00</mn><mo> </mo><mtext>am</mtext></math> is historically known to follow a Poisson distribution with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>4</mn></math> cars per minute.</p>
<p>A new road is built near the town. It is claimed that the new road has decreased the number of cars arriving at the junction.</p>
<p>To test the claim, the number of cars, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>, arriving at the junction between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>:</mo><mn>00</mn><mo> </mo><mtext>am</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>:</mo><mn>00</mn><mo> </mo><mtext>am</mtext></math> on a particular day will be recorded. The test will have the following hypotheses:</p>
<p style="padding-left: 90px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>0</mn></msub><mo> </mo><mo>:</mo></math> the mean number of cars arriving at the junction has not changed,<br><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>H</mtext><mn>1</mn></msub><mo> </mo><mo>:</mo></math> the mean number of cars arriving at the junction has decreased.</p>
<p>The alternative hypothesis will be accepted if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>≤</mo><mn>300</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming the null hypothesis to be true, state the distribution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of a Type I error.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability of a Type II error, if the number of cars now follows a Poisson distribution with a mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn></math> cars per minute.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider two events <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> defined in the same sample space.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = \frac{4}{9},{\text{ P}}(B|A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪<!-- ∪ --></mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B|A') = \frac{1}{6}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(A' \cap B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo>∪</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo>∩</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(A) = \frac{1}{3}">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span>;</p>
<p>(ii) hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(B)">
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi>B</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Sue sometimes goes out for lunch. If she goes out for lunch on a particular day then the probability that she will go out for lunch on the following day is 0.4. If she does not go out for lunch on a particular day then the probability she will go out for lunch on the following day is 0.3.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the transition matrix for this Markov chain.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>We know that she went out for lunch on a particular Sunday, find the probability that she went out for lunch on the following Tuesday.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the steady state probability vector for this Markov chain.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of telephone calls received by a helpline over 80 one-minute periods are summarized in the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Find the exact value of the mean of this distribution.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Test, at the 5% level of significance, whether or not the data can be modelled by a Poisson distribution.</p>
<div class="marks">[12]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram below shows part of the screen from a weather forecasting website showing the data for town <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. The percentages on the bottom row represent the likelihood of some rain during the hour leading up to the time given. For example there is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>69</mn><mo>%</mo></math> chance (a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>69</mn></math>) of rain falling on any point in town <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0900</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn></math>.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">Paula works at a building site in the area covered by this page of the website from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0900</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1700</mn></math>. She has lunch from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1300</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1400</mn></math>.</p>
</div>
<div class="specification">
<p>In the following parts you may assume all probabilities are independent.</p>
<p>Paula needs to work outside between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1300</mn></math> and will also spend her lunchtime outside.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the probability it rains during Paula’s lunch break.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability it will not rain while Paula is outside.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability it will rain at least once while Paula is outside.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given it rains at least once while Paula is outside find the probability that it rains during her lunch hour.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> be a random variable which follows a normal distribution with mean <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ<!-- μ --></mi>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < \mu - 5} \right) = 0.2">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mi>μ<!-- μ --></mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.2</mn>
</math></span> , find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X > \mu + 5} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {X < \mu + 5\,\left| {\,X > \mu - 5} \right.} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>X</mi>
<mo><</mo>
<mi>μ</mi>
<mo>+</mo>
<mn>5</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>|</mo>
<mrow>
<mspace width="thinmathspace"></mspace>
<mi>X</mi>
<mo>></mo>
<mi>μ</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>George goes fishing. From experience he knows that the mean number of fish he catches per hour is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>1</mn></math>. It is assumed that the number of fish he catches can be modelled by a Poisson distribution.</p>
<p>On a day in which George spends <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> hours fishing, find the probability that he will catch more than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> fish.</p>
</div>
<br><hr><br><div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> has eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>A switch has two states, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>. Each second it either remains in the same state or moves according to the following rule: If it is in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> it will move to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> with a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn></math> and if it is in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> it will move to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> with a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>7</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an eigenvector corresponding to the eigenvalue of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using your answer to (a), or otherwise, find the long-term probability of the switch being in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>c</mi><mi>d</mi></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The faces of a fair six-sided die are numbered 1, 2, 2, 4, 4, 6. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> be the discrete random variable that models the score obtained when this die is rolled.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the probability distribution table for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
<p><img src="images/Schermafbeelding_2017-02-28_om_11.16.45.png" alt="N16/5/MATHL/HP1/ENG/TZ0/02.a"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the expected value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The strength of earthquakes is measured on the Richter magnitude scale, with values typically between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> is the most severe.</p>
<p>The Gutenberg–Richter equation gives the average number of earthquakes per year, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>, which have a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>. For a particular region the equation is</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>N</mi><mo>=</mo><mi>a</mi><mo>-</mo><mi>M</mi></math>, for some <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>This region has an average of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> earthquakes per year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>The equation for this region can also be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfrac><mi>b</mi><msup><mn>10</mn><mi>M</mi></msup></mfrac></math>.</p>
</div>
<div class="specification">
<p>Within this region the most severe earthquake recorded had a magnitude of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>The number of earthquakes in a given year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> can be modelled by a Poisson distribution, with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>. The number of earthquakes in one year is independent of the number of earthquakes in any other year.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> be the number of years between the earthquake of magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> and the next earthquake of at least this magnitude.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the average number of earthquakes in a year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mi>Y</mi><mo>></mo><mn>100</mn><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Find the coordinates of the point of intersection of the planes defined by the equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + z = 3,{\text{ }}x - y + z = 5">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + 2z = 6">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>6</mn>
</math></span>.</p>
</div>
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