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</div><h2>SL Paper 2</h2><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The heights of certain plants are normally distributed. The plants are classified into three categories. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The shortest \(12.92\% \) are in category A. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The tallest \(10.38\% \) are in category C. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> All the other plants are in category B with heights between \(r{\text{ cm}}\) and \(t{\text{ cm}}\) . </span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Complete the following diagram to represent this information.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/big_bang.png" alt></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 style="font-family: times new roman,times; font-size: medium;">Given that the mean height is \(6.84{\text{ cm}}\) and the standard deviation \(0.25{\text{ cm}}\) , find the value of <em>r</em> and of <em>t</em>. </span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/big_bang_2.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Notes</strong>: Award <em><strong>A1</strong></em> for three regions (may be shown by lines or shading), </span><span style="font-family: times new roman,times; font-size: medium;"><strong><em>A1</em></strong> for clear labelling of two regions (may be shown by percentages or categories). <em>r</em> and <em>t</em> need not be labelled, but if they are, they may be interchanged.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X < r) = 0.1292\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(r = 6.56\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(1 - 0.1038\) (= 0.8962) (may be seen later) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X < t) = 0.8962\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(t = 7.16\) <em><strong>A1 N2</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding <em>z</em>-values \( - 1.130 \ldots{\text{, }}1.260 \ldots \) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of setting up one standardised equation <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{r - 6.84}}{{0.25}} = - 1.13 \ldots \) , \(t = 1.260 \times 0.25 + 6.84\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(r = 6.56\) , \(t = 7.16\) <em><strong>A1A1 N2N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates shaded or otherwise correctly labelled the appropriate regions in </span><span style="font-family: times new roman,times; font-size: medium;">the normal curve.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Although many candidates shaded or otherwise correctly labelled the appropriate regions in </span><span style="font-family: times new roman,times; font-size: medium;">the normal curve, far fewer could apply techniques of normal probabilities to achieve correct </span><span style="font-family: times new roman,times; font-size: medium;">results in part (b). Many set the standardized formula equal to the probabilities instead of the </span><span style="font-family: times new roman,times; font-size: medium;">appropriate <em>z</em>-scores, which can be found either by the use of tables or the GDC. Others </span><span style="font-family: times new roman,times; font-size: medium;">simply left this part blank, which suggests a lack of preparation for such “inverse” types of </span><span style="font-family: times new roman,times; font-size: medium;">questions in a normal distribution.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(A\) and \(B\) be independent events, where \({\text{P}}(A) = 0.3\) and \({\text{P}}(B) = 0.6\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\text{P}}(A \cap B)\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\text{P}}(A \cup B)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">On the following Venn diagram, shade the region that represents \(A \cap B'\).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><img src="images/maths_4c.png" alt></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\text{P}}(A \cap B')\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c(ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.3 \times 0.6\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(A \cap B) = 0.18\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(A \cup B) = 0.3 + 0.6 - 0.18\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(A \cup B) = 0.72\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times;"><span style="font-size: medium;"><img src="images/maths_4c_markscheme.png" alt> <strong><em>A1 N1</em></strong></span></span></p>
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">appropriate approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.3 - 0.18,{\text{ P}}(A) \times {\text{P}}(B')\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(A \cap B') = 0.12\) (may be seen in Venn diagram) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c(ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following frequency distribution of marks has mean 4.5.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bbc3.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>x</em>.</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><span style="font-family: times new roman,times; font-size: medium;">Write down the standard deviation.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sum {fx = 1(2) + 2(4) + \ldots + 7(4)} \) , \(\sum {fx = 146 + 5x} \) (seen anywhere) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting into mean \(\frac{{\sum {fx} }}{{\sum f }}\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{146 + 5x}}{{34 + x}} = 4.5\) , \(146 + 5x = 4.5(34 + x)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 14\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sigma = 1.54\) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Surprisingly, this question was not well done by many candidates. A good number of </span><span style="font-family: times new roman,times; font-size: medium;">candidates understood the importance of the frequencies in calculating mean. Some neglected </span><span style="font-family: times new roman,times; font-size: medium;">to sum the frequencies for the denominator, which often led to a negative value for a </span><span style="font-family: times new roman,times; font-size: medium;">frequency. Unfortunately, candidates did not appreciate the unreasonableness of this result.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Surprisingly, this question was not well done by many candidates. A good number of </span><span style="font-family: times new roman,times; font-size: medium;">candidates understood the importance of the frequencies in calculating mean. Some neglected </span><span style="font-family: times new roman,times; font-size: medium;">to sum the frequencies for the denominator, which often led to a negative value for a </span><span style="font-family: times new roman,times; font-size: medium;">frequency. Unfortunately, candidates did not appreciate the unreasonableness of this result. In </span><span style="font-family: times new roman,times; font-size: medium;">part (b), many candidates could not find the standard deviation in their GDC, often trying to </span><span style="font-family: times new roman,times; font-size: medium;">calculate it by hand with no success. Further, many could not distinguish between the sample </span><span style="font-family: times new roman,times; font-size: medium;">and the population standard deviation given in the GDC.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let the random variable <em>X</em> be normally distributed with mean 25, as shown in the </span><span style="font-family: times new roman,times; font-size: medium;">following diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/son.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The shaded region between 25 and 27 represents \(30\% \) of the distribution.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(X > 27)\) .</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 style="font-family: times new roman,times; font-size: medium;">Find the standard deviation of <em>X</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">symmetry of normal curve <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(X < 25) = 0.5\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X > 27) = 0.2\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding standardized value <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{27 - 25}}{\sigma }\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of complement <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - p\) , \({\rm{P}}(X < 27)\) , 0.8</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding <em>z</em>-score <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(z = 0.84 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up equation involving the standardized value <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.84 = \frac{{27 - 25}}{\sigma }\) , \(0.84 = \frac{{X - \mu }}{\sigma }\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\sigma = 2.38\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3</span></strong></em></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">set up using normal CDF function and probability <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(25 < X < 27) = 0.3\) , \({\rm{P}}(X < 27) = 0.8\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <strong><em>A2</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(25 < X < 27) = 0.3\) , \({\rm{P}}(X > 27) = 0.2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve the equation using GDC <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. solver, graph, trial and error (more than two trials must be shown)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span lang="EN-US">\(\sigma = 2.38\) <em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3</span></strong></em></span></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question proved challenging for many candidates. A surprising number did not use the symmetry of the normal curve to find the probability required in (a). While many students were able to set up a standardized equation in (b), far fewer were able to use the complement to find the correct <em>z</em>-score. Others used 0.8 as the <em>z</em>-score. A common confusion when approaching parts (a) and (b) was whether to use a probability or a <em>z</em>-score. Additionally, many candidates seemed unsure of appropriate notation on this problem which would have allowed them to better demonstrate their method. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question proved challenging for many candidates. A surprising number did not use the symmetry of the normal curve to find the probability required in (a). While many students were able to set up a standardized equation in (b), far fewer were able to use the complement to find the correct <em>z</em>-score. Others used 0.8 as the <em>z</em>-score. A common confusion when approaching parts (a) and (b) was whether to use a probability or a <em>z</em>-score. Additionally, many candidates seemed unsure of appropriate notation on this problem which would have allowed them to better demonstrate their method. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following table shows the average weights ( <em>y </em>kg) for given heights (<em>x </em>cm) in a population of men.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"> </p>
<table class="block_black_border" style="height: 106px; width: 347px;" border="0">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Heights (<em>x</em> cm)</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">165</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">170</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">175</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">180</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">185</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Weights (<em>y</em> kg)</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">67.8</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">70.0</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">72.7</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">75.5</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">77.2</span></td>
</tr>
</tbody>
</table>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The relationship between the variables is modelled by the regression equation \(y = ax + b\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(a\) and of \(b\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The relationship between the variables is modelled by the regression equation \(y = ax + b\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, estimate the weight of a man whose height is 172 cm.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the correlation coefficient.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">State which <strong>two </strong>of the following describe the correlation between the variables.</span></p>
<table border="0">
<tbody>
<tr>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">strong </span></td>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">zero </span></td>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">positive</span></td>
</tr>
<tr>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">negative </span></td>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">no correlation </span></td>
<td><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal;">weak</span></td>
</tr>
</tbody>
</table>
<div class="marks">[2]</div>
<div class="question_part_label">b(ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 0.486\) (exact) <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(b = - 12.41\) (exact), \(-12.4\) <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.486(172) - 12.41\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(71.182\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(71.2\) (kg) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = 0.997276\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = 0.997\) <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">strong, positive (must have both correct) <strong><em>A2 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b(ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A company produces a large number of water containers. Each container has </span><span style="font-family: times new roman,times; font-size: medium;">two parts, a bottle and a cap. The bottles and caps are tested to check that they are </span><span style="font-family: times new roman,times; font-size: medium;">not defective.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A cap has a probability of 0.012 of being defective. A random sample of 10 caps is </span><span style="font-family: times new roman,times; font-size: medium;">selected for inspection.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that exactly one cap in the sample will be defective.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The sample of caps passes inspection if at most one cap is defective. Find the </span><span style="font-family: times new roman,times; font-size: medium;">probability that the sample passes inspection.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The heights of the bottles are normally distributed with a mean of \(22{\text{ cm}}\) and a </span><span style="font-family: times new roman,times; font-size: medium;">standard deviation of \(0.3{\text{ cm}}\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Copy and complete the following diagram, shading the region representing </span><span style="font-family: times new roman,times; font-size: medium;">where the heights are less than \(22.63{\text{ cm}}\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/flynn.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the probability that the height of a bottle is less than \(22.63{\text{ cm}}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) A bottle is accepted if its height lies between \(21.37{\text{ cm}}\) and \(22.63{\text{ cm}}\). </span><span style="font-family: times new roman,times; font-size: medium;">Find the probability that a bottle selected at random is accepted.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) A sample of 10 bottles passes inspection if all of the bottles in the sample </span><span style="font-family: times new roman,times; font-size: medium;">are accepted. Find the probability that the sample passes inspection.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The bottles and caps are manufactured separately. A sample of 10 bottles and </span><span style="font-family: times new roman,times; font-size: medium;">a sample of 10 caps are randomly selected for testing. Find the probability that </span><span style="font-family: times new roman,times; font-size: medium;">both samples pass inspection.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> There may be slight differences in answers, depending on whether candidates use </span><span style="font-family: times new roman,times; font-size: medium;">tables or GDCs, or their 3 sf answers in subsequent parts. Do not penalise answers that </span><span style="font-size: medium;"><span style="font-family: times new roman,times;">are consistent with <strong>their</strong></span> <span style="font-family: times new roman,times;">working and check carefully for <strong><em>FT</em></strong>.</span></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing binomial (seen anywhere in the question) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(_n{C_r}{p^r}{q^{n - r}}\) , \({\text{B}}(n{\text{, }}p)\) , \(^{10}{C_1}{(0.012)^1}{(0.988)^9}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.108\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(X \le 1)\) , \(0.88627 \ldots + 0.10764 \ldots \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.994\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/rapunzel.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1 N2</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for vertical line to right of mean, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong> </em>for shading to left of <strong>their</strong> vertical line.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(X < 22.63)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">working to find standardized value <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{22.63 - 22}}{{0.3}}\) , 2.1</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.982\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(21.37 < X < 22.63)\) , \({\rm{P}}( - 2.1 < z < 2.1)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.982 - (1 - 0.982)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.964\) <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) correct working <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(X \sim {\rm{B}}(10,0.964)\) , \({(0.964)^{10}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.695\) (accept 0.694 from tables) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(A \cap B) = {\rm{P}}(A){\rm{P}}(B)\) , \((0.994) \times {(0.964)^{10}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.691\) (accept \(0.690\) from tables) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many stronger candidates were completely successful with this question, employing technology efficiently. A number of candidates did not recognize the binomial probability in parts (a) and (b), and in part (b) a proportion of candidates just subtracted their part (a) answer from one. Candidates had more success with the normal distribution and many obtained follow-through marks in part (e) after an error made in part (b). Many candidates did not appreciate the independence in part (e) and added probabilities rather than multiplying them. A number of candidates were penalised for not giving their answers to 3 significant figures. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many stronger candidates were completely successful with this question, employing technology efficiently. A number of candidates did not recognize the binomial probability in parts (a) and (b), and in part (b) a proportion of candidates just subtracted their part (a) answer from one. Candidates had more success with the normal distribution and many obtained follow-through marks in part (e) after an error made in part (b). Many candidates did not appreciate the independence in part (e) and added probabilities rather than multiplying them. A number of candidates were penalised for not giving their answers to 3 significant figures. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many stronger candidates were completely successful with this question, employing technology efficiently. A number of candidates did not recognize the binomial probability in parts (a) and (b), and in part (b) a proportion of candidates just subtracted their part (a) answer from one. Candidates had more success with the normal distribution and many obtained follow-through marks in part (e) after an error made in part (b). Many candidates did not appreciate the independence in part (e) and added probabilities rather than multiplying them. A number of candidates were penalised for not giving their answers to 3 significant figures. </span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many stronger candidates were completely successful with this question, employing technology efficiently. A number of candidates did not recognize the binomial probability in parts (a) and (b), and in part (b) a proportion of candidates just subtracted their part (a) answer from one. Candidates had more success with the normal distribution and many obtained follow-through marks in part (e) after an error made in part (b). Many candidates did not appreciate the independence in part (e) and added probabilities rather than multiplying them. A number of candidates were penalised for not giving their answers to 3 significant figures. </span></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many stronger candidates were completely successful with this question, employing technology efficiently. A number of candidates did not recognize the binomial probability in parts (a) and (b), and in part (b) a proportion of candidates just subtracted their part (a) answer from one. Candidates had more success with the normal distribution and many obtained follow-through marks in part (e) after an error made in part (b). Many candidates did not appreciate the independence in part (e) and added probabilities rather than multiplying them. A number of candidates were penalised for not giving their answers to 3 significant figures. </span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A box contains a large number of biscuits. The weights of biscuits are normally distributed with mean \(7{\text{ g}}\) and standard deviation \(0.5{\text{ g}}\) . </span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">One biscuit is chosen at random from the box. Find the probability that this biscuit </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) weighs less than \(8{\text{ g}}\) ; </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) weighs between \(6{\text{ g}}\) and \(8{\text{ g}}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Five percent of the biscuits in the box weigh less than<em> d</em> grams. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (i) Copy and complete the following normal distribution diagram, to represent this information, by indicating <em>d</em>, and shading the appropriate region. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/8b.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of <em>d</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The weights of biscuits in another box are normally distributed with mean \(\mu \) and standard deviation \(0.5{\text{ g}}\). It is known that \(20\% \) of the biscuits in this second box weight less than \(5{\text{ g}}\). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\mu \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(X \sim {\text{N}}(7{\text{, }}{0.5^2})\)<br></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(z = 2\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X < 8) = {\rm{P}}(Z < 2) = 0.977\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of appropriate approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. symmetry, \(z = - 2\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(6 < X < 8) = 0.954\) (tables 0.955) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>M1A1(AP)</strong></em> if candidates refer to 2 standard deviations from </span><span style="font-family: times new roman,times; font-size: medium;">the mean, leading to 0.95.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/last_one.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for <em>d</em> to the left of the mean, <strong><em>A1</em></strong> for area to the left of <em>d</em> shaded.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(z = - 1.645\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{d - 7}}{{0.5}} = - 1.645\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(d = 6.18\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(Y \sim {\text{N}}(\mu {\text{, }}{0.5^2})\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(Y < 5) = 0.2\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(z = - 0.84162 \ldots \) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{5 - \mu }}{{0.5}} = - 0.8416\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\mu = 5.42\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Those that understood the normal distribution did well on parts (a) and (bi).</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Those that understood the normal distribution did well on parts (a) and (bi). Parts (bii) and (c) </span><span style="font-family: times new roman,times; font-size: medium;">proved to be a little more difficult. In particular, in part (bii) the z-score was incorrectly set </span><span style="font-family: times new roman,times; font-size: medium;">equal to 0.05 and in part (c), 0.2 was used instead of the <em>z</em>-score. For those who had a good </span><span style="font-family: times new roman,times; font-size: medium;">grasp of the concept of normal distributions the entire question was quite accessible and full </span><span style="font-family: times new roman,times; font-size: medium;">marks were gained.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Those that understood the normal distribution did well on parts (a) and (bi). Parts (bii) and (c) </span><span style="font-family: times new roman,times; font-size: medium;">proved to be a little more difficult. In particular, in part (bii) the <em>z</em>-score was incorrectly set </span><span style="font-family: times new roman,times; font-size: medium;">equal to 0.05 and in part (c), 0.2 was used instead of the <em>z</em>-score. For those who had a good </span><span style="font-family: times new roman,times; font-size: medium;">grasp of the concept of normal distributions the entire question was quite accessible and full </span><span style="font-family: times new roman,times; font-size: medium;">marks were gained.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The histogram below shows the time <em>T</em> seconds taken by 93 children to solve a puzzle.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sunburnt.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The following is the frequency distribution for <em>T</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/cats.png" alt></span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>p</em> and of <em>q</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the median class.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A child is selected at random. Find the probability that the child takes less than </span><span style="font-family: times new roman,times; font-size: medium;">95 seconds to solve the puzzle.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the class interval \(45 \le T < 55\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the interval width.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the mid-interval value.</span></p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Hence find an estimate for the</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) mean;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) standard deviation.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">John assumes that <em>T</em> is normally distributed and uses this to estimate the probability </span><span style="font-family: times new roman,times; font-size: medium;">that a child takes less than 95 seconds to solve the puzzle.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find John’s estimate.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(p = 17\) , \(q = 11\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(75 \le T < 85\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. adding frequencies </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(\frac{{76}}{{93}} = 0.8172043 \ldots \)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(T < 95) = \frac{{76}}{{93}} = 0.817\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) 10 <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) 50 <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of approach using mid-interval values (may be seen in part (ii)) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(79.1397849\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\overline x = 79.1\) <em><strong>A2 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(16.4386061\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sigma = 16.4\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. standardizing, \(z = 0.9648 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.8326812\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(T < 95) = 0.833\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well done. The terms "median" and "median class" were often confused. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well done. The terms "median" and "median class" were often confused. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c) some candidates had problems with the term "interval width" and there were some rather interesting mid-interval values noted. </span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d), candidates often ignored the "hence" command and estimated values from the graph rather than from the information in part (c). </span></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Those who correctly obtained the mean and standard deviation had little difficulty with part (e) although candidates often used unfamiliar calculator notation as their working or used the mid-interval value as the mean of the distribution. </span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the following cumulative frequency table.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/carly.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(p\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) the mean;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) the variance.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(35 - 26\) , \(26 + p = 36\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 9\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) </span><span style="font-family: times new roman,times; font-size: medium;">mean \( = 26.7\) <em><strong>A2 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) recognizing that variance is (sd)<sup>2</sup> <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(11.021{ \ldots ^2}\) , \(\sigma = \sqrt {{\mathop{\rm var}} } \) , \(11.158{ \ldots ^2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\sigma ^2} = 121\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">Candidates had little problem determining a missing frequency from a cumulative frequency table.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), few used the GDC to their advantage to correctly find the mean and variance. There were numerous unsuccessful attempts at using the formulae for mean and variance, most resulting in algebraic errors along the way. Candidates recognized the concept of variance but were often unable to determine what value should be squared.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Each day, a factory recorded the number ( \(x\) ) of boxes it produces and the total </span><span style="font-family: times new roman,times; font-size: medium;">production cost ( \(y\) ) dollars. The results for nine days are shown in the following table.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="images/plates.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the equation of the regression line of <em>y</em> on <em>x</em> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use your regression line from part (a) as a model to answer the following.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Interpret the meaning of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the gradient;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the <em>y</em>-intercept.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Estimate the cost of producing 60 boxes.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The factory sells the boxes for $19.99 each. Find the least number of boxes that </span><span style="font-family: times new roman,times; font-size: medium;">the factory should produce in one day in order to make a profit.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Comment on the appropriateness of using your model to</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) estimate the cost of producing 5000 boxes;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) estimate the number of boxes produced when the total production cost </span><span style="font-family: times new roman,times; font-size: medium;">is $540.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(y = 10.7x + 121\) <em><strong>A1A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) additional cost per box (unit cost) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) fixed costs <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute into regression equation <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y = 10.7 \times 60 + 121\) , \(y = 760.12 \ldots \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{cost}} = \$ 760\) (accept \(\$ 763\) from 3 s.f. values) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">setting up inequality (accept equation) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(19.99x > 10.7x + 121\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x > 12.94 \ldots \)<em><strong> A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">13 boxes (accept 14 from \(x > 13.02\) , using 3 s.f. values) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Exception to the <em><strong>FT</strong></em> rule: if working shown, award the final <em><strong>A1</strong></em> for a </span><span style="font-family: times new roman,times; font-size: medium;">correct integer solution for their value of <em>x</em>.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) this would be extrapolation, not appropriate <em><strong>R1R1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) this regression line cannot predict <em>x</em> from <em>y</em>, not appropriate <em><strong>R1R1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="JUSTIFY"><span style="font-family: times new roman,times; font-size: medium;">A factory makes switches. The probability that a switch is defective is 0.04. </span><span style="font-family: times new roman,times; font-size: medium;">The factory tests a random sample of 100 switches. </span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the mean number of defective switches in the sample.</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 style="font-family: times new roman,times; font-size: medium;">Find the probability that there are exactly six defective switches in the sample.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that there is at least one defective switch in the sample.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of binomial distribution (may be seen in parts (b) or (c)) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. <em>np</em>, \(100 \times 0.04\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{mean}} = 4\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 6) = \left( {\begin{array}{*{20}{c}}<br>{100}\\<br>6<br>\end{array}} \right){(0.04)^6}{(0.96)^{94}}\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 0.105\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">for evidence of appropriate approach <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. complement, \(1 - {\rm{P}}(X = 0)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 0) = {(0.96)^{100}} = 0.01687 \ldots \) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \ge 1) = 0.983\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was handled well by most students.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Although this question was a rather </span><span style="font-family: times new roman,times; font-size: medium;">straightforward question on binomial distribution, parts (b) and(c) seemed to cause much </span><span style="font-family: times new roman,times; font-size: medium;">difficulty.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Although this question was a rather </span><span style="font-family: times new roman,times; font-size: medium;">straightforward question on binomial distribution, parts (b) and(c) seemed to cause much </span><span style="font-family: times new roman,times; font-size: medium;">difficulty. In part (c), finding at least one defective switch, many forgot to take the complement.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A multiple choice test consists of ten questions. Each question has five answers. </span><span style="font-family: times new roman,times; font-size: medium;">Only one of the answers is correct. For each question, Jose randomly chooses one of </span><span style="font-family: times new roman,times; font-size: medium;">the five answers.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the expected number of questions Jose answers correctly.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that Jose answers exactly three questions correctly.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that Jose answers more than three questions correctly.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}(X) = 2\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach involving binomial <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>{10}\\<br>3<br>\end{array}} \right){(0.2)^3}\) , \({(0.2)^3}{(0.8)^7}\) , \(X \sim {\rm{B}}(10,0.2)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 3) = 0.201\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \le 3) = 0.10737 + 0.26844 + 0.30199 + 0.20133\) \(( = 0.87912 \ldots )\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using the complement (seen anywhere) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - \) any probability , \({\rm{P}}(X > 3) = 1 - {\rm{P}}(X \le 3)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X > 3) = 0.121\) <em><strong>A1 N2</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that \({\rm{P}}(X > 3) = {\rm{P}}(X \ge 4)\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. summing probabilities from \(X = 4\) to \(X = 10\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression or values <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\sum\limits_{r = 4}^{10} {\left( {\begin{array}{*{20}{c}}<br>{10}\\<br>r<br>\end{array}} \right)} {(0.2)^{10 - r}}{(0.8)^r}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.08808 + 0.02642 + 0.005505 + 0.000786 + 0.0000737 + 0.000004 + 0.0000001\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X > 3) = 0.121\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to find the mean by applying various methods. Although many recognised binomial probability, fewer were able to use the GDC effectively. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to find the mean by applying various methods. Although many recognised binomial probability, fewer were able to use the GDC effectively. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (c) was problematic in some cases but most candidates recognized that either a sum of probabilities or the complement was required. Many misinterpreted "more than three" as inclusive of three, and so obtained incorrect answers. When adding individual probabilities, some candidates used three or fewer significant figures, which resulted in an incorrect final answer due to premature rounding. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The following is a cumulative frequency diagram for the time <em>t</em>, in minutes, taken by </span><span style="font-family: times new roman,times; font-size: medium;">80 students to complete a task.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sunday.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the median.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the interquartile range.</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><span style="font-family: times new roman,times; font-size: medium;">Complete the frequency table below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/oned.png" alt></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">median \(m = 32\) <em><strong>A1 N1</strong> </em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"><em>[1 mark]</em></span></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">lower quartile \({Q_1} = 22\) , upper quartile \({Q_3} = 40\) <em><strong>(A1)(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{interquartile range}} = 18\) <em><strong>A1 N3</strong> </em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"><em>[3 marks]</em></span></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/dad.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1 N2 </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was answered successfully by a majority of candidates. A common error was to use values of 20 and 60 for the lower and upper quartiles. Some were careless when reading the graph scale and wrote incorrect answers as a result. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was answered successfully by a majority of candidates. A common error was to use values of 20 and 60 for the lower and upper quartiles. Some were careless when reading the graph scale and wrote incorrect answers as a result. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was answered successfully by a majority of candidates. A common error was to use values of 20 and 60 for the lower and upper quartiles. Some were careless when reading the graph scale and wrote incorrect answers as a result. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The price of a used car depends partly on the distance it has travelled. The following table shows the distance and the price for seven cars on <span class="s1">1 </span>January <span class="s1">2010</span>.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_17.58.43.png" alt="M16/5/MATME/SP2/ENG/TZ2/08"></p>
<p class="p1">The relationship between \(x\) and \(y\) can be modelled by the regression equation \(y = ax + b\).</p>
</div>
<div class="specification">
<p class="p1">On <span class="s1">1 </span>January <span class="s1">2010</span>, Lina buys a car which has travelled \(11\,000{\text{ km}}\).</p>
</div>
<div class="specification">
<p class="p1">The price of a car decreases by <span class="s1">5% </span>each year.</p>
</div>
<div class="specification">
<p class="p1">Lina will sell her car when its price reaches \(10\,000\)<span class="s1"> </span>dollars.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the correlation coefficient.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the value of \(a\) and of \(b\).</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 class="p1">Use the regression equation to estimate the price of Lina’s car, giving your answer to the nearest <span class="s1">100 </span>dollars.</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 class="p1">Calculate the price of Lina’s car after <span class="s1">6 </span>years.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the year when Lina sells her car.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>There may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>valid approach <strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)correct value for \(r\) (or for \(a\) or \(b\) <span class="s1">seen in (ii))</span></p>
<p class="p2">\( - 0.994347\)</p>
<p class="p1"><span class="s1">\(r = - 0.994\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2">(ii) <span class="Apple-converted-space"> \( - 1.58095,{\text{ }}33480.3\)</span></p>
<p class="p1"><span class="s1">\(a = - 1.58,{\text{ }}b = 33500\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>There may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</p>
<p class="p1">correct substitution into <strong>their </strong>regression equation</p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\( - 1.58095(11000){\text{ }} + 33480.3\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\(16\,089.85{\text{ }}(16\,120{\text{ from 3sf}})\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="s1">\({\text{price}} = 16\,100{\text{ }}({\text{dollars}})\) </span>(must be rounded to the nearest 100 <span class="s1">dollars) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>There may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</p>
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(P \times {({\text{rate}})^t}\)</p>
<p class="p1">\({\text{rate}} = 0.95\) (may be seen in their expression) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">correct expression <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(16100 \times {0.95^6}\)</p>
<p class="p1">\(11\,834.97\)</p>
<p class="p1"><span class="Apple-converted-space">\(11\,800{\text{ }}({\text{dollars}})\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">attempt to find all six terms <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(\left( {\left( {(16\,100 \times 0.95) \times 0.95} \right) \ldots } \right) \times 0.95\)</span>, table of values</p>
<p class="p2">5 correct values (accept values that round correctly to the nearest dollar)</p>
<p class="p2"><span class="Apple-converted-space">\(15\,295,{\text{ }}14\,530,{\text{ }}13\,804,{\text{ }}13\,114,{\text{ }}12\,458\) </span><span class="s1"><strong><em>A2</em></strong></span></p>
<p class="p2">\(11\,835\)</p>
<p class="p1"><span class="s2">\(11\,800{\text{ }}({\text{dollars}})\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>There may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</p>
<p class="p1"> </p>
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">correct equation <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(16\,100 \times {0.95^x}{\text{ = }}10\,000\)</p>
<p class="p1">valid attempt to solve <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)<img src="images/Schermafbeelding_2017-02-03_om_08.51.35.png" alt="M16/5/MATME/SP2/ENG/TZ2/08.d/M"></span>, using logs</p>
<p class="p2">9.28453 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2">year 2019 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">valid approach using table of values <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><strong>both </strong>crossover values (accept values that round correctly to the nearest dollar) <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({\text{P}} = 10\,147{\text{ }}({\text{1 Jan 2019}}),{\text{ P}} = 9\,639.7{\text{ }}({\text{1 Jan 2020}})\)</p>
<p class="p2">year 2019 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Although the question talked about the regression equation, a few students tried to find the values of <em>a </em>and <em>b </em>by forming two equations with the coordinates of two points from the table. A considerable number of candidates did not write the value of the correlation coefficient or gave an incorrect one. It can be that a GDC feature (Diagnostics) from some calculators was turned off.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was generally well done, with many candidates earning follow through marks. There were some difficulties in rounding the answer to the nearest 100 dollars.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (c) was attempted in two different ways: recognizing the correct rate 0.95 and then finding the price of the car after 6 years. Some of these candidates used a formula similar to the one for terms of a geometric sequence, \(P \times {({\text{rate)}}^{t - 1}}\), but substituted \(t\) by 6 and hence, got an incorrect result.</p>
<p class="p1">Others listed all six values to obtain the answer. When using this method, the problem was using less accurate intermediate results and hence, not getting the first 5 correct values of the car.</p>
<p class="p1">Many candidates either missed out questions 8 (c) and (d) or multiplied either \(0.05 \times 6 \times 16\,100\) or \(0.95 \times 6 \times 16\,100\) and failed to notice that the answer did not make sense. Other students tried to use the sum formula for a geometric series.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Many candidates either missed out questions 8 (c) and (d) or multiplied either \(0.05 \times 6 \times 16\,100\) or \(0.95 \times 6 \times 16\,100\) and failed to notice that the answer did not make sense. Other students tried to use the sum formula for a geometric series.</p>
<p class="p1">Part (d) was attempted using a graphical approach as well as analytically using logarithms to find the year in which Lina would sell the car, though many failed in giving the correct year. Common answers were “in the ninth year” or “in 2020”. The same happened to those candidates who used a table of values and found the price of the car after 9 years and 10 years. These candidates should be reminded to show both “crossover” values for a table method to be valid.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A discrete random variable \(X\) has the following probability distribution.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_10.34.18.png" alt="N17/5/MATME/SP2/ENG/TZ0/04"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(k\).</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>Write down \({\text{P}}(X = 2)\).</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>Find \({\text{P}}(X = 2|X > 0)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)total probability = 1</p>
<p>correct equation <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0.475 + 2{k^2} + \frac{k}{{10}} + 6{k^2} = 1,{\text{ }}8{k^2} + 0.1k - 0.525 = 0\)</p>
<p>\(k = 0.25\) <strong><em>A2 N3</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(X = 2) = 0.025\) <strong><em>A1 N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach for finding \({\text{P}}(X > 0)\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(1 - 0.475,{\text{ }}2({0.25^2}) + 0.025 + 6({0.25^2}),{\text{ }}1 - {\text{P}}(X = 0),{\text{ }}2{k^2} + \frac{k}{{10}} + 6{k^2}\)</p>
<p>correct substitution into formula for conditional probability <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{0.025}}{{1 - 0.475}},{\text{ }}\frac{{0.025}}{{0.525}}\)</p>
<p>0.0476190</p>
<p>\({\text{P}}(X = 2|X > 0) = \frac{1}{{21}}\) (exact), 0.0476 <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The maximum temperature \(T\), in degrees Celsius, in a park on six randomly selected days is shown in the following table. The table also shows the number of visitors, \(N\), to the park on each of those six days.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_17.34.22.png" alt="M17/5/MATME/SP2/ENG/TZ2/02"></p>
<p>The relationship between the variables can be modelled by the regression equation \(N = aT + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(a\) and of \(b\).</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>Write down the value of \(r\).</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>Use the regression equation to estimate the number of visitors on a day when the maximum temperature is 15 °C.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of set up <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)correct value for \(a\) or \(b\)</p>
<p>0.667315, 22.2117</p>
<p>\(a = 0.667,{\text{ }}b = 22.2\) <strong><em>A1A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.922958</p>
<p>\(r = 0.923\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0.667(15) + 22.2,{\text{ }}N(15)\)</p>
<p>32.2214 <strong><em>(A1)</em></strong></p>
<p>32 (visitors) (must be an integer) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A test has five questions. To pass the test, at least three of the questions must be </span><span style="font-family: times new roman,times; font-size: medium;">answered correctly.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The probability that Mark answers a question correctly is \(\frac{1}{5}\) </span><span style="font-family: times new roman,times; font-size: medium;">. Let <em>X</em> be the number of </span><span style="font-family: times new roman,times; font-size: medium;">questions that Mark answers correctly.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Bill also takes the test. Let <em>Y</em> be the number of questions that Bill answers correctly.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The following table is the probability distribution for <em>Y</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/molly.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find E(<em>X</em> ) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the probability that Mark passes the test.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(4a + 2b = 0.24\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Given that \({\rm{E}}(Y) = 1\) , find <em>a</em> and <em>b</em> .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find which student is more likely to pass the test.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(np\) , \(5 \times \frac{1}{5}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}(X) = 1\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of appropriate approach involving binomial <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(X \sim B\left( {5,\frac{1}{5}} \right)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing that Mark needs to answer 3 <strong>or more</strong> questions correctly <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(X \ge 3)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - {\rm{P}}(X \le 2)\) , \({\rm{P}}(X = 3) + {\rm{P}}(X = 4) + {\rm{P}}(X = 5)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(pass)}} = 0.0579\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of summing probabilities to 1 <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.67 + 0.05 + (a + 2b) + \ldots + 0.04 = 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">some simplification that clearly leads to required answer</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.76 + 4a + 2b = 1\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(4a + 2b = 0.24\) <em><strong>AG N0</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into the formula for expected value <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0(0.67) + 1(0.05) + \ldots + 5(0.04)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">some simplification <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.05 + 2a + 4b + \ldots + 5(0.04) = 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(13a + 5b = 0.75\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of solving <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = 0.05\) , \(b = 0.02\) <em><strong>A1A1 N4</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find probability Bill passes <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(Y \ge 3)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct value 0.19 <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Bill</strong> (is more likely to pass) <em><strong>A1 N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was wide spectrum of success on this problem. Candidates could normally find E(<em>X</em>) using \(n \times p\) but many failed to recognize that the "experiment" was binomial or that for Mark to the pass the test, he needed to answer either 3, 4 or 5 questions correctly. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (b) was generally well done although there were a number of algebraic errors particularly in part (b) (ii), leading to incorrect values of <em>a</em> and <em>b</em>. Again, appropriate use of the GDC here would have eliminated these errors. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In (c), candidates had trouble with the command term, "find" and often just wrote down either "Mark" or "Bill". </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows a probability distribution for the random variable \(X\), where \({\text{E}}(X) = 1.2\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.18.09.png" alt="M17/5/MATME/SP2/ENG/TZ2/10"></p>
</div>
<div class="specification">
<p>A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable \(X\).</p>
</div>
<div class="specification">
<p>A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(q\).</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 \(p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the probability of drawing three blue marbles.</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>Explain why the probability of drawing three white marbles is \(\frac{1}{6}\).</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>The bag contains a total of ten marbles of which \(w\) are white. Find \(w\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into \({\text{E}}(X)\) formula <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0(p) + 1(0.5) + 2(0.3) + 3(q) = 1.2\)</p>
<p>\(q = \frac{1}{{30}}\), 0.0333 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of summing probabilities to 1 <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(p + 0.5 + 0.3 + q = 1\)</p>
<p>\(p = \frac{1}{6},{\text{ }}0.167\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P (3 blue)}} = \frac{1}{{30}},{\text{ }}0.0333\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid reasoning <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P (3 white)}} = {\text{P(0 blue)}}\)</p>
<p>\({\text{P(3 white)}} = \frac{1}{6}\) <strong><em>AG</em></strong> <strong><em>N0</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid method <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P(3 white)}} = \frac{w}{{10}} \times \frac{{w - 1}}{9} \times \frac{{w - 2}}{8},{\text{ }}\frac{{_w{C_3}}}{{_{10}{C_3}}}\)</p>
<p>correct equation <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{w}{{10}} \times \frac{{w - 1}}{9} \times \frac{{w - 2}}{8} = \frac{1}{6},{\text{ }}\frac{{_w{C_3}}}{{_{10}{C_3}}} = 0.167\)</p>
<p>\(w = 6\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing one prize in first seven attempts <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ 1 \end{array}} \right),{\text{ }}{\left( {\frac{1}{6}} \right)^1}{\left( {\frac{5}{6}} \right)^6}\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ 1 \end{array}} \right){\left( {\frac{1}{6}} \right)^1}{\left( {\frac{5}{6}} \right)^6},{\text{ }}0.390714\)</p>
<p>correct approach <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ 1 \end{array}} \right){\left( {\frac{1}{6}} \right)^1}{\left( {\frac{5}{6}} \right)^6} \times \frac{1}{6}\)</p>
<p>0.065119</p>
<p>0.0651 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">In a school with 125 girls, each student is tested to see how many sit-up exercises (sit-ups) she can do in one minute. The results are given in the table below. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/margam.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>p</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the value of <em>q</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the median number of sit-ups.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the mean number of sit-ups.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(p = 65\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) for evidence of using sum is 125 (or \(99 - p\) ) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(q = 34\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of median position <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. 63rd student, \(\frac{{125}}{2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">median is 17 (sit-ups) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting into \(\frac{{\sum {fx} }}{{125}}\) </span><span style="font-family: times new roman,times; font-size: medium;"> <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{15(11) + 16(21) + 17(33) + 18(34) + 19(18) + 20(8)}}{{125}}\) , \(\frac{{2176}}{{125}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">mean \(= 17.4\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) of this question was well done.</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Finding the median seemed to be the most difficult for </span><span style="font-family: times new roman,times; font-size: medium;">the candidates. Most had the idea that it was in the middle but did not know how to find the </span><span style="font-family: times new roman,times; font-size: medium;">value.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">When calculating the mean, many ignored the frequencies.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following frequency table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_09.52.57.png" alt="M17/5/MATME/SP2/ENG/TZ1/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the mode.</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>Find the value of the range.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the mean.</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 the variance.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({\text{mode}} = 10\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({x_{\max }} - {x_{\min }}\), interval 2 to 11</p>
<p>\({\text{range}} = 9\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>7.14666</p>
<p>\({\text{mean}} = 7.15\) <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing that variance is \({({\text{sd}})^2}\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\operatorname{var} = {\sigma ^2},{\text{ 2.9060}}{{\text{5}}^2},{\text{ }}{2.92562^2}\)</p>
<p>\({\sigma ^2} = 8.44515\)</p>
<p>\({\sigma ^2} = 8.45\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Paula goes to work three days a week. On any day, the probability that she goes on a red bus is \(\frac{1}{4}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the expected number of times that Paula goes to work on a red bus in one week.</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 style="font-family: times new roman,times; font-size: medium;">In one week, find the probability that she goes to work on a red bus on exactly two days.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In one week, find the probability that she goes to work on a red bus on at least one day.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of binomial distribution (seen anywhere) <em><strong> (M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(X \sim {\text{B}}\left( {3{\text{, }}\frac{1}{4}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{mean}} = \frac{3}{4}\) (\(= 0.75\)) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="color: #000000;">\({\rm{P}}(X = 2) = \left( {\begin{array}{*{20}{c}}<br>3\\<br>2<br>\end{array}} \right){\left( {\frac{1}{4}} \right)^2}\left( {\frac{3}{4}} \right)\) </span></span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\rm{P}}(X = 2) = 0.141\) \(\left( { = \frac{9}{{64}}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. complement, \(1 - {\rm{P}}(X = 0)\) , adding probabilities</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(X = 0) = {(0.75)^3}\) \(\left( { = 0.422,\frac{{27}}{{64}}} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\rm{P}}(X \ge 1) = 0.578\) \(\left( { = \frac{{37}}{{64}}} \right)\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates did not recognize the binomial nature of this question, suggesting an overall </span><span style="font-family: times new roman,times; font-size: medium;">lack of preparation with this topic. Many used 7 days instead of 3 but could still earn marks in </span><span style="font-family: times new roman,times; font-size: medium;">follow-through if working was shown. Those who could use their GDC effectively often </span><span style="font-family: times new roman,times; font-size: medium;">answered correctly.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates did not recognize the binomial nature of this question, suggesting an overall </span><span style="font-family: times new roman,times; font-size: medium;">lack of preparation with this topic. Many used 7 days instead of 3 but could still earn marks in </span><span style="font-family: times new roman,times; font-size: medium;">follow-through if working was shown. Those who could use their GDC effectively often </span><span style="font-family: times new roman,times; font-size: medium;">answered correctly.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates did not recognize the binomial nature of this question, suggesting an overall </span><span style="font-family: times new roman,times; font-size: medium;">lack of preparation with this topic. Many used 7 days instead of 3 but could still earn marks in </span><span style="font-family: times new roman,times; font-size: medium;">follow-through if working was shown. Those who could use their GDC effectively often </span><span style="font-family: times new roman,times; font-size: medium;">answered correctly, although in part (c) some candidates misinterpreted the meaning of “at </span><span style="font-family: times new roman,times; font-size: medium;">least one” and found either \({\rm{P}}(X \le 1)\) or \(1 - {\rm{P}}(X \le 1)\) .</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Two fair 4-sided dice, one red and one green, are thrown. For each die, the faces are </span><span style="font-family: times new roman,times; font-size: medium;">labelled 1, 2, 3, 4. The score for each die is the number which lands face down.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">List the pairs of scores that give a sum of 6.</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><span style="font-family: times new roman,times; font-size: medium;">The probability distribution for the sum of the scores on the two dice is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bbw.png" alt></span></p>
<p><span style="font-size: medium;"><span style="font-family: times new roman,times;">Find the value of <em>p</em></span> , </span><span style="font-family: times new roman,times; font-size: medium;">of <em>q</em> , and of <em>r</em> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Fred plays a game. He throws two fair 4-sided dice four times. He wins a prize if the </span><span style="font-family: times new roman,times; font-size: medium;">sum is 5 on three or more throws.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that Fred wins a prize.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">three correct pairs <em><strong>A1A1A1 N3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. (2, 4), (3, 3), (4, 2) , <em>R</em>2<em>G</em>4, <em>R</em>3<em>G</em>3, <em>R</em>4<em>G</em>2</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = \frac{1}{{16}}\) , \(q = \frac{2}{{16}}\) , \(r = \frac{2}{{16}}\) <em><strong>A1A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">let <em>X</em> be the number of times the sum of the dice is 5</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(X \sim {\rm{B}}(n{\text{, }}p)\) , tree diagram, 5 sets of outcomes produce a win</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>one</strong> correct parameter <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(n = 4\) , \(p = 0.25\) , \(q = 0.75\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Fred wins prize is \({\rm{P}}(X \ge 3)\) <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">appropriate approach to find probability <strong><em>M1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. complement, summing probabilities, using a CDF function</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - 0.949 \ldots \) , \(1 - \frac{{243}}{{256}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , \(0.046875 + 0.00390625\) , \(\frac{{12}}{{256}} + \frac{1}{{256}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{probability of winning}} = 0.0508\) \(\left( {\frac{{13}}{{256}}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">All but the weakest candidates managed to score full marks for parts (a) and (b). An occasional error in part (a) was including additional pair(s) or listing (3, 3) twice. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">All but the weakest candidates managed to score full marks for parts (a) and (b). </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates found part (c) challenging, as they failed to recognize the binomial probability. Successful candidates generally used either the binomial CDF function or the sum of two binomial probabilities. Some used approaches like multiplying probabilities or tree diagrams, but these were less successful.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The weights, \(W\), of newborn babies in Australia are normally distributed with a mean <span class="s1">3.41 kg </span>and standard deviation <span class="s1">0.57 kg</span>. A newborn baby has a low birth weight if it weighs less than \(w\) <span class="s1">kg</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Given that </span><span class="s2">5.3% </span>of newborn babies have a low birth weight, find \(w\).</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="p1">A newborn baby has a low birth weight.</p>
<p class="p2">Find the probability that the baby weighs at least <span class="s1">2.15 kg</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(z = - 1.61643\), <img src="images/Schermafbeelding_2017-03-06_om_06.21.13.png" alt="N16/5/MATME/SP2/ENG/TZ0/05.a/M"></p>
<p class="p3">2.48863</p>
<p class="p4"><span class="s2">\(w = 2.49{\text{ (kg)}}\) <span class="Apple-converted-space"> </span></span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p4"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct value or expression (seen anywhere)</p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(0.053 - {\text{P}}(X \leqslant 2.15),{\text{ }}0.039465\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">evidence of conditional probability <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\text{P}}(2.15 \leqslant X \leqslant w}}{{{\text{P}}(X \leqslant w)}},{\text{ }}\frac{{0.039465}}{{0.053}}\)</p>
<p class="p3">0.744631</p>
<p class="p3">0.745 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Let \(C\) and \(D\) </span>be independent events, with \({\text{P}}(C) = 2k\) and \({\text{P}}(D) = 3{k^2}\), where \(0 < k < 0.5\)<span class="s1">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down an expression for \({\text{P}}(C \cap D)\) <span class="s1">in terms of \(k\).</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 class="p1">Given that \({\text{P}}(C \cap D) = 0.162\) <span class="s1">find \(k\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}(C'|D)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(C \cap D) = 2k \times 3{k^2}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\({\text{P}}(C \cap D) = 6{k^3}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>their </strong>correct equation <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;2k \times 3{k^2} = 0.162,{\text{ }}6{k^3} = 0.162\)</p>
<p class="p1">\(k = 0.3\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">finding <strong>their </strong>\({\text{P}}(C' \cap D)\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s1"><em>eg </em></span>\(0.4 \times 0.27,0.27 - 0.162,0.108\)</p>
<p class="p1">correct substitution into conditional probability formula <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s1"><em>eg</em></span>\(\;\;\;{\text{P}}(C'|D) = \frac{{{\text{P}}(C' \cap D)}}{{0.27}},{\text{ }}\frac{{(1 - 2k)(3{k^2})}}{{3{k^2}}}\)</p>
<p class="p1">\({\text{P}}(C'|D) = 0.4\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">recognizing \({\text{P}}(C'|D) = {\text{P}}(C')\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">finding <strong>their </strong>\({\text{P}}(C') = 1 - {\text{P}}(C)\) (only if first line seen) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;1 - 2k,{\text{ }}1 - 0.6\)</p>
<p class="p1">\({\text{P}}(C'|D) = 0.4\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A factory has two machines, <span class="s1">A </span>and <span class="s1">B</span>. The number of breakdowns of each machine is independent from day to day.</p>
<p class="p1">Let \(A\) be the number of breakdowns of Machine <span class="s1">A </span>on any given day. The probability distribution for \(A\) can be modelled by the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.41.35.png" alt="M16/5/MATME/SP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p class="p1">Let \(B\) be the number of breakdowns of Machine <span class="s1">B </span>on any given day. The probability distribution for \(B\) can be modelled by the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.43.25.png" alt="M16/5/MATME/SP2/ENG/TZ1/08.c+d"></p>
</div>
<div class="specification">
<p class="p1">On Tuesday, the factory uses both Machine <span class="s1">A </span>and Machine <span class="s1">B</span>. The variables \(A\) and \(B\) are independent.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(k\).</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="p1">(i) A day is chosen at random. Write down the probability that Machine <span class="s1">A </span>has no breakdowns.</p>
<p class="p1">(ii) Five days are chosen at random. Find the probability that Machine <span class="s1">A </span>has no breakdowns on exactly four of these days.</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 class="p1">Find \({\text{E}}(B)\).</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 class="p1">(i) Find the probability that there are exactly two breakdowns on Tuesday.</p>
<p class="p1">(ii) Given that there are exactly two breakdowns on Tuesday, find the probability that both breakdowns are of Machine <span class="s1">A</span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of summing to 1 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(0.55 + 0.3 + 0.1 + k = 1\)</p>
<p class="p2"><span class="Apple-converted-space">\(k = 0.05{\text{ (exact)}}\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>0.55 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>recognizing binomial probability <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(X:{\text{ }}B(n,{\text{ }}p),{\text{ }}\left( {\begin{array}{*{20}{c}} 5 \\ 4 \end{array}} \right),{\text{ }}{(0.55)^4}(1 - 0.55),{\text{ }}\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){p^r}{q^{n - r}}\)</p>
<p class="p2">\(P(X = 4) = 0.205889\)</p>
<p class="p2"><span class="Apple-converted-space">\(P(X = 4) = 0.206\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution into formula for \({\text{E}}(X)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(0.2 + (2 \times 0.08) + (3 \times 0.02)\)</p>
<p class="p2"><span class="Apple-converted-space">\({\text{E}}(B) = 0.42{\text{ (exact)}}\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>valid attempt to find one possible way of having 2 <span class="s1">breakdowns <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(2A,{\text{ }}2B,{\text{ }}1A\)<span class="s2"> and \(1B\), tree diagram</span></p>
<p class="p1">one correct calculation for 1 <span class="s1">way (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(0.1 \times 0.7,{\text{ }}0.55 \times 0.08,{\text{ }}0.3 \times 0.2\)</p>
<p class="p1">recognizing there are 3 ways of having 2 <span class="s1">breakdowns <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)<span class="s2">A twice or B </span>twice or one breakdown each</p>
<p class="p2">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\((0.1 \times 0.7) + (0.55 \times 0.08) + (0.3 \times 0.2)\)</p>
<p class="p3">\({\text{P(2 breakdowns)}} = 0.174{\text{ (exact)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>recognizing conditional probability <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(A|B),{\text{ P}}(2A|{\text{2breakdowns}})\)</p>
<p class="p2">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{0.1 \times 0.7}}{{0.174}}\)</p>
<p class="p2">\({\text{P}}(A = 2|{\text{two breakdowns}}) = 0.402298\)</p>
<p class="p2">\({\text{P}}(A = 2|{\text{two breakdowns}}) = 0.402\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates generally found parts (a), (b)(i) and (c) of this question the most straightforward and those who recognised the binomial distribution in (b)(ii) were usually able to obtain the required solution using their GDCs. Part (d)(i) proved to be more problematic with many candidates identifying one possible way of having two breakdowns (usually 1A and 1B), but not recognising three ways of having two breakdowns. Furthermore, many were not able to successfully calculate the probability of two breakdowns on one machine (and none on the other)<span class="s1">. </span>The conditional probability in (d)(ii) was generally recognised though and those who showed their working in full were able to score follow through marks in this part.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates generally found parts (a), (b)(i) and (c) of this question the most straightforward and those who recognised the binomial distribution in (b)(ii) were usually able to obtain the required solution using their GDCs. Part (d)(i) proved to be more problematic with many candidates identifying one possible way of having two breakdowns (usually 1A and 1B), but not recognising three ways of having two breakdowns. Furthermore, many were not able to successfully calculate the probability of two breakdowns on one machine (and none on the other)<span class="s1">. </span>The conditional probability in (d)(ii) was generally recognised though and those who showed their working in full were able to score follow through marks in this part.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates generally found parts (a), (b)(i) and (c) of this question the most straightforward and those who recognised the binomial distribution in (b)(ii) were usually able to obtain the required solution using their GDCs. Part (d)(i) proved to be more problematic with many candidates identifying one possible way of having two breakdowns (usually 1A and 1B), but not recognising three ways of having two breakdowns. Furthermore, many were not able to successfully calculate the probability of two breakdowns on one machine (and none on the other)<span class="s1">. </span>The conditional probability in (d)(ii) was generally recognised though and those who showed their working in full were able to score follow through marks in this part.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates generally found parts (a), (b)(i) and (c) of this question the most straightforward and those who recognised the binomial distribution in (b)(ii) were usually able to obtain the required solution using their GDCs. Part (d)(i) proved to be more problematic with many candidates identifying one possible way of having two breakdowns (usually 1A and 1B), but not recognising three ways of having two breakdowns. Furthermore, many were not able to successfully calculate the probability of two breakdowns on one machine (and none on the other)<span class="s1">. </span>The conditional probability in (d)(ii) was generally recognised though and those who showed their working in full were able to score follow through marks in this part.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A fisherman catches 200 fish to sell. He measures the lengths, <em>l</em> cm of these fish, and the results are shown in the frequency table below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/despic.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Calculate an estimate for the standard deviation of the lengths of the fish.</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><span style="font-family: times new roman,times; font-size: medium;">A cumulative frequency diagram is given below for the lengths of the fish.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/flags.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use the graph to answer the following.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Estimate the interquartile range.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Given that \(40\% \) of the fish have a length more than \(k{\text{ cm}}\), find the value of <em>k</em>.</span></p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In order to sell the fish, the fisherman classifies them as small, medium or large.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Small fish have a length less than \(20{\text{ cm}}\).</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Medium fish have a length greater than or equal to </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(20{\text{ cm}}\)</span> but less than </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(60{\text{ cm}}\)</span>.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Large fish have a length greater than or equal to </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(60{\text{ cm}}\)</span>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the probability that a fish is small.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The cost of a small fish is \(\$ 4\), a medium fish \(\$ 10\), and a large fish \(\$ 12\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Copy and complete the following table, which gives a probability distribution </span><span style="font-family: times new roman,times; font-size: medium;">for the cost \(\$ X\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/kirk.png" alt></span></p>
<p align="LEFT"> </p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\text{E}}(X)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using mid-interval values (5, 15, 25, 35, 50, 67.5, 87.5) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sigma = 19.8\) (cm) <em><strong>A2 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \({Q_1} = 15\) , \({Q_3} = 40\) <em><strong>(A1)(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(IQR = 25\) (accept any notation that suggests the interval 15 to 40) <em><strong>A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>METHOD 1 </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(60\% \) have a length less than <em>k</em> <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.6 \times 200 = 120\) <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 30\) (cm) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.4 \times 200 = 80\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(200 - 80 = 120\) <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 30\) (cm) <em><strong>A1 N2</strong></em> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(l < 20{\text{ cm}} \Rightarrow 70{\text{ fish}}\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P(small)}} = \frac{{70}}{{200}}( = 0.35)\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/kirk2.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1 N2</strong> </span></em></p>
<p><em> <span style="font-family: times new roman,times; font-size: medium;"><strong>[2 marks]</strong> </span></em></p>
<p> </p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution (of their <em>p</em> values) into formula for </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({\text{E}}(X)\)</span> <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(4 \times 0.35 + 10 \times 0.565 + 12 \times 0.085\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{E}}(X) = 8.07\) (accept \(\$ 8.07\)) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) defeated the vast majority of candidates who clearly had not been taught data entry. Some schools had attempted to teach how to use a formula rather than the GDC to find the standard deviation and their students invariably used this formula incorrectly. Use of the GDC was not only expected but should be emphasized as stated in the syllabus.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (b) revealed poor understanding of cumulative frequency and the IQR was often reported as an interval. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This was generally answered well although a number of candidates had difficulty with using the formula for expected value. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This was generally answered well although a number of candidates had difficulty with using the formula for expected value. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This was generally answered well although a number of candidates had difficulty with using the formula for expected value. </span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A random variable <em>X</em> is distributed normally with a mean of 20 and variance 9.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(X \le 24.5)\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \({\rm{P}}(X \le k) = 0.85\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Represent this information on the following diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/lowe.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of <em>k</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\sigma = 3\) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of attempt to find \({\rm{P}}(X \le 24.5)\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(z = 1.5\) , \(\frac{{24.5 - 20}}{3}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \le 24.5) = 0.933\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/NY.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1 N2</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> with shading that clearly extends to right of the mean, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for any correct label, either <em>k</em>, area or their value of <em>k</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(z = 1.03(64338)\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to set up an equation <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{k - 20}}{3} = 1.0364\) , \(\frac{{k - 20}}{3} = 0.85\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(k = 23.1\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question clearly demonstrated that some centres are still not giving adequate treatment to this topic. A great many candidates neglected to find the standard deviation and used the variance throughout. More still did not leave their answers to the required accuracy. Ignoring the use of the variance, responses to part (a) demonstrated that most candidates were comfortable finding the required probability using their calculator or setting up a suitable standardized equation. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question clearly demonstrated that some centres are still not giving adequate treatment to this topic. A great many candidates neglected to find the standard deviation and used the variance throughout. More still did not leave their answers to the required accuracy. Ignoring the use of the variance, responses to part (a) demonstrated that most candidates were comfortable finding the required probability using their calculator or setting up a suitable standardized equation. In part (b) (i), the sketch was often poorly shaded or incorrectly labelled. In (b) (ii), candidates frequently confused the <em>z</em>-score with the given probability of 0.85. Calculator approaches were more successful than working by hand but candidates should remember to avoid the use of calculator notation in their working, as it is not correct mathematical notation. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p>Adam is a beekeeper who collected data about monthly honey production in his bee hives. The data for six of his hives is shown in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_10.46.13.png" alt="N17/5/MATME/SP2/ENG/TZ0/08"></p>
<p>The relationship between the variables is modelled by the regression line with equation \(P = aN + b\).</p>
</div>
<div class="specification">
<p>Adam has 200 hives in total. He collects data on the monthly honey production of all the hives. This data is shown in the following cumulative frequency graph.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_10.49.33.png" alt="N17/5/MATME/SP2/ENG/TZ0/08.c.d.e"></p>
<p>Adam’s hives are labelled as low, regular or high production, as defined in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_10.51.25.png" alt="N17/5/MATME/SP2/ENG/TZ0/08.c.d.e_02"></p>
</div>
<div class="specification">
<p>Adam knows that 128 of his hives have a regular production.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(a\) and of \(b\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this regression line to estimate the monthly honey production from a hive that has 270 bees.</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>Write down the number of low production hives.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(k\);</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of hives that have a high production.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75 that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regular production hives.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of setup <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)correct value for \(a\) or \(b\)</p>
<p>\(a = 6.96103,{\text{ }}b = - 454.805\)</p>
<p>\(a = 6.96,{\text{ }}b = - 455{\text{ (accept }}6.96x - 455)\) <strong><em>A1A1 N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting \(N = 270\) into <strong>their</strong> equation <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(6.96(270) - 455\)</p>
<p>1424.67</p>
<p>\(P = 1420{\text{ (g)}}\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>40 (hives) <strong><em>A1 N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(128 + 40\)</p>
<p>168 hives have a production less than \(k\) <strong><em>(A1)</em></strong></p>
<p>\(k = 1640\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(200 - 168\)</p>
<p>32 (hives) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognize binomial distribution (seen anywhere) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(X \sim {\text{B}}(n,{\text{ }}p),{\text{ }}\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){p^r}{(1 - p)^{n - r}}\)</p>
<p>correct values <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(n = 40\) (check <em><strong>FT</strong></em>) and \(p = 0.75\) and \(r = 30,{\text{ }}\left( {\begin{array}{*{20}{c}} {40} \\ {30} \end{array}} \right){0.75^{30}}{(1 - 0.75)^{10}}\)</p>
<p>0.144364</p>
<p>0.144 <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following table shows the sales, \(y\) millions of dollars, of a company, \(x\) <span class="s1">years after it opened.</span></p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_10.36.15.png" alt></p>
<p class="p2">The relationship between the variables is modelled by the regression line with equation \(y = ax + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \(a\) and of \(b\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the value of \(r\).</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 class="p1">Hence estimate the sales in millions of dollars after seven years.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>evidence of set up <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)correct value for \(a\), \(b\) or \(r\)</p>
<p class="p1">\(a = 4.8,{\text{ }}b = 1.2\) <span class="Apple-converted-space"> </span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(r = 0.988064\)</p>
<p class="p1">\(r = 0.988\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution into <strong>their </strong>regression equation <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;4.8 \times 7 + 1.2\)</p>
<p class="p1">\(34.8\) (millions of dollars) (accept \(35\) and \({\text{34}}\,{\text{800}}\,{\text{000}}\)) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2 </em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<p class="p1"><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many answered this question completely correct, showing familiarity with the GDC operation for finding the equation of the line and coefficient. It was not uncommon to see \(a = 5.05\) and \(b = - 0.488\), which indicates incorrect use of the GDC lists to find the values.</p>
<p class="p1">Some candidates attempted an algebraic approach to finding the regression line and a few seemed to not recognize that \(r\) represents the coefficient of correlation.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many answered this question completely correct, showing familiarity with the GDC operation for finding the equation of the line and coefficient. It was not uncommon to see \(a = 5.05\) and \(b = - 0.488\), which indicates incorrect use of the GDC lists to find the values.</p>
<p class="p1">Some candidates attempted an algebraic approach to finding the regression line and a few seemed to not recognize that \(r\) represents the coefficient of correlation.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The heights of a group of seven-year-old children are normally distributed with mean \(117{\text{ cm}}\)</span><span style="font-family: times new roman,times;"><span style="font-size: medium;"> and standard deviation \(5{\text{ cm}}\). A child is chosen at random from the group.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that this child is taller than \(122.5{\text{ cm}}\).</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 align="LEFT"><span style="font-family: times new roman,times;"><span style="font-size: medium;">The heights of a group of seven-year-old children are normally distributed with mean \(117{\text{ cm}}\)</span><span style="font-size: medium;"> and standard deviation \(5{\text{ cm}}\). A child is chosen at random from the group.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The probability that this child is shorter than \(k{\text{ cm}}\) is \(0.65\). Find the value of k .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate method <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(z = \frac{{122.5 - 117}}{5}\) , sketch of normal curve showing mean and \(122.5\), \(1.1\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(Z < 1.1) = 0.8643\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.135666\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P(H}} > 122.5) = 0.136\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(z = 0.3853\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">set up equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{X - 117}}{5} = 0.3853\) , sketch </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 118.926602\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 199\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were many completely successful attempts at this question, with good use of formulae and calculator features. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were many completely successful attempts at this question, with good use of formulae and calculator features. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">However, in part (b) some candidates did not recognize the need to find the standardized value and set their equation equal to the probability given in the question, thus earning only one mark. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In a large university the probability that a student is left handed is 0.08. A sample of 150 students is randomly selected from the university. Let \(k\) be the expected number of left-handed students in this sample.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(k\).</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>Hence, find the probability that exactly \(k\) students are left handed;</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>Hence, find the probability that fewer than \(k\) students are left handed.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of binomial distribution (may be seen in part (b)) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(np,{\text{ }}150 \times 0.08\)</p>
<p>\(k = 12\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}\left( {X = 12} \right) = \left( {\begin{array}{*{20}{c}}<br> {150} \\ <br> {12} <br>\end{array}} \right){\left( {0.08} \right)^{12}}{\left( {0.92} \right)^{138}}\) <strong><em>(A1)</em></strong></p>
<p>0.119231</p>
<p>probability \( = 0.119\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition that \(X \leqslant 11\) <strong><em>(M1)</em></strong></p>
<p>0.456800</p>
<p>\({\text{P}}(X < 12) = 0.457\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Ten students were surveyed about the number of hours, \(x\), they spent browsing the Internet during week <span class="s1">1 </span>of the school year. The results of the survey are given below.</p>
<p class="p1">\[\sum\limits_{i = 1}^{10} {{x_i} = 252,{\text{ }}\sigma = 5{\text{ and median}} = 27.} \]</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">During week </span><span class="s2">4</span>, the survey was extended to all <span class="s2">200 </span>students in the school. The results are shown in the cumulative frequency graph:</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_16.35.16.png" alt="N16/5/MATME/SP2/ENG/TZ0/08.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the mean number of hours spent browsing the Internet.</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="p1"><span class="s1">During week </span><span class="s2">2</span>, the students worked on a major project and they each spent an additional five hours browsing the Internet. For week <span class="s2">2</span><span class="s3">, write down</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>the mean;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the standard deviation.</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 class="p1"><span class="s1">During week </span><span class="s2">3 </span>each student spent <span class="s2">5% </span>less time browsing the Internet than during week <span class="s2">1</span>. For week <span class="s2">3</span><span class="s3">, find</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>the median;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the variance.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the number of students who spent between <span class="s1">25 </span>and <span class="s1">30 </span><span class="s2">hours browsing the Internet.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Given that <span class="s1">10% </span>of the students spent more than <span class="s1"><em>k </em></span>hours browsing the Internet, find the maximum value of \(k\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to substitute into formula for mean <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{\Sigma x}}{{10}},{\text{ }}\frac{{252}}{n},{\text{ }}\frac{{252}}{{10}}\)</p>
<p class="p1">mean \( = 25.2{\text{ (hours)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>mean \( = 30.2{\text{ (hours)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 N1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(\sigma = 5{\text{ (hours)}}\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></span></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)</span>95%, 5% of 27</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(0.95 \times 27,{\text{ }}27 - (5\% {\text{ of }}27)\)</p>
<p class="p1">median \( = 25.65{\text{ (exact), }}25.7{\text{ (hours)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span><strong>METHOD 1</strong></p>
<p class="p1">variance \( = {({\text{standard deviation}})^2}\) (seen anywhere) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">valid attempt to find new standard deviation <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\({\sigma _{new}} = 0.95 \times 5,{\text{ }}4.75\)</p>
<p class="p1">variance \( = 22.5625{\text{ }}({\text{exact}}),{\text{ }}22.6\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">variance \( = {({\text{standard deviation}})^2}\) (seen anywhere) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">valid attempt to find new variance <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\({0.95^2}{\text{ }},{\text{ }}0.9025 \times {\sigma ^2}\)</p>
<p class="p1">new variance \( = 22.5625{\text{ }}({\text{exact}}),{\text{ }}22.6\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p3"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>both correct frequencies <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)</span>80, 150</p>
<p class="p1">subtracting <strong>their </strong>frequencies in either order <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(150 - 80,{\text{ }}80 - 150\)</p>
<p class="p1"><span class="s2">70 </span>(students) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>evidence of a valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)10% of 200, 90%</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(0.90 \times 200,{\text{ }}200 - 20\)</span><span class="s2">, 180 </span>students</p>
<p class="p3"><span class="s3">\(k = 35\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p3"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A random variable <strong><em>X</em></strong> is distributed normally with mean 450. It is known that </span><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X > a) = 0.27\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Represent all this information on the following diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/camel.png" alt></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that the standard deviation is 20, find <em>a</em> . Give your answer correct to the </span><span style="font-family: times new roman,times; font-size: medium;">nearest whole number.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><em><strong><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/hump.png" alt> A1A1A1 N3</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for 450 , <em><strong>A1</strong></em> for a to the right of the mean, <em><strong>A1</strong></em> for area 0.27 .</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong> M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(X < a) = 1 - {\rm{P}}(X > a)\) , 0.73</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = 462.256 \ldots \) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = 462\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The weights of fish in a lake are normally distributed with a mean of \(760\) g and standard deviation \(\sigma \). It is known that \(78.87\% \) of the fish have weights between \(705\) g and \(815\) g.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Write down the probability that a fish weighs more than \(760\) g.</p>
<p class="p1">(ii) Find the probability that a fish weighs less than \(815\) g.</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 class="p1">(i) <span class="Apple-converted-space"> </span>Write down the standardized value for \(815\) g.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find <span class="s1">\(\sigma \)</span>.</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 class="p1">A fishing contest takes place in the lake. Small fish, called tiddlers, are thrown back into the lake. The maximum weight of a tiddler is \(1.5\) standard deviations below the mean.</p>
<p class="p1">Find the maximum weight of a tiddler.</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 class="p1">A fish is caught at random. Find the probability that it is a tiddler.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">\(25\% \) of the fish in the lake are salmon. \(10\% \) of the salmon are tiddlers. Given that a fish caught at random is a tiddler, find the probability that it is a salmon.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> There may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</p>
<p> </p>
<p>(i) \({\text{P}}(X > 760) = 0.5{\text{ (exact), }}[0.499,{\text{ }}0.500]{\text{ }}\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p>(ii) evidence of valid approach <strong><em>(M1)</em></strong></p>
<p>recognising symmetry, \(\frac{{0.7887}}{2},{\text{ }}1 - {\text{P}}(W < 815),{\text{ }}\frac{{21.13}}{2} + 78.87\% \)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\(0.5 + 0.39435,{\text{ }}1 - 0.10565,\) <img src="image.html" alt></p>
<p>\(0.89435{\text{ (exact)}},{\text{ }}0.894{\text{ }}[0.894,{\text{ }}0.895]\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \(1.24999\) <strong><em>A1 N1</em></strong></p>
<p>\(z = 1.25{\text{ }}[1.24,{\text{ }}1.25]\)</p>
<p>(ii) evidence of appropriate approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\(\sigma = \frac{{x - \mu }}{{1.25}},{\text{ }}\frac{{815 - 760}}{\sigma }\)</p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\(1.25 = \frac{{815 - 760}}{\sigma },{\text{ }}\frac{{815 - 760}}{{1.24999}}\)</p>
<p>\(44.0003\)</p>
<p>\(\sigma = 44.0{\text{ }}[44.0,{\text{ }}44.1]{\text{ (g)}}\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(760 - 1.5 \times 44\)</p>
<p class="p1">\(693.999\)</p>
<p class="p1">\(694{\text{ }}[693,{\text{ }}694]{\text{ (g)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(0.0668056\)</p>
<p class="p1">\({\text{P}}(X < 694) = 0.0668{\text{ }}[0.0668,{\text{ }}0.0669]\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing conditional probability (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\({\text{P}}({\text{A}}|{\text{B}}),{\text{ }}\frac{{0.025}}{{0.0668}}\)</p>
<p class="p1">appropriate approach involving conditional probability <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\({\text{P}}(S|T) = \frac{{{\text{P}}(S{\text{ and }}T)}}{{{\text{P}}(T)}}\),</p>
<p class="p1">correct working</p>
<p class="p1"><em>eg</em>\(\;\;\;\)<span class="s1">P (salmon and tiddler) </span>\( = 0.25 \times 0.1,{\text{ }}\frac{{0.25 \times 0.1}}{{0.0668}}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(0.374220\)</p>
<p class="p1">\(0.374{\text{ }}[0.374,{\text{ }}0.375]\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<p class="p1"><strong><em>Total [16 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">There was a wide range of ability shown by candidates in this question. While the majority knew how to find probabilities, very few understood the concepts behind the normal distribution, including the answer to the straightforward question (ai). Quite a few students did not yet recognize the instruction “write down”, spending considerable time trying to find the 0.5 answer in (ai) or the standardised value in (bi).</p>
<p class="p1">Many candidates did not understand question (bi), giving either a probability value as the z-value or finding the correct value later on in part (bii) in the calculation of the standard deviation (without recognising its significance). For many of those who did understand these concepts, the context of the question was not a real challenge and a number of candidates managed to answer the entire question correctly.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">There was a wide range of ability shown by candidates in this question. While the majority knew how to find probabilities, very few understood the concepts behind the normal distribution, including the answer to the straightforward question (ai). Quite a few students did not yet recognize the instruction “write down”, spending considerable time trying to find the 0.5 answer in (ai) or the standardised value in (bi).</p>
<p class="p1">Many candidates did not understand question (bi), giving either a probability value as the z-value or finding the correct value later on in part (bii) in the calculation of the standard deviation (without recognising its significance). For many of those who did understand these concepts, the context of the question was not a real challenge and a number of candidates managed to answer the entire question correctly.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">There was a wide range of ability shown by candidates in this question. While the majority knew how to find probabilities, very few understood the concepts behind the normal distribution, including the answer to the straightforward question (ai). Quite a few students did not yet recognize the instruction “write down”, spending considerable time trying to find the 0.5 answer in (ai) or the standardised value in (bi).</p>
<p class="p1">Many candidates did not understand question (bi), giving either a probability value as the z-value or finding the correct value later on in part (bii) in the calculation of the standard deviation (without recognising its significance). For many of those who did understand these concepts, the context of the question was not a real challenge and a number of candidates managed to answer the entire question correctly.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">There was a wide range of ability shown by candidates in this question. While the majority knew how to find probabilities, very few understood the concepts behind the normal distribution, including the answer to the straightforward question (ai). Quite a few students did not yet recognize the instruction “write down”, spending considerable time trying to find the 0.5 answer in (ai) or the standardised value in (bi).</p>
<p class="p1">Many candidates did not understand question (bi), giving either a probability value as the z-value or finding the correct value later on in part (bii) in the calculation of the standard deviation (without recognising its significance). For many of those who did understand these concepts, the context of the question was not a real challenge and a number of candidates managed to answer the entire question correctly.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">There was a wide range of ability shown by candidates in this question. While the majority knew how to find probabilities, very few understood the concepts behind the normal distribution, including the answer to the straightforward question (ai). Quite a few students did not yet recognize the instruction “write down”, spending considerable time trying to find the 0.5 answer in (ai) or the standardised value in (bi).</p>
<p class="p1">Many candidates did not understand question (bi), giving either a probability value as the z-value or finding the correct value later on in part (bii) in the calculation of the standard deviation (without recognising its significance). For many of those who did understand these concepts, the context of the question was not a real challenge and a number of candidates managed to answer the entire question correctly.</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A competition consists of two independent events, shooting at 100 <span class="s1">targets and running for one hour.</span></p>
<p class="p2">The number of targets a contestant hits is the \(S\) score. The \(S\) <span class="s2">scores are normally distributed with mean 65 and standard deviation 10</span>.</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The distance in km </span>that a contestant runs in one hour is the \(R\) score. The \(R\) <span class="s1">scores are normally distributed with mean 12 and standard deviation 2.5</span>. The \(R\) score is independent of the \(S\) score.</p>
<p class="p1">Contestants are disqualified if their \(S\) <span class="s1">score is less than 50 </span><strong>and </strong>their \(R\) score is less than \(x\) <span class="s1">km</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A contestant is chosen at random. Find the probability that their \(S\) <span class="s1">score is less than 50</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 class="p1"><span class="s1">Given that 1% </span>of the contestants are disqualified, find the value of \(x\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">0.0668072</p>
<p class="p2"><span class="s1">\({\text{P}}(S < 50) = 0.0668{\text{ }}({\text{accept P}}(S \leqslant 49) = 0.0548)\) <span class="Apple-converted-space"> </span></span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>Eg</em>\(\,\,\,\,\,\)\({\text{P}}(S < 50) \times {\text{P}}(R < x)\)</p>
<p class="p1">correct equation (accept any variable) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(S < 50) \times {\text{P}}(R < x) = 1\% ,{\text{ }}0.0668072 \times p = 0.01,{\text{ P}}(R < x) = \frac{{0.01}}{{0.0668}}\)</p>
<p class="p1">finding the value of \({\text{P}}(R < x)\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{0.01}}{{0.0668}},{\text{ }}0.149684\)</p>
<p class="p2">9.40553</p>
<p class="p1"><span class="s1">\(x = 9.41{\text{ }}({\text{accept }}x = 9.74{\text{ from }}0.0548)\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">The first part of this question was a direct application of the normal distribution and most candidates who attempted the question obtained the correct value. In some cases, candidates gave the answer to 2 or 1 sf, losing a mark and taking the risk of obtaining an incorrect answer in the following question.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part b) proved challenging for various reasons. Many did not recognize that 0.01 was the probability of an intersection. Others did not know how to find that probability using the fact that the events were independent. Some candidates thought that the independence formula was \({\text{P}}(A) + {\text{P}}(B) = 0.01\) instead of \({\text{P}}(A) \times {\text{P}}(B) = 0.01\).</p>
<p class="p1">Of those that were able to find the correct value of \({\text{P}}(R < x)\), only some continued to find the value of \(x\).</p>
<p class="p1">Premature rounding in the answer to (a) sometimes caused the final mark in (b) to be lost unnecessarily.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The weights, in grams, of oranges grown in an orchard, are normally distributed with a mean of 297 g. It is known that 79 % of the oranges weigh more than 289 g and 9.5 % of the oranges weigh more than 310 g.</p>
</div>
<div class="specification">
<p>The weights of the oranges have a standard deviation of σ.</p>
</div>
<div class="specification">
<p>The grocer at a local grocery store will buy the oranges whose weights exceed the 35th percentile.</p>
</div>
<div class="specification">
<p>The orchard packs oranges in boxes of 36.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that an orange weighs between 289 g and 310 g.</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 standardized value for 289 g.</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>Hence, find the value of σ.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>To the nearest gram, find the minimum weight of an orange that the grocer will buy.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the grocer buys more than half the oranges in a box selected at random.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The grocer selects two boxes at random.</p>
<p>Find the probability that the grocer buys more than half the oranges in each box.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct approach indicating subtraction <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 0.79 − 0.095, appropriate shading in diagram</p>
<p>P(289 < <em>w</em> < 310) = 0.695 (exact), 69.5 % <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p>eg 1 − <em>p</em>, 21</p>
<p>−0.806421</p>
<p><em>z</em> = −0.806 <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>(i) & (ii)</p>
<p>correct expression for <em>z</em> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\(\frac{{289 - u}}{\sigma }\)</p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em> 1 − <em>p</em>, 21</p>
<p>−0.806421</p>
<p><em>z</em> = −0.806 (seen anywhere) <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to standardize <em><strong>(M1)</strong></em></p>
<p>eg \(\sigma = \frac{{289 - 297}}{z},\,\,\frac{{289 - 297}}{\sigma }\)</p>
<p>correct substitution with their <em>z</em> (do not accept a probability) <em><strong>A1</strong></em></p>
<p>eg \( - 0.806 = \frac{{289 - 297}}{\sigma },\,\,\frac{{289 - 297}}{{ - 0.806}}\)</p>
<p>9.92037</p>
<p>σ = 9.92 <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>(i) & (ii)</p>
<p>correct expression for <em>z</em> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\(\frac{{289 - u}}{\sigma }\)</p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em> 1 − <em>p</em>, 21</p>
<p>−0.806421</p>
<p><em>z</em> = −0.806 (seen anywhere) <em><strong>A1 N2</strong></em></p>
<p>valid attempt to set up an equation with <strong>their</strong> <em>z</em> (do not accept a probability) <em><strong>(M1)</strong></em></p>
<p>eg \( - 0.806 = \frac{{289 - 297}}{\sigma },\,\,\frac{{289 - 297}}{{ - 0.806}}\)</p>
<p>9.92037</p>
<p>σ = 9.92 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em> P(<em>W</em> < <em>w</em>) = 0.35, −0.338520 (accept 0.385320), diagram showing values in a standard normal distribution</p>
<p>correct score at the 35th percentile <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 293.177</p>
<p>294 (g) <em><strong>A1 N2</strong></em></p>
<p><strong>Note:</strong> If working shown, award <em><strong>(M1)(A1)A0</strong></em> for 293.<br>If no working shown, award <em><strong>N1</strong></em> for 293.177, <em><strong>N1</strong></em> for 293.</p>
<p>Exception to the <em><strong>FT</strong> </em>rule: If the score is incorrect, and working shown, award <em><strong>A1FT</strong></em> for correctly finding their minimum weight (by rounding up)</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of recognizing binomial (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(X \sim {\text{B}}\left( {36,\,\,p} \right),\,\,{}_n{C_a} \times {p^a} \times {q^{n - a}}\)</p>
<p>correct probability (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 0.65</p>
<p><strong>EITHER</strong></p>
<p>finding P(<em>X</em> ≤ 18) from GDC <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 0.045720</p>
<p>evidence of using complement <em><strong>(M1)</strong></em></p>
<p><em>eg</em> 1−P(<em>X</em> ≤ 18)</p>
<p>0.954279</p>
<p>P(<em>X</em> > 18) = 0.954 <em><strong>A1 N2</strong></em></p>
<p><strong>OR</strong></p>
<p>recognizing P(<em>X</em> > 18) = P(<em>X</em> ≥ 19) <em><strong>(M1)</strong></em></p>
<p>summing terms from 19 to 36 <em><strong>(A1)</strong></em></p>
<p><em>eg</em> P(<em>X</em> = 19) + P(<em>X</em> = 20) + … + P(<em>X</em> = 36)</p>
<p>0.954279</p>
<p>P(<em>X</em> > 18) = 0.954 <em><strong>A1 N2</strong></em></p>
<p><strong>[<em>5</em><em> marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct calculation <em><strong>(A1)</strong></em></p>
<p>\({0.954^2},\,\,\left( \begin{gathered}<br> 2 \hfill \\<br> 2 \hfill \\ <br>\end{gathered} \right){0.954^2}{\left( {1 - 0.954} \right)^0}\)</p>
<p>0.910650</p>
<p>0.911 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The weights in grams of 80 rats are shown in the following cumulative frequency diagram.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_8.png" alt></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>Do </em><strong><em>NOT </em></strong><em>write solutions on this page.</em></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the median weight of the rats.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the percentage of rats that weigh 70 grams or less.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The same data is presented in the following table.</span></p>
<table class="block_black_border" style="height: 72px; width: 802px;" border="0">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Weights </strong>\(w\)<strong> grams<br></strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0 \leqslant w \leqslant 30\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(30 < w \leqslant 60\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(60 < w \leqslant 90\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(90 < w \leqslant 120\)</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Frequency</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(45\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(q\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5\)</span></td>
</tr>
</tbody>
</table>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(p\).</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The same data is presented in the following table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"> </p>
<table class="block_black_border" style="height: 72px; width: 802px;" border="0">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Weights </strong>\(w\)<strong> grams<br></strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0 \leqslant w \leqslant 30\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(30 < w \leqslant 60\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(60 < w \leqslant 90\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(90 < w \leqslant 120\)</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Frequency</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(45\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(q\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5\)</span></td>
</tr>
</tbody>
</table>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(q\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-size: medium; font-family: 'times new roman', times;">The same data is presented in the following table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"> </p>
<table class="block_black_border" style="height: 72px; width: 802px;" border="0">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Weights </strong>\(w\)<strong> grams<br></strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0 \leqslant w \leqslant 30\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(30 < w \leqslant 60\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(60 < w \leqslant 90\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(90 < w \leqslant 120\)</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Frequency</strong></span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(45\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(q\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5\)</span></td>
</tr>
</tbody>
</table>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-size: medium; font-family: 'times new roman', times;">Use the values from the table to estimate the mean and standard deviation of the weights.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Assume that the weights of these rats are normally distributed with the mean and standard deviation estimated in part (c).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the percentage of rats that weigh 70 grams or less.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Assume that the weights of these rats are normally distributed with the mean and standard deviation estimated in part (c).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A sample of five rats is chosen at random. Find the probability that at most three rats weigh 70 grams or less.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">50 (g) <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">65 rats weigh less than 70 grams <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find a percentage <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{65}}{{80}},{\text{ }}\frac{{65}}{{80}} \times 100\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">81.25 (%) (exact), 81.3 <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(p = 10\) <strong><em>A2 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">subtracting to find \(q\)<em> </em><strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(75 - 45 - 10\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(q = 20\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of mid-interval values <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>\(15, 45, 75, 105\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\overline x = 52.5\) (exact), \(\sigma = 22.5\) (exact) <strong><em>A1A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">0.781650</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">78.2 (%) <strong><em>A2 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">recognize binomial probability <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(X \sim {\text{B}}(n,{\text{ }}p)\), \(\left( \begin{array}{c}5\\r\end{array} \right)\) \( \times {0.782^r} \times {0.218^{5 - r}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>\({\text{P}}(X \leqslant 3)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.30067\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.301\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A factory makes lamps. The probability that a lamp is defective is 0.05. A random </span><span style="font-family: times new roman,times; font-size: medium;">sample of 30 lamps is tested.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that there is at least one defective lamp in the sample.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A factory makes lamps. The probability that a lamp is defective is 0.05. A random </span><span style="font-family: times new roman,times; font-size: medium;">sample of 30 lamps is tested.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that there is at least one defective lamp in the sample, find the probability </span><span style="font-family: times new roman,times; font-size: medium;">that there are at most two defective lamps.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing binomial (seen anywhere) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{B}}(n{\text{, }}p)\), \({0.95^{30}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding \({\rm{P}}(X = 0) = 0.21463876\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. complement, summing probabilities </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.785361\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">probability is \(0.785\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">identifying correct outcomes (seen anywhere) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>e.g.</em> \({\rm{P}}(X = 1) + {\rm{P}}(X = 2)\) , 1 or 2 defective, \(0.3389 \ldots + 0.2586 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing conditional probability (seen anywhere) <em><strong>R1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>e.g.</em> \({\rm{P}}(A|B)\) , \({\rm{P}}(X \le 2|X \ge 1)\) , P(at most 2|at least 1)<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate approach involving conditional probability <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>e.g.</em> \(\frac{{{\rm{P}}(X = 1) + {\rm{P}}(X = 2)}}{{{\rm{P}}(X \ge 1)}}\) , \(\frac{{0.3389 \ldots + 0.2586 \ldots }}{{0.785 \ldots }}\) , \(\frac{{1{\text{ or }}2}}{{0.785}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.760847\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">probability is \(0.761\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[4 marks] </strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Although candidates seemed more confident in attempting binomial probabilities than in previous years, some of them failed to recognize the binomial nature of the question in part (a). Many knew that the complement was required, but often used \(1 - {\rm{P}}(X = 1)\) or \(1 - {\rm{P}}(X \le 1)\) instead of \(1 - {\rm{P}}(X = 0)\) .</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (b) was poorly answered. While some candidates recognized that it was a conditional probability, very few were able to correctly apply the formula, identify the outcomes and follow on to achieve the correct result. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Only a few could find the intersection of the events correctly. Several thought the numerator was a product (i.e. \({\rm{P}}({\text{at most 2}}) \times {\rm{P({\text{at least 1}})}}\)), and then cancelled common factors with the denominator. Others realized that \(x = 1\) and \(x = 2\) were required but multiplied their probabilities. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">This was the most commonly missed out question from Section A. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A random variable <em>X</em> is distributed normally with mean 450 and standard deviation 20.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(X \le 475)\) .</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 style="font-family: times new roman,times; font-size: medium;">Given that \({\rm{P}}(X > a) = 0.27\) , find \(a\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of attempt to find \({\rm{P}}(X \le 475)\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(Z \le 1.25)\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(X \le 475) = 0.894\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using the complement <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 0.73, \(1 - p\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(z = 0.6128\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{a - 450}}{{20}} = 0.6128\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 462\) <em><strong> A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">It remains very clear that some centres still do not give appropriate attention to the normal distribution. This is a major cause for concern. Most candidates had been taught the topic but many had difficulty understanding the difference between \(z\), \(F(z)\), \(a\) and \(x\) . Very little working was shown which demonstrated understanding. Although the GDC was used extensively, candidates often worked with the wrong tail and did not write their answers correct to 3 significant figures. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;">It remains very clear that some centres still do not give appropriate attention to the normal distribution. This is a major cause for concern. Most candidates had been taught the topic but many had difficulty understanding the difference between \(z\), \(F(z)\), \(a\) and \(x\) . Very little working was shown which demonstrated understanding. Although the GDC was used extensively, candidates often worked with the wrong tail and did not write their answers correct to 3 significant figures</span>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates had trouble with part (b), a majority never found the complement, instead using their GDCs to calculate the result, which many times was finding a for \(P(X \leqslant a) = 0.27\) instead of for \(P(X \geqslant a) = 0.27\) . Many others substituted the values of \(0.27\) or \(0.73\) into the equation, instead of the \(z\)-scores.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A forest has a large number of tall trees. The heights of the trees are normally distributed with a mean of \(53\) metres and a standard deviation of \(8\) metres. Trees are classified as giant trees if they are more than \(60\) metres tall.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A tree is selected at random from the forest.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that this tree is a giant.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A tree is selected at random from the forest.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that this tree is a giant, find the probability that it is taller than \(70\) metres.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Two trees are selected at random. Find the probability that they are both giants.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(100\) trees are selected at random.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the expected number of these trees that are giants.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(100\) trees are selected at random.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that at least \(25\) of these trees are giants.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(G) = {\text{P}}(H > 60,{\text{ }}z = 0.875,{\text{ P}}(H > 60) = 1 - 0.809,{\text{ N}}\left( {53, {8^2}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.190786\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(G) = 0.191\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">finding \({\text{P}}(H > 70) = 0.01679\) (seen anywhere) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing conditional probability <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(A\left| {B),{\text{ P}}(H > 70\left| {H > 60)} \right.} \right.\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{0.01679}}{{0.191}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.0880209\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X > 70\left| {G) = 0.0880} \right.\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to square their \({\text{P}}(G)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({0.191^2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.0363996\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}({\text{both }}G) = 0.0364\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into formula for \({\text{E}}(X)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(100(0.191)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(G) = 19.1{\text{ }}[19.0,{\text{ }}19.1]\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing binomial probability (may be seen in part (c)(i)) <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(X \sim {\text{B}}(n,{\text{ }}p)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach (seen anywhere) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(X \geqslant 25) = 1 - {\text{P}}(X \leqslant 24),{\text{ }}1 - {\text{P}}(X < a)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(X \leqslant 24) = 0.913 \ldots ,{\text{ }}1 - 0.913 \ldots \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.0869002\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \geqslant 25) = 0.0869\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c(ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Evan likes to play two games of chance, A and B.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">For game A, the probability that Evan wins is 0.9. He plays game A seven times.</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">For game B, the probability that Evan wins is <em>p</em> . He plays game B seven times.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that he wins exactly four games.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down an expression, in terms of <em>p</em> , for the probability that he wins exactly </span><span style="font-family: times new roman,times; font-size: medium;">four games.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Hence, find the values of <em>p</em> such that the probability that he wins exactly four </span><span style="font-family: times new roman,times; font-size: medium;">games is 0.15.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing binomial probability (may be seen in (b) or (c)) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. probability \( = \left( {\begin{array}{*{20}{c}}<br>7\\<br>4<br>\end{array}} \right){(0.9)^4}{(0.1)^3}\) , \(X \sim {\rm{B}}(7,0.9)\) , complementary probabilities </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">probability \(= 0.0230\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>A1A1 N2 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>7\\<br>4<br>\end{array}} \right){p^4}{(1 - p)^3}\) , \(35{p^4}{(1 - p)^3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <strong>A1</strong> for binomial coefficient (accept \(\left( {\begin{array}{*{20}{c}}<br>7\\<br>3<br>\end{array}} \right)\) ) , <em><strong>A1</strong></em> for \({p^4}{(1 - p)^3}\) . </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of attempting to solve <strong>their</strong> equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>7\\<br>4<br>\end{array}} \right){p^4}{(1 - p)^3} = 0.15\) , sketch </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.356\), \(0.770\) <em><strong>A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Parts of this question were handled very well by a great many candidates. Most were able to </span><span style="font-family: times new roman,times; font-size: medium;">recognize the binomial condition and had little difficulty with part (a). However, more than a </span><span style="font-family: times new roman,times; font-size: medium;">few reported the answer as 0.23, thus incurring the accuracy penalty.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Those candidates that </span><span style="font-family: times new roman,times; font-size: medium;">were successful in part (a) could easily write the required expression for part (b).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (c), many candidates set up the question correctly or set their expression from (b) </span><span style="font-family: times new roman,times; font-size: medium;">equal to 0.15, however few candidates considered the GDC as a method to solve the equation. </span><span style="font-family: times new roman,times; font-size: medium;">Rather, those who attempted usually tried to expand the polynomial, and <strong>still</strong> did not use the </span><span style="font-family: times new roman,times; font-size: medium;">GDC to solve <strong>this</strong> equation. A graphical approach to the solution would reveal that there are </span><span style="font-family: times new roman,times; font-size: medium;">two solutions for <em>p</em>, but few caught this subtlety.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A company makes containers of yogurt. The volume of yogurt in the containers is normally distributed with a mean of \(260\) ml and standard deviation of \(6\) ml.</p>
<p class="p1">A container which contains less than \(250\) ml of yogurt is <strong>underfilled</strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A container is chosen at random. Find the probability that it is underfilled.</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="p1">The company decides that the probability of a container being underfilled should be reduced to \(0.02\). It decreases the standard deviation to \(\sigma \) and leaves the mean unchanged.</p>
<p class="p1">Find \(\sigma \).</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 class="p1">The company changes to the new standard deviation, \(\sigma \), and leaves the mean unchanged.</p>
<p class="p1">A container is chosen at random for inspection. It passes inspection if its volume of yogurt is between \(250\) and \(271\) ml.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the probability that it passes inspection.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Given that the container is <strong>not</strong> underfilled, find the probability that it passes inspection.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A sample of \(50\) containers is chosen at random. Find the probability that \(48\) or more of the containers pass inspection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(0.0477903\)</p>
<p class="p1">probability \( = 0.0478\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}({\text{volume}} < 250) = 0.02\) <strong><em>(M1)</em></strong></p>
<p>\(z = - 2.05374\;\;\;\)(may be seen in equation) <strong><em>A1</em></strong></p>
<p>attempt to set up equation with \(z\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{{\mu - 260}}{\sigma } = z,{\text{ }}260 - 2.05(\sigma ) = 250\)</p>
<p>\(4.86914\)</p>
<p>\(\sigma = 4.87{\text{ (ml)}}\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(0.968062\)</p>
<p class="p1">\({\text{P}}(250 < {\text{Vol}} < 271) = 0.968\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>recognizing conditional probability (seen anywhere, including in correct working) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">eg\(\;\;\;{\text{P}}(A|B),{\text{ }}\frac{{{\text{P}}(A \cap B)}}{{{\text{P}}(B)}},{\text{ P}}(A \cap B) = {\text{P}}(A|B){\text{P}}(B)\)</p>
<p class="p1">correct value or expression for \(P\) (not underfilled) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">eg\(\;\;\;0.98,1 - 0.02,{\text{ }}1 - {\text{P}}(X < 250)\)</p>
<p class="p1">probability \( = \frac{{0.968}}{{0.98}}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(0.987818\)</p>
<p class="p1">probability \( = 0.988\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">evidence of recognizing binomial distribution (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;X\;\;\;{\text{B}}(50,{\text{ }}0.968),{\text{ binomial cdf, }}p = 0.968,{\text{ }}r = 47\)</p>
<p class="p1">\({\text{P}}(X \le 47\)) = 0.214106\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">evidence of using complement <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;1 - {\text{P}}(X \le 47\))</p>
<p class="p1">\(0.785894\)</p>
<p class="p1">probability \( = 0.786\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">evidence of recognizing binomial distribution (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;X\;\;\;{\text{B}}(50,{\text{ }}0.968),{\text{ binomial cdf, }}p = 0.968,{\text{ }}r = 47\)</p>
<p class="p1">\({\text{P(not pass)}} = 1 - {\text{P(pass)}} = 0.0319378\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">evidence of attempt to find \(P\) (\(2\) or fewer fail) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(0\), \(1\), or \(2\) not pass, \({\text{B}}(50,{\text{ }}2)\)</p>
<p class="p1">\(0.785894\)</p>
<p class="p1">probability \( = 0.786\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong>METHOD 3</strong></p>
<p class="p1">evidence of recognizing binomial distribution (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;X\;\;\;{\text{B}}(50,{\text{ }}0.968),{\text{ binomial cdf, }}p = 0.968,{\text{ }}r = 47\)</p>
<p class="p1">evidence of summing probabilities <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{\text{P}}(X = 48) + {\text{P}}(X = 49) + {\text{P}}(X = 50)\)</p>
<p class="p1">correct working</p>
<p class="p1"><em>eg</em>\(\;\;\;0.263088 + 0.325488 + 0.197317\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(0.785894\)</p>
<p class="p1">probability \( = 0.786\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<p class="p1"><strong><em>Total [16 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question saw many candidates competently using their GDCs to obtain required values, although a surprising number in part (b) chose to use an inefficient ‘guess and check’ method to try and obtain the standard deviation. Those using a correct approach often used a rounded z-score to find \(\sigma \) leading to an inaccurate final answer. In part (c), some candidates did not recognize or understand how to apply the given condition. In part (d), the binomial distribution, although often recognized, was not applied successfully.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question saw many candidates competently using their GDCs to obtain required values, although a surprising number in part (b) chose to use an inefficient ‘guess and check’ method to try and obtain the standard deviation. Those using a correct approach often used a rounded z-score to find \(\sigma \) leading to an inaccurate final answer. In part (c), some candidates did not recognize or understand how to apply the given condition. In part (d), the binomial distribution, although often recognized, was not applied successfully.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question saw many candidates competently using their GDCs to obtain required values, although a surprising number in part (b) chose to use an inefficient ‘guess and check’ method to try and obtain the standard deviation. Those using a correct approach often used a rounded z-score to find \(\sigma \) leading to an inaccurate final answer. In part (c), some candidates did not recognize or understand how to apply the given condition. In part (d), the binomial distribution, although often recognized, was not applied successfully.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question saw many candidates competently using their GDCs to obtain required values, although a surprising number in part (b) chose to use an inefficient ‘guess and check’ method to try and obtain the standard deviation. Those using a correct approach often used a rounded z-score to find \(\sigma \) leading to an inaccurate final answer. In part (c), some candidates did not recognize or understand how to apply the given condition. In part (d), the binomial distribution, although often recognized, was not applied successfully.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Samantha goes to school five days a week. When it rains, the probability that she goes to school by bus is 0.5. When it does not rain, the probability that she goes to school by bus is 0.3. The probability that it rains on any given day is 0.2.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">On a randomly selected school day, find the probability that Samantha goes to school by bus.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that Samantha went to school by bus on Monday, find the probability that it was raining.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">In a randomly chosen school week, find the probability that Samantha goes to school by bus on exactly three days.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">After \(n\) school days, the probability that Samantha goes to school by bus at least once is greater than \(0.95\). Find the smallest value of \(n\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">appropriate approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(R \cap B) + {\text{P}}(R' \cap B)\), tree diagram,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">one correct multiplication <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.2 \times 0.5,{\text{ }}0.24\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.2 \times 0.5 + 0.8 \times 0.3,{\text{ }}0.1 + 0.24\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P(bus)}} = 0.34 {\text{(exact)}}\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing conditional probability <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(A|B) = \frac{{{\text{P}}(A \cap B)}}{{{\text{P}}(B)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{0.2 \times 0.5}}{{0.34}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(R|B) = \frac{5}{{17}},{\text{ }}0.294\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing binomial probability <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(X \sim {\text{B}}(n,{\text{ }}p)\), \(\left( \begin{array}{c}5\\3\end{array} \right)\) \({(0.34)^3},{\text{ }}{(0.34)^3}{(1 - 0.34)^2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 3) = 0.171\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of using complement (seen anywhere) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(1 - {\text{P (none), }}1 - 0.95\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(1 - {\text{P (none)}} > 0.95,{\text{ P (none)}} < 0.05,{\text{ }}1 - {\text{P (none)}} = 0.95\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct inequality (accept equation) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(1 - {(0.66)^n} > 0.95,{\text{ }}{(0.66)^n} = 0.05\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(n > 7.209{\text{ (accept }}n = 7.209{\text{)}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 8\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach using guess and check/trial and error <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> finding \({\text{P}}(X \geqslant 1)\) for various values of <em>n</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">seeing the “cross over” values for the probabilities <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 7,{\text{ P}}(X \geqslant 1) = 0.9454,{\text{ }}n = 8,{\text{ P}}(X \geqslant 1) = 0.939\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognising \(0.9639 > 0.95\) <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 8\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A standard die is rolled 36 times. The results are shown in the following table.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bedtime.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the standard deviation.</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 style="font-family: times new roman,times; font-size: medium;">Write down the median score.</span></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><span style="font-family: times new roman,times; font-size: medium;">Find the interquartile range.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sigma = 1.61\) <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">median \( = 4.5\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({Q_1} = 3\) , \({Q_3} = 5\) (may be seen in a box plot) <em><strong>(A1)(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{IQR}} = 2\) (accept any notation that suggests the interval 3 to 5) <em><strong> A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Surprisingly, this question was not answered well primarily due to incorrect GDC use and a lack of understanding of the terms "median" and "interquartile range". Many candidates opted for an analytical approach in part (a) which always resulted in mistakes. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Some candidates wrote the down the mean instead of the median in part (b). </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Surprisingly, this question was not answered well primarily due to incorrect GDC use and a lack of understanding of the terms "median" and "interquartile range". </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following table shows the average number of hours per day spent watching television by seven mothers and each mother’s youngest child.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-13_om_17.56.59.png" alt></p>
<p class="p1">The relationship can be modelled by the regression line with equation \(y = ax + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the correlation coefficient.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the value of \(a\) and of \(b\).</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 class="p1">Elizabeth watches television for an average of \(3.7\) hours per day.</p>
<p class="p1">Use your regression line to predict the average number of hours of television watched per day by Elizabeth’s youngest child. Give your answer correct to one decimal place.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>evidence of valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(1\) correct value for \(r\), (or for \(a\) or \(b\), seen in (ii))</p>
<p class="p1">\(0.946591\)</p>
<p class="p1">\(r = 0.947\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(a = 0.500957,{\text{ }}b = 0.803544\)</p>
<p class="p1">\(a = 0.501,{\text{ }}b = 0.804\) <span class="Apple-converted-space"> </span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">substituting \(x = 3.7\) into <strong>their </strong>equation <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;0.501(3.7) + 0.804\)</p>
<p class="p1">\(2.65708\;\;\;\)(\(2\) hours \(39.4252\) minutes) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(y = 2.7\) (hours) (<strong>must </strong>be correct \(1\) dp, accept \(2\) hours \(39.4\) minutes) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [7 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates continue to have difficulty using their GDCs to find and correctly identify the coefficients of a linear regression. Both the \(r\) and \({r^2}\) values were often given as candidates were hedging their bets and were not entirely clear which one to give. Candidates frequently were unable to find the correct values for \(a\) and \(b\) suggesting a lack of familiarity working with GDCs. It was also surprising to see so many candidates leave these values to only one significant figure sacrificing all the marks for this part. Subsequent use of their line to find \(y\) for a given \(x\) was not difficult for most, but answers were not often given to the required accuracy of one decimal place.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates continue to have difficulty using their GDCs to find and correctly identify the coefficients of a linear regression. Both the \(r\) and \({r^2}\) values were often given as candidates were hedging their bets and were not entirely clear which one to give. Candidates frequently were unable to find the correct values for \(a\) and \(b\) suggesting a lack of familiarity working with GDCs. It was also surprising to see so many candidates leave these values to only one significant figure sacrificing all the marks for this part. Subsequent use of their line to find \(y\) for a given \(x\) was not difficult for most, but answers were not often given to the required accuracy of one decimal place.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The probability of obtaining “tails” when a biased coin is tossed is \(0.57\). The coin is </span><span style="font-family: times new roman,times; font-size: medium;">tossed ten times. Find the probability of obtaining </span><span style="font-family: times new roman,times; font-size: medium;"><strong>at least</strong> four tails.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The probability of obtaining “tails” when a biased coin is tossed is 0.57. The coin is </span><span style="font-family: times new roman,times; font-size: medium;">tossed ten times. Find the probability of obtaining </span><span style="font-family: times new roman,times; font-size: medium;">the fourth tail on the tenth toss.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing binomial distribution <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(X \sim {\rm{B}}(10,0.57)\) , \(p = 0.57\) , \(q = 0.43\)</span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">EITHER </span></strong></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(X \le 3) = 2.16 \times {10^{ - 4}} + 0.00286 + 0.01709 + 0.06041\) \(( = 0.08057)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1) </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using complement <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - \) any probability, \({\rm{P}}(X \ge 4) = 1 - {\rm{P}}(X \le 3)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.919423 \ldots \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \ge 4) = 0.919\) <em><strong>A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>OR</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">summing the probabilities from \(X = 4\) to \(X = 10\) <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression or values <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\sum\limits_{r = 4}^{10} {\left( {\begin{array}{*{20}{c}}<br>{10}\\<br>r<br>\end{array}} \right)} {(0.57)^r}{(0.43)^{10 - r}}\) , \(0.14013 + 0.2229 + \ldots + 0.02731 + 0.00362\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">0.919424 </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \ge 4) = 0.919\) </span><em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> e.g. three tails in nine tosses, \(\left( {\begin{array}{*{20}{c}}<br>9\\<br>3<br>\end{array}} \right){(0.57)^3}{(0.43)^6}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> correct calculation </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> e.g. \(\left( {\begin{array}{*{20}{c}}<br>9\\<br>3<br>\end{array}} \right){(0.57)^3}{(0.43)^6} \times 0.57\) , \(0.09834 \times 0.57\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.05605178 \ldots \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\text{P(4th tail on 10th toss)}} = 0.0561\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> [3 marks]</strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This was an accessible problem that created some difficulties for candidates. Most were able to recognize the binomial nature of the problem but were confused by the phrase "at least four tails" which was often interpreted as the complement of four or less. Poor algebraic manipulation also led to unnecessary errors that the calculator approach would have avoided. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This was an accessible problem that created some difficulties for candidates. Most were able to recognize the binomial nature of the problem but were confused by the phrase "at least four tails" which was often interpreted as the complement of four or less. Poor algebraic manipulation also led to unnecessary errors that the calculator approach would have avoided. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the mean weight, <em>y</em> kg , of children who are <em>x</em> years old.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The relationship between the variables is modelled by the regression line with equation \(y = ax + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of<em> a</em> and of <em>b</em>.</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>Write down the correlation coefficient.</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>Use your equation to estimate the mean weight of a child that is 1.95 years old.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> correct value for <em>a</em> or <em>b</em> (or for <em>r</em> seen in (ii))</p>
<p><em>a</em> = 1.91966 <em>b</em> = 7.97717</p>
<p><em>a</em> = 1.92, <em>b</em> = 7.98 <em><strong>A1A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.984674</p>
<p><em>r </em>= 0.985 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into their equation <em><strong>(A1)</strong></em><br><em>eg</em> 1.92 × 1.95 + 7.98</p>
<p>11.7205</p>
<p>11.7 (kg) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The weights of players in a sports league are normally distributed with a mean of \(76.6{\text{ kg}}\), </span><span style="font-family: times new roman,times; font-size: medium;">(correct to three significant figures). It is known that \(80\% \) of the players have weights </span><span style="font-family: times new roman,times; font-size: medium;">between \(68{\text{ kg}}\) and \(82{\text{ kg}}\). The probability that a player weighs less than \(68{\text{ kg}}\) is 0.05.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that a player weighs more than \(82{\text{ kg}}\).</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the standardized value, <em>z</em>, for \(68{\text{ kg}}\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, find the standard deviation of weights.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">To take part in a tournament, a player’s weight must be within 1.5 standard deviations </span><span style="font-family: times new roman,times; font-size: medium;">of the mean.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the set of all possible weights of players that take part in the </span><span style="font-family: times new roman,times; font-size: medium;">tournament.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) A player is selected at random. Find the probability that the player takes </span><span style="font-family: times new roman,times; font-size: medium;">part in the tournament.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Of the players in the league, \(25\% \) are women. Of the women, \(70\% \) take part in </span><span style="font-family: times new roman,times; font-size: medium;">the tournament.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that a player selected at random takes part in the tournament, find the </span><span style="font-family: times new roman,times; font-size: medium;">probability that the selected player is a woman.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - 0.85\) , diagram showing values in a normal curve </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(w \ge 82) = 0.15\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(z = - 1.64\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of appropriate approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 1.64 = \frac{{x - \mu }}{\sigma }\) , \(\frac{{68 - 76.6}}{\sigma }\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 1.64 = \frac{{68 - 76.6}}{\sigma }\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sigma = 5.23\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(68.8 \le {\rm{weight}} \le 84.4\) <em><strong>A1A1A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for 68.8, <em><strong>A1</strong></em> for 84.4, <em><strong>A1</strong></em> for giving answer as an interval. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of appropriate approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}( - 1.5 \le z \le 1.5)\) , \({\rm{P}}(68.76 < y < 84.44)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(qualify)}} = 0.866\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing conditional probability <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(A|B) = \frac{{{\rm{P}}(A \cap B)}}{{{\rm{P}}(B)}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}({\text{woman and qualify}}) = 0.25 \times 0.7\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}({\rm{woman}}|{\rm{qualify}}) = \frac{{0.25 \times 0.7}}{{0.866}}\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}({\rm{woman}}|{\rm{qualify}}) = 0.202\) <em><strong>A1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This question was quite accessible to those candidates in centres where this topic is given the </span><span style="font-family: times new roman,times; font-size: medium;">attention that it deserves. Most candidates handled part (a) well using the basic properties of a </span><span style="font-family: times new roman,times; font-size: medium;">normal distribution.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (b) (i), candidates often confused the <em>z</em>-score with the area in the </span><span style="font-family: times new roman,times; font-size: medium;">table which led to a standard deviation that was less than zero in part (b) (ii). At this point, </span><span style="font-family: times new roman,times; font-size: medium;">candidates “fudged” results in order to continue with the remaining parts of the question. In </span><span style="font-family: times new roman,times; font-size: medium;">(b) (ii), the “hence” command was used expecting candidates to use the results of (b) (i) to </span><span style="font-family: times new roman,times; font-size: medium;">find a standard deviation of 4.86. Unfortunately, many decided to use their answers and the </span><span style="font-family: times new roman,times; font-size: medium;">information from part (a) resulting in quite a different standard deviation of 5.79. Recognizing </span><span style="font-family: times new roman,times; font-size: medium;">the inconsistency in the question, full marks were awarded for this approach, as well as full </span><span style="font-family: times new roman,times; font-size: medium;">follow-through in subsequent parts of the question.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Candidates could obtain full marks easily in part (c) with little understanding of a normal </span><span style="font-family: times new roman,times; font-size: medium;">distribution but they often confused <em>z</em>-scores with data values, adding and subtracting 1.5 </span><span style="font-family: times new roman,times; font-size: medium;">from the mean of 76.</span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (d), few recognized the conditional nature of the question and only determined the </span><span style="font-family: times new roman,times; font-size: medium;">probability that a woman qualifies <strong>and</strong> takes part in the tournament.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The probability of obtaining heads on a biased coin is 0.18. The coin is tossed </span><span style="font-family: times new roman,times; font-size: medium;">seven times.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability of obtaining <strong>exactly</strong> two heads.</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 style="font-family: times new roman,times; font-size: medium;">Find the probability of obtaining <strong>at least</strong> two heads.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using binomial probability <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(X = 2) = \left( {\begin{array}{*{20}{c}}<br>7\\<br>2<br>\end{array}} \right){(0.18)^2}{(0.82)^5}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 2) = 0.252\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using the complement <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - ({\rm{P}}(X \le 1))\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X \le 1) = 0.632\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X \ge 2) = 0.368\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of attempting to sum probabilities <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\text{P(2 heads) + P(3 heads)}} + \ldots + {\text{P(7 heads)}}\) , \(0.252 + 0.0923 + \ldots \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct values for each probability <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.252 + 0.0923 + 0.0203 + 0.00267 + 0.0002 + 0.0000061\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(}}X \ge {\rm{2) = 0}}{\rm{.368}}\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[3 marks]</strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates who recognized binomial probability answered this question very well, using their GDC to perform the final calculations. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Some candidates misinterpreted the meaning of "at least two" in part (b), and instead found \({\text{P(}}X > {\rm{2)}}\) . Others wrote down a correct interpretation but accumulated to in their GDC (e.g. <em>binomcdf</em> (7, 0.18, 2)). Still, the number of candidates who either left this question blank or approached the question without binomial considerations suggests that this topic continues to be neglected in some centres. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The probability of obtaining heads on a biased coin is 0.4. The coin is tossed </span><span style="font-family: times new roman,times; font-size: medium;">600 times.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the mean number of heads.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the standard deviation of the number of heads.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the probability that the number of heads obtained is less than one standard </span><span style="font-family: times new roman,times; font-size: medium;">deviation away from the mean.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) recognizing binomial with \(n = 600\) , \(p = 0.4\) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}(X) = 240\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into formula for variance or standard deviation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. 144, \(\sqrt {600 \times 0.4 \times 0.6} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">sd = 12 <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find range of values <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(240 \pm 12\) \(228 < X < 252\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correct approach <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(X \le 251) - {\rm{P}}(X \le 228)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(228 < X < 252) = 0.662\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A random variable \(X\) is normally distributed with \(\mu = 150\) and \(\sigma = 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the interquartile range of \(X\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing one quartile probability (may be seen in a sketch) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{P}}(X < {Q_3}) = 0.75\) , \(0.25\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding standardized value for either quartile <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em>\(z = 0.67448 \ldots \) , \(z = - 0.67448 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up equation (must be with \(z\)-values) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(0.67 = \frac{{{Q_3} - 150}}{{10}}\) , \( - 0.67448 = \frac{{x - 150}}{{10}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">one correct quartile </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({Q_3} = 156.74 \ldots \) , \({Q_1} = 143.25 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> other correct quartile, \({Q_3} - \mu = 6.744 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach for IQR (seen anywhere) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({Q_3} - {Q_1}\) , \(2({Q_3} - \mu )\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">IQR \( = 13.5\) <em><strong>A1 N4 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">This was an accessible problem that created difficulties for candidates. Although they recognized and often wrote down a formula for IQR, most did not understand the conceptual nature of the first and third quartiles. Those who did could solve the problem effectively using their GDC in relatively few steps. Candidates that were able to start this question often drew the normal curve and gave quartile values at \(140\) and \(160\). This generally led to a solution which while wrong, was also clearly inadequate for the indicated 7 marks.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The mass \(M\) <span class="s1">of a decaying substance is measured at one minute intervals. The points \((t,{\text{ }}\ln M)\) are plotted for \(0 \leqslant t \leqslant 10\)</span>, where \(t\) is in minutes. The line of best fit is drawn. This is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.21.06.png" alt="M16/5/MATME/SP2/ENG/TZ1/05"></p>
<p class="p1">The correlation coefficient for this linear model is \(r = - 0.998\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">State <strong>two </strong></span>words that describe the linear correlation between \(\ln M\) <span class="s1">and \(t\).</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 class="p1">The equation of the line of best fit is \(\ln M = - 0.12t + 4.67\)<span class="s1">. Given that \(M = a \times {b^t}\), find the value of \(b\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">strong, negative (both required) <strong><em>A2 N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({{\text{e}}^{\ln M}} = {{\text{e}}^{ - 0.12t + 4.67}}\)</p>
<p class="p1">correct use of exponent laws for \({{\text{e}}^{ - 0.12t + 4.67}}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({{\text{e}}^{ - 0.12t}} \times {{\text{e}}^{4.67}}\)</p>
<p class="p1">comparing coefficients/terms <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({b^t} = {{\text{e}}^{ - 0.12t}}\)</p>
<p class="p1"><span class="Apple-converted-space">\(b = {{\text{e}}^{ - 0.12}}{\text{ (exact), }}0.887\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln (a \times {b^t}) = - 0.12t + 4.67\)</p>
<p class="p1">correct use of log laws for \(\ln (a{b^t})\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln a + t\ln b\)</p>
<p class="p1">comparing coefficients <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\( - 0.12 = \ln b\)</p>
<p class="p1"><span class="Apple-converted-space">\(b = {{\text{e}}^{ - 0.12}}{\text{ (exact), }}0.887\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This turned out to be one of the more challenging questions on the paper. Although many candidates correctly described the linear correlation in part (a), a surprisingly large number of candidates were unable to do so.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was not well done with many candidates unable to transfer their knowledge of exponentials and/or log manipulation to the question. After rewriting the line of best fit as \( = {{\text{e}}^{ - 0.12t + 4.67}}\), candidates were neither able to apply the rules for exponentials correctly nor were they familiar with the method of comparing coefficients to find the answer. There were numerous failed attempts of trying to estimate points from the graph and substitute these into the equation, while others arbitrarily chose values for \(t\) in an effort to set up a system of equations, the latter having some measure of success.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In any given season, a soccer team plays 65 % of their games at home.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When the team plays at home, they win 83 % of their games.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When they play away from home, they win 26 % of their games.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The team plays one game.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that the team wins the game.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If the team does not win the game, find the probability that the game was played </span><span style="font-family: times new roman,times; font-size: medium;">at home.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. tree diagram or a table </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P(win)}} = {\rm{P}}(H \cap W) + {\rm{P}}(A \cap W)\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = (0.65)(0.83) + (0.35)(0.26)\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 0.6305\) (or 0.631) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using complement <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 - p\) , 0.3695</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">choosing a formula for conditional probability <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(H|W') = \frac{{{\rm{P}}(W' \cap H)}}{{{\rm{P}}(W')}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{(0.65)(0.17)}}{{0.3695}}\) \(\left( { = \frac{{0.1105}}{{0.3695}}} \right)\) <em><strong>A1</strong></em></span></p>
<p><span style="font-size: medium;"><span style="font-family: times new roman,times;"> P(home) = 0.299 </span><em><strong><span style="font-family: times new roman,times;">A1 N3 </span></strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was nearly always correctly answered by those who attempted the question, but part (b) (conditional probability) was poorly done. A surprisingly small number of students drew a tree diagram in part (a) and those who did answered this part and part (b) well. Many found the correct complement in part (b) but could not make any further progress. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (b) (conditional probability) was poorly done. A surprisingly small number of students drew a tree diagram in part (a) and those who did answered this part and part (b) well. Many found the correct complement in part (b) but could not make any further progress. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A biased four-sided die is rolled. The following table gives the probability of each score.</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> k</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>Calculate the expected value of the score.</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 die is rolled 80 times. On how many rolls would you expect to obtain a three?</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of summing to 1 <em><strong>(M1)</strong></em></p>
<p><em>eg </em>0.28 + <em>k</em> + 1.5 + 0.3 = 1, 0.73 + <em>k</em> = 1</p>
<p><em>k</em> = 0.27 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into formula for E (<em>X</em>) <em><strong>(A1)</strong></em><br>eg 1 × 0.28 + 2 × <em>k</em> + 3 × 0.15 + 4 × 0.3</p>
<p>E (<em>X</em>) = 2.47 (exact) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg np</em>, 80 × 0.15</p>
<p>12 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A random variable \(X\) is normally distributed with mean, \(\mu \). In the following diagram, the shaded region between 9 and \(\mu \) represents 30% of the distribution.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_10.15.49.png" alt="M17/5/MATME/SP2/ENG/TZ1/09"></p>
</div>
<div class="specification">
<p>The standard deviation of \(X\) is 2.1.</p>
</div>
<div class="specification">
<p>The random variable \(Y\) is normally distributed with mean \(\lambda \) and standard deviation 3.5. The events \(X > 9\) and \(Y > 9\) are independent, and \(P\left( {(X > 9) \cap (Y > 9)} \right) = 0.4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(X < 9)\).</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 \(\mu \).</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 \(\lambda \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(Y > 9\), find \({\text{P}}(Y < 13)\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(X < \mu ) = 0.5,{\text{ }}0.5 - 0.3\)</p>
<p>\({\text{P}}(X < 9) = 0.2\) (exact) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(z = - 0.841621\) (may be seen in equation) <strong><em>(A1)</em></strong></p>
<p>valid attempt to set up an equation with <strong>their</strong> \(z\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\( - 0.842 = \frac{{\mu - X}}{\sigma },{\text{ }} - 0.842 = \frac{{X - \mu }}{\sigma },{\text{ }}z = \frac{{9 - \mu }}{{2.1}}\)</p>
<p>10.7674</p>
<p>\(\mu = 10.8\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(X > 9) = 0.8\) (seen anywhere) <strong><em>(A1)</em></strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(A) \times {\text{P}}(B)\)</p>
<p>correct equation <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0.8 \times {\text{P}}(Y > 9) = 0.4\)</p>
<p>\({\text{P}}(Y > 9) = 0.5\) <strong><em>A1</em></strong></p>
<p>\(\lambda = 9\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finding \({\text{P}}(9 < Y < 13) = 0.373450\) (seen anywhere) <strong><em>(A2)</em></strong></p>
<p>recognizing conditional probability <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(A|B),{\text{ P}}(Y < 13|Y > 9)\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\text{0.373}}}}{{0.5}}\)</p>
<p>0.746901</p>
<p>0.747 <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the independent events <em>A</em> and <em>B</em> . Given that \({\rm{P}}(B) = 2{\rm{P}}(A)\) , and \({\rm{P}}(A \cup B) = 0.52\) , </span><span style="font-family: times new roman,times; font-size: medium;">find \({\rm{P}}(B)\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">for independence \({\rm{P}}(A \cap B) = {\rm{P}}(A) \times {\rm{P}}(B)\) <em><strong> (R1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">expression for \({\rm{P}}(A \cap B)\) , indicating \({\rm{P}}(B) = 2{\rm{P}}(A)\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(A) \times 2{\rm{P}}(A)\) , \(x \times 2x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting into \({\rm{P}}(A \cup B) = {\rm{P}}(A) + {\rm{P}}(B) - {\rm{P}}(A \cap B)\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>correct</strong> substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.52 = x + 2x - 2{x^2}\) , \(0.52 = {\rm{P}}(A) + 2{\rm{P}}(A) - 2{\rm{P}}(A){\rm{P}}(A)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct solutions to the equation <em><strong>(A2)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.2\), \(1.3\) (accept the single answer \(0.2\))</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(B) = 0.4\) <em><strong>A1 N6</strong></em></span></p>
<p align="LEFT"><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">for independence \({\rm{P}}(A \cap B) = {\rm{P}}(A) \times {\rm{P}}(B)\) <em><strong>(R1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">expression for \({\rm{P}}(A \cap B)\) , indicating \({\rm{P}}(A) = \frac{1}{2}{\rm{P}}(B)\) <em><strong> </strong></em></span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(B) \times \frac{1}{2}{\rm{P}}(B)\) , \(x \times \frac{1}{2}x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting into \({\rm{P}}(A \cup B) = {\rm{P}}(A) + {\rm{P}}(B) - {\rm{P}}(A \cap B)\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>correct</strong> substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.52 = 0.5x + x - 0.5{x^2}\) , \(0.52 = 0.5{\rm{P}}(B) + {\rm{P}}(B) - 0.5{\rm{P}}(B){\rm{P}}(B)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct solutions to the equation <em><strong>(A2)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. 0.4, 2.6 (accept the single answer 0.4)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(B) = 0.4\) (accept \(x = 0.4\) if <em>x</em> set up as \({\rm{P}}(B)\) ) <em><strong>A1 N6</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates confused the concept of independence of events with mutual exclusivity, mistakenly trying to use the formula \({\rm{P}}(A \cup B) = {\rm{P}}(A) + {\rm{P}}(B)\) . Those who did recognize that \({\rm{P}}(A \cap B) = {\rm{P}}(A) \times {\rm{P}}(B)\) were often able to find the correct equation, but many were unable to use their GDC to solve it. A few provided two answers without discarding the value greater than one. </span></p>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A company uses two machines, A and B, to make boxes. Machine A makes \(60\% \) of </span><span style="font-family: times new roman,times; font-size: medium;">the boxes.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(80\% \) of the boxes made by machine A pass inspection.</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(90\% \) of the boxes made by machine B pass inspection.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A box is selected at random.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that it passes inspection.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The company would like the probability that a box passes inspection to be 0.87.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the percentage of boxes that should be made by machine B to achieve this.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach involving <em>A</em> and <em>B</em> <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(A \cap {\rm{pass}}) + {\rm{P}}(B \cap {\rm{pass}})\) , tree diagram</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}({\rm{pass}}) = 0.6 \times 0.8 + 0.4 \times 0.9\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}({\rm{pass}}) = 0.84\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing complement (seen anywhere) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(B) = x\) , \({\rm{P}}(A) = 1 - x\) , \(1 - {\rm{P}}(B)\) , \(100 - x\) , \(x + y = 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.8(1 - x) + 0.9x\) , \(0.8x + 0.9y\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.87 = 0.8(1 - x) + 0.9x\) , \(0.8 \times 0.3 + 0.9 \times 0.7 = 0.87\) , \(0.8x + 0.9y = 0.87\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(70\% \) from B <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was usually well done. Those candidates that did not succeed with this part often did not show a correct tree diagram indicating that they did not really understand the problem or indeed how to start it. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many successful attempts to (b) relied on "guess and check" or intuitive solutions while a surprising number of candidates could not manage to systematically set up an appropriate algebraic expression involving a complement. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows a probability distribution for the random variable \(X\), where \({\text{E}}(X) = 1.2\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.18.09.png" alt="M17/5/MATME/SP2/ENG/TZ2/10"></p>
</div>
<div class="specification">
<p>A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable \(X\).</p>
</div>
<div class="specification">
<p>A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.</p>
</div>
<div class="question">
<p>Jill plays the game nine times. Find the probability that she wins exactly two prizes.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{B}}(n,{\text{ }}p),{\text{ }}\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){p^r}{q^{n - r}},{\text{ }}{(0.167)^2}{(0.833)^7},{\text{ }}\left( {\begin{array}{*{20}{c}} 9 \\ 2 \end{array}} \right)\)</p>
<p>0.279081</p>
<p>0.279 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p class="p1">A random variable \(X\) is distributed normally with a mean of <span class="s1">20 </span>and standard deviation of <span class="s1">4</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On the following diagram, shade the region representing \({\text{P}}(X \leqslant 25)\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_16.44.01.png" alt="M16/5/MATME/SP2/ENG/TZ1/01"></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="p1"><span class="s1">Write down \({\text{P}}(X \leqslant 25)\)</span>, correct to two decimal places.</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 class="p1"><span class="s1">Let \({\text{P}}(X \leqslant c) = 0.7\)</span>. Write down the value of \(c\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-02-01_om_16.47.04.png" alt="M16/5/MATME/SP2/ENG/TZ1/01.a/M"> <strong><em>A1A1 N2</em></strong></p>
<p> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A1 </em></strong>for vertical line clearly to right of mean,</p>
<p class="p1"><strong><em>A1 </em></strong>for shading to left of their vertical line.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X \leqslant 25) = 0.894350\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="s2">\({\text{P}}(X \leqslant 25) = 0.89\) </span><span class="s3">(must be 2 </span>d.p.) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(c = 22.0976\)</p>
<p class="p1"><span class="Apple-converted-space">\(c = 22.1\) </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">There was a mixed response to this question: candidates either very confidently answered each part, or were completely lost. Most were able to shade the correct region in part (a) and in part (b), answers were not always left to two decimal places. It was not uncommon to see candidates converting to standard normal form in part (c) before using their calculators, suggesting that they were not only unfamiliar with the meaning of the command term “write down” but were unclear as to what the given information was actually asking for.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">There was a mixed response to this question: candidates either very confidently answered each part, or were completely lost. Most were able to shade the correct region in part (a) and in part (b), answers were not always left to two decimal places. It was not uncommon to see candidates converting to standard normal form in part (c) before using their calculators, suggesting that they were not only unfamiliar with the meaning of the command term “write down” but were unclear as to what the given information was actually asking for.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">There was a mixed response to this question: candidates either very confidently answered each part, or were completely lost. Most were able to shade the correct region in part (a) and in part (b), answers were not always left to two decimal places. It was not uncommon to see candidates converting to standard normal form in part (c) before using their calculators, suggesting that they were not only unfamiliar with the meaning of the command term “write down” but were unclear as to what the given information was actually asking for.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A van can take either Route A or Route B for a particular journey.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If Route A is taken, the journey time may be assumed to be normally distributed </span><span style="font-family: times new roman,times; font-size: medium;">with mean 46 minutes and a standard deviation 10 minutes.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">If Route B is taken, the journey time may be assumed to be normally distributed </span><span style="font-family: times new roman,times; font-size: medium;">with mean \(\mu \) minutes and standard deviation 12 minutes.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For Route A, find the probability that the journey takes <strong>more</strong> than \(60\) minutes.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For Route B, the probability that the journey takes <strong>less</strong> than \(60\) minutes is \(0.85\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\mu \) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The van sets out at 06:00 and needs to arrive before 07:00.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Which route should it take?</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Justify your answer.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">On five consecutive days the van sets out at 06:00 and takes Route B. Find the </span><span style="font-family: times new roman,times; font-size: medium;">probability that</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) it arrives before 07:00 on all five days;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) it arrives before 07:00 on at least three days.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A \sim N(46{\text{, }}{10^2})\) \(B \sim N(\mu {\text{, }}{12^2})\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(A > 60) = 0.0808\) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct approach <em><strong>(A1) </strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}\left( {Z < \frac{{60 - \mu }}{{12}}} \right) = 0.85\) , sketch </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{60 - \mu }}{{12}} = 1.036 \ldots \) <em><strong>(A1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\mu = 47.6\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) route A <em><strong>A1 N1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>METHOD 1</strong> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(A < 60) = 1 - 0.0808 = 0.9192\) <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reason <em><strong>R1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. probability of <em>A</em> getting there on time is greater than probability of <em>B</em> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(0.9192 > 0.85\) <em><strong>N2</strong> </em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2 </span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(B > 60) = 1 - 0.85 = 0.15\) <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reason <em><strong>R1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. probability of <em>A</em> getting there late is less than probability of <em>B </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(0.0808 < 0.15\) <em><strong>N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) let <em>X</em> be the number of days when the van arrives before 07:00 </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 5) = {(0.85)^5}\) <em><strong>(A1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( = 0.444\) <em><strong>A1 N2</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>METHOD 1</strong> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of adding correct probabilities <em><strong>(M1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(X \ge 3) = {\rm{P}}(X = 3) + {\rm{P}}(X = 4) + {\rm{P}}(X = 5)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct values \(0.1382 + 0.3915 + 0.4437\) <em><strong>(A1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \ge 3) = 0.973\) <em><strong>A1 N3</strong> </em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2 </span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using the complement <em><strong>(M1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(X \ge 3) = 1 - {\rm{P}}(X \le 2)\) , \(1 - p\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct values \(1 - 0.02661\) <em><strong>(A1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \ge 3) = 0.973\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A significant number of students clearly understood what was asked in part (a) and used the GDC to find the result. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), many candidates set the standardized formula equal to the probability (\(0.85\)), instead of using the corresponding <em>z</em>-score. Other candidates used the solver on their GDC with the inverse norm function. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A common incorrect approach in part (c) was to attempt to use the means and standard deviations for justification, although many candidates successfully considered probabilities. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A pleasing number of candidates recognized the binomial probability and made progress on part (d). </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Jan plays a game where she tosses two fair six-sided dice. She wins a prize if the </span><span style="font-family: times new roman,times; font-size: medium;">sum of her scores is 5.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Jan tosses the two dice once. Find the probability that she wins a prize.</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><span style="font-family: times new roman,times; font-size: medium;">Jan tosses the two dice 8 times. Find the probability that she wins 3 prizes.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">36 outcomes (seen anywhere, even in denominator) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach of listing ways to get sum of 5, showing at least two pairs <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. (1, 4)(2, 3), (1, 4)(4, 1), (1, 4)(4, 1), (2, 3)(3, 2) , lattice diagram</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P(prize)}} = \frac{4}{{36}}\) \(\left( { = \frac{1}{9}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing binomial probability <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{B}}\left( {8,\frac{1}{9}} \right)\)</span><span style="font-family: times new roman,times; font-size: medium;"> , binomial pdf, \(\left( {\begin{array}{*{20}{c}}<br>8\\<br>3<br>\end{array}} \right){\left( {\frac{1}{9}} \right)^3}{\left( {\frac{8}{9}} \right)^5}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(3 prizes)}} = 0.0426\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">While many candidates were successful at part (a), far fewer recognized the binomial </span><span style="font-family: times new roman,times; font-size: medium;">distribution in the second part of the problem.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">While many candidates were successful at part (a), far fewer recognized the binomial </span><span style="font-family: times new roman,times; font-size: medium;">distribution in the second part of the problem.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Those who did not obtain the correct answer at part (a) often scored partial credit by either </span><span style="font-family: times new roman,times; font-size: medium;">drawing a table to represent the sample space or by noting relevant pairs.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following table shows the probability distribution of a discrete random variable <em>X</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/alice.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>k</em>.</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><span style="font-family: times new roman,times; font-size: medium;">Find the expected value of <em>X</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using \(\sum {{p_i} = 1} \) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(10{k^2} + 3k + 0.6 = 1\) , \(10{k^2} + 3k - 0.4 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 0.1\) <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using \({\rm{E}}(X) = \sum {{p_i}{x_i}} \) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 1 \times 0.2 + 2 \times 0.4 + 3 \times 0.3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}(X) = 1.5\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A good number of candidates answered this question well, although some incorrectly set the </span><span style="font-family: times new roman,times; font-size: medium;">sum of the probabilities to zero instead of one, suggesting rote recognition of a quadratic </span><span style="font-family: times new roman,times; font-size: medium;">equal to zero. Many candidates recognized that only the positive value for <em>k</em> was appropriate </span><span style="font-family: times new roman,times; font-size: medium;">and correctly indicated this in their working. Many went on to find the correct expected value </span><span style="font-family: times new roman,times; font-size: medium;">as well, although at times candidates wrote the formula from the information booklet without </span><span style="font-family: times new roman,times; font-size: medium;">making use of it, thus earning no marks.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A good number of candidates answered this question well, although some incorrectly set the </span><span style="font-family: times new roman,times; font-size: medium;">sum of the probabilities to zero instead of one, suggesting rote recognition of a quadratic </span><span style="font-family: times new roman,times; font-size: medium;">equal to zero. Many candidates recognized that only the positive value for <em>k</em> was appropriate </span><span style="font-family: times new roman,times; font-size: medium;">and correctly indicated this in their working. Many went on to find the correct expected value </span><span style="font-family: times new roman,times; font-size: medium;">as well, although at times candidates wrote the formula from the information booklet without </span><span style="font-family: times new roman,times; font-size: medium;">making use of it, thus earning no marks.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times;"><span style="font-size: medium;">The scores of a test given to students are normally distributed with a mean of 21. \(80\% \)</span><span style="font-size: medium;"> of the students have scores less than 23.7.</span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the standard deviation of the scores.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A student is chosen at random. This student has the same probability of having a score </span><span style="font-family: times new roman,times; font-size: medium;">less than 25.4 as having a score greater than <em>b</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the probability the student has a score less than 25.4.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of <em>b</em>.</span></p>
<p align="LEFT"> </p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. finding \(0.84 \ldots \), using \(\frac{{23.7 - 21}}{\sigma }\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.84 \ldots = \frac{{23.7 - 21}}{\sigma }\) , graph <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sigma = 3.21\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of attempting to find \({\text{P}}(X < 25.4)\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. using \(z = 1.37\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X < 25.4) = 0.915\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of recognizing symmetry <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(b = 21 - 4.4\) , using \(z = - 1.37\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates who clearly understood the nature of normal probability answered this question cleanly. A common misunderstanding was to use the value of 0.8 as a <em>z</em>-score when finding the standard deviation. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many correctly used their GDC to find the probability in part (b). Fewer used some aspect of the symmetry of the curve to find a value for <em>b</em>. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">At a large school, students are required to learn at least one language, Spanish </span><span style="font-family: times new roman,times; font-size: medium;">or French. It is known that \(75\% \) of the students learn Spanish, and \(40\% \) learn French.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the percentage of students who learn <strong>both</strong> Spanish and French.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">At a large school, students are required to learn at least one language, Spanish </span><span style="font-family: times new roman,times; font-size: medium;">or French. It is known that \(75\% \) of the students learn Spanish, and \(40\% \) learn French.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the percentage of students who learn Spanish, but not French.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">At a large school, students are required to learn at least one language, Spanish </span><span style="font-family: times new roman,times; font-size: medium;">or French. It is known that \(75\% \) of the students learn Spanish, and \(40\% \) learn French.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">At this school, \(52\% \) of the students are girls, and \(85\% \) of the girls learn Spanish.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A student is chosen at random. Let <em>G</em> be the event that the student is a girl, and </span><span style="font-family: times new roman,times; font-size: medium;">let <em>S</em> be the event that the student learns Spanish.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find \({\rm{P}}(G \cap S)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that <em>G</em> and <em>S</em> are <strong>not</strong> independent.</span></p>
<p> </p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">At a large school, students are required to learn at least one language, Spanish </span><span style="font-family: times new roman,times; font-size: medium;">or French. It is known that \(75\% \) of the students learn Spanish, and \(40\% \) learn French.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">At this school, \(52\% \) of the students are girls, and \(85\% \) of the girls learn Spanish.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A boy is chosen at random. Find the probability that he learns Spanish.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. Venn diagram with intersection, union formula,</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(S \cap F) = 0.75 + 0.40 - 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(15\) (accept \(15\% \)) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach involving subtraction <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. Venn diagram, \(75 - 15\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">60 (accept \(60\% \)) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) valid approach <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. tree diagram, multiplying probabilities, \({\rm{P}}(S|G) \times {\rm{P(}}G{\rm{)}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct calculation <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.52 \times 0.85\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(G \cap S) = 0.442\) (exact) <strong><em>A1 N3</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) valid reasoning, with words, symbols or numbers (seen anywhere) <em><strong>R1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P(}}G{\rm{)}} \times {\rm{P}}(S) \ne {\rm{P}}(G \cap S)\) , \({\rm{P}}(S|G) \ne {\rm{P(}}S{\rm{)}}\) , not equal, </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">one correct value <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(G) \times {\rm{P}}(S) = 0.39\) , \({\rm{P}}(S|G) = 0.85\) , \(0.39 \ne 0.442\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>G</em> and <em>S</em> are not independent <em><strong>AG N0</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(48\% \) are boys (seen anywhere) <strong><em>A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(B) = 0.48\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate approach <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\text{P(girl and Spanish)}} + {\text{P(boy and Spanish)}} = {\text{P(Spanish)}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct approach to find P(boy and Spanish) <strong><em> (A1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P(}}B \cap S{\rm{) = P(}}S{\rm{)}} - {\rm{P}}(G \cap S)\) , \({\rm{P(}}B \cap S{\rm{) = P(}}S|B) \times {\rm{P}}(B)\) , 0.308</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <strong><em> (A1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.442 + 0.48x = 0.75\) , \(0.48x = 0.308\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct manipulation <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}(S|B) = \frac{{0.308}}{{0.48}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}({\rm{Spanish}}|{\rm{boy}}) = 0.641666 \ldots \) , \(0.641\bar 6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}({\rm{Spanish}}|{\rm{boy}}) = 0.642\) \([0.641{\text{, }}0.642]\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(48\% \) are boys (seen anywhere) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 0.48 used in tree diagram </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate approach <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. tree diagram </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correctly labelled branches on tree diagram <strong><em>(A1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. first branches are boy/girl, second branches are Spanish/not Spanish </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.442 + 0.48x = 0.75\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct manipulation <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.48x = 0.308\) , \({\rm{P}}(S|B) = \frac{{0.308}}{{0.48}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}({\rm{Spanish}}|{\rm{boy}}) = 0.641666 \ldots \) , \(0.641\bar 6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}({\rm{Spanish}}|{\rm{boy}}) = 0.642\) \([0.641{\text{, }}0.642]\)</span></p>
<p><em> <span style="font-family: times new roman,times; font-size: medium;"><strong>[6 marks]</strong> </span></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally done well although some candidates left answers as decimals rather than the required percentages. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally done well although some candidates left answers as decimals rather than the required percentages. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c) (i), candidates failed to find the intersection of the events as, in general, they multiplied probabilities, assuming the events were independent or they incorrectly attempted to use the union formula. Independence in (c) (ii) caused difficulty with some candidates attempting to use the conditions for mutually exclusive events while others assumed the events were independent in part (i) and then found \({\rm{P}}(G \cap S)\) by multiplying \({\rm{P}}(S|G) \times {\rm{P(}}G{\rm{)}}\) . </span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) proved quite challenging as a great majority could only find the probability of being a boy. Those who did attempt it, and successfully connected the problem with conditional probability, often had difficulties in reaching the correct final answer. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A bag contains four gold balls and six silver balls.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Two balls are drawn at random from the bag, with replacement. Let \(X\) be the </span><span style="font-family: times new roman,times; font-size: medium;">number of gold balls drawn from the bag.</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Fourteen balls are drawn from the bag, with replacement.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find \({\rm{P}}(X = 0)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \({\rm{P}}(X = 1)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Hence, find \({\rm{E}}(X)\) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, find \({\rm{E}}(X)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that exactly five of the balls are gold.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that at most five of the balls are gold.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that at most five of the balls are gold, find the probability that exactly five </span><span style="font-family: times new roman,times; font-size: medium;">of the balls are gold. Give the answer correct to two decimal places.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) appropriate approach <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{6}{{10}} \times \frac{6}{{10}}\) , \(\frac{6}{{10}} \times \frac{5}{9}\) , \(\frac{6}{{10}} \times \frac{5}{{10}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(X = 0) = \frac{9}{{25}} = 0.36\) </span><strong><span style="font-family: times new roman,times; font-size: medium;"><em>A1 N2</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) multiplying one pair of gold and silver probabilities <strong><em> (M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{6}{{10}} \times \frac{4}{{10}}\) , \(\frac{6}{{10}} \times \frac{4}{9}\) , 0.24</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">adding the product of both pairs of gold and silver probabilities <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{6}{{10}} \times \frac{4}{{10}} \times 2\) , \(\frac{6}{{10}} \times \frac{4}{9} + \frac{4}{{10}} \times \frac{6}{9}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(X = 1) = \frac{{12}}{{25}} = 0.48\) </span><strong><span style="font-family: times new roman,times; font-size: medium;"><em>A1 N3</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 2) = 0.16\) (seen anywhere) (A1)<br><br>correct substitution into formula for \({\rm{E}}(X)\) (A1)<br><br>eg \(0 \times 0.36 + 1 \times 0.48 + 2 \times 0.16\) , \(0.48 + 0.32\)<br><br>\({\rm{E}}(X) = \frac{4}{5} = 0.8\) A1 N3</span></p>
<p> </p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of recognizing binomial (may be seen in part (ii)) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(X \sim {\rm{B}}(2,0.6)\) , \(\left( \begin{array}{l}<br>2\\<br>0<br>\end{array} \right){(0.4)^2}{(0.6)^0}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct probability for use in binomial <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(p = 0.4\) , \(X \sim {\rm{B}}(2,0.4)\) , \(^2{C_0}{(0.4)^0}{(0.6)^2}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(X = 0) = \frac{9}{{25}} = 0.36\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1 N3</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct set up <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(_2{C_1}{(0.4)^1}{(0.6)^1}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{P}}(X = 1) = \frac{{12}}{{25}} = 0.48\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute into \(np\) (M1)<br><br>eg \(2 \times 0.6\)<br><br>correct substitution into \(np\) (A1)<br><br>eg \(2 \times 0.4\)<br><br>\({\rm{E}}(X) = \frac{4}{5} = 0.8\) A1 N3<br></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;"> [8 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 2) = 0.16\) (seen anywhere) <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into formula for \({\rm{E}}(X)\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(0 \times 0.36 + 1 \times 0.48 + 2 \times 0.16\) , \(0.48 + 0.32\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{E}}(X) = \frac{4}{5} = 0.8\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3</span></strong></em></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute into \(np\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(2 \times 0.6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into \(np\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(2 \times 0.4\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\rm{E}}(X) = \frac{4}{5} = 0.8\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1 N3</span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;"> </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks]<br></span></em></strong></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(Y\) be the number of gold balls drawn from the bag. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing binomial (seen anywhere) <strong><em> (M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(_{14}{C_5}{(0.4)^5}{(0.6)^9}\) , \({\rm{B}}(14,0.4)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(Y = 5) = 0.207\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognize need to find \({\rm{P}}(Y \le 5)\) <strong><em>(M1)</em> </strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(Y \le 5) = 0.486\) <strong><em>A1 N2 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(Y\) be the number of gold balls drawn from the bag. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing conditional probability <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{P}}(A|B)\) , \({\rm{P}}(Y = 5|Y \le 5)\) , \(\frac{{{\rm{P}}(Y = 5)}}{{{\rm{P}}(Y \le 5)}}\) , \(\frac{{0.207}}{{0.486}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(Y = 5|Y \le 5) = 0.42522518\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(Y = 5|Y \le 5) = 0.43\) (to \(2\) dp) <strong><em>A1 N2 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a)(i) and (ii) were generally well done, with candidates either using a tree diagram or a binomial approach. Part (a)(iii) proved difficult, with many either having trouble finding \({\rm{P}}(X = 2)\) or using \({\rm{E}}(X) = np\) .<br></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a)(iii) proved difficult, with many either having trouble finding \({\rm{P}}(X = 2)\) or using \({\rm{E}}(X) = np\) .</span></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A great majority were confident solving part (b) with the GDC, although some did write the binomial term. <br></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Those candidates who did not use the binomial function on the GDC had more difficulty in part (c), although a pleasing number were still able to identify that they were seeking \({\rm{P}}(X \leqslant 5)\) .</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">While most candidate knew to use conditional probability in part (d), fewer were able to do so successfully, and even fewer still correctly rounded their answer to two decimal places. The most common error was to multiply probabilities to find the intersection needed for the conditional probability formula. Overall, candidates seemed better prepared for probability.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Two events \(A\) and \(B\) are such that \({\text{P}}(A) = 0.2\) and \({\text{P}}(A \cup B) = 0.5\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(A\) and \(B\) are mutually exclusive, find \({\text{P}}(B)\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(A\) and \(B\) are independent, find \({\text{P}}(B)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct approach <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.5 = 0.2 + {\text{P}}(B),{\text{ P}}(A \cap B) = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(B) = 0.3\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Correct expression for \({\text{P}}(A \cap B)\) (seen anywhere) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(A \cap B) = 0.2{\text{P}}(B),{\text{ }}0.2x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to substitute into correct formula for \({\text{P}}(A \cup B)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(A \cup B) = 0.2 + {\text{P}}(B) - {\text{P}}(A \cap B),{\text{ P}}(A \cup B) = 0.2 + x - 0.2x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.5 = 0.2 + {\text{P}}(B) - 0.2{\text{P}}(B),{\text{ }}0.8x = 0.3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(B) = \frac{3}{8}{\text{ }}( = 0.375,{\text{ exact}})\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following table shows the Diploma score \(x\) and university entrance mark \(y\) for seven IB Diploma students.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_08.02.07.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the correlation coefficient.</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="p1">The relationship can be modelled by the regression line with equation \(y = ax + b\).</p>
<p class="p1">Write down the value of \(a\) and of \(b\).</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 class="p1">Rita scored a total of \(26\) in her IB Diploma.</p>
<p class="p1">Use your regression line to estimate Rita’s university entrance mark.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of set up <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)correct value for \(r\) (or for \(a\) or \(r\), seen in (b))</p>
<p class="p1">\(0.996010\)</p>
<p class="p1">\(r = 0.996\;\;\;[0.996,{\text{ }}0.997]\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(a = 3.15037,{\text{ }}b = - 15.4393\)</p>
<p>\(a = 3.15{\text{ }}[3.15,{\text{ }}3.16],{\text{ }}b = - 15.4{\text{ }}[ - 15.5,{\text{ }} - 15.4]\) <strong><em>A1A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">substituting \(26\) into <strong>their </strong>equation <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(y = 3.15(26) - 15.4\)</p>
<p class="p1">\(66.4704\)</p>
<p class="p1">\(66.5{\text{ }}[66.4,{\text{ }}66.5]\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<p class="p1"><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (b) and (c) of this question were correctly answered by most candidates.</p>
<p class="p1">However, a few students did not recognize that this question involved linear regression. And for those who did, not all of them knew what the correlation coefficient was. Some of them left this part of the question blank, and others wrote the value of \({r^2}\).</p>
<p class="p1">A number of students tried to find the values of \(a\) and \(b\) by forming two linear equations with two points from the table and solving them.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (b) and (c) of this question were correctly answered by most candidates.</p>
<p class="p1">However, a few students did not recognize that this question involved linear regression. And for those who did, not all of them knew what the correlation coefficient was. Some of them left this part of the question blank, and others wrote the value of \({r^2}\).</p>
<p class="p1">A number of students tried to find the values of \(a\) and \(b\) by forming two linear equations with two points from the table and solving them.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (b) and (c) of this question were correctly answered by most candidates.</p>
<p class="p1">However, a few students did not recognize that this question involved linear regression. And for those who did, not all of them knew what the correlation coefficient was. Some of them left this part of the question blank, and others wrote the value of \({r^2}\).</p>
<p class="p1">A number of students tried to find the values of \(a\) and \(b\) by forming two linear equations with two points from the table and solving them.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The cumulative frequency curve below represents the heights of 200 sixteen-year-old boys.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/donald.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Use the graph to answer the following.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the median value.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A boy is chosen at random. Find the probability that he is shorter than \(161{\text{ cm}}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(82\% \) of the boys are taller than \(h{\text{ cm}}\), find <em>h</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{median}} = 174 {\text{(cm)}}\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find number shorter than 161 <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. line on graph, 12 boys</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = \frac{{12}}{{200}}( = 0.06)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(18\% \) have a height less than<em> h</em> <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(0.18 \times 200 = 36\) (36 may be seen as a line on the graph) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(h = 166\) (cm) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><em><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(0.82 \times 200 = 164\) (164 may be seen as a line on the graph) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(200 - 164 = 36\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(h = 166\)</span> (cm) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well done. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well done. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Some candidates could only earn the first mark in part (c) for finding \(82\% \) of 200. Others gave the answer as 164, neglecting to subtract this value from the total of 200. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following cumulative frequency graph shows the monthly income, \(I\) dollars, of \(2000\) families.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_10.22.12.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the median monthly income.</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="p1">(i) <span class="Apple-converted-space"> </span>Write down the number of families who have a monthly income of \(2000\) dollars or less.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the number of families who have a monthly income of more than \(4000\) dollars.</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 class="p1">The \(2000\) families live in two different types of housing. The following table gives information about the number of families living in each type of housing and their monthly income \(I\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_11.02.20.png" alt></p>
<p class="p1">Find the value of \(p\).</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 class="p1">A family is chosen at random.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the probability that this family lives in an apartment.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the probability that this family lives in an apartment, given that its monthly income is greater than \(4000\) dollars.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Estimate the mean monthly income for families living in a villa.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing that the median is at half the total frequency <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(\frac{{2000}}{2}\)</p>
<p class="p1">\(m = 2500{\text{ (dollars)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(500\) families have a monthly income less than \(2000\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>correct cumulative frequency, <span class="s1">\(1850\) <span class="Apple-converted-space"> </span></span><strong><em>(A1)</em></strong></p>
<p class="p1">subtracting <strong>their </strong>cumulative frequency from \(2000\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(2000 - 1850\)</p>
<p class="p1">\(150\) families have a monthly income of more than \(4000\) dollars <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>Note:</strong> If working shown, award <em><strong>M1A1A1</strong></em> for \(128{\rm{ }} + {\rm{ }}22{\rm{ }} = {\rm{ }}150\), using the table.</p>
<p class="p1"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct calculation <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(2000 - (436 + 64 + 765 + 28 + 122),{\text{ }}1850 - 500 - 765\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(p = 585\) <span class="Apple-converted-space"> </span><strong><em>A1 N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(436 + 765 + 28\)</p>
<p class="p1">\(0.6145\;\;\;\)(exact) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">\(\frac{{1229}}{{2000}},{\text{ }}0.615{\text{ }}[0.614,{\text{ }}0.615]\)</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>correct working/probability for number of families <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(122 + 28,{\text{ }}\frac{{150}}{{2000}},{\text{ 0.075}}\)</p>
<p class="p1">\(0.186666\)</p>
<p class="p1">\(\frac{{28}}{{150}}\;\;\;\left( { = \frac{{14}}{{75}}} \right),{\text{ }}0.187{\text{ }}[0.186,{\text{ }}0.187]\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of using correct mid-interval values (\(1500,{\rm{ }}3000,{\rm{ }}4500\)) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">attempt to substitute into \(\frac{{\sum {fx} }}{{\sum f }}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(\frac{{1500 \times 64 + 3000 \times p + 4500 \times 122}}{{64 + 585 + 122}}\)</p>
<p class="p1">\(3112.84\)</p>
<p class="p1">\(3110{\text{ }}[3110,{\text{ }}3120]{\text{ (dollars)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [15 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was well handled by most candidates. Except for miscalculations and incorrect readings from the cumulative frequency graph, the processes and concepts seemed to be well understood by the majority.</p>
<p class="p1">A number of students did not gain full marks in parts (bii) and (e), for not showing their process. In part (c), some candidates wrote things like “using GDC”, without showing relevant work, and so lost marks. Those who chose a formulaic approach to the conditional probability question in (dii) were often not as successful as those who could interpret the question in terms of the table values.</p>
<p class="p1">A large number of candidates could not find the mean value in (e). Some used the incorrect mid-interval values and others did not consider their use.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was well handled by most candidates. Except for miscalculations and incorrect readings from the cumulative frequency graph, the processes and concepts seemed to be well understood by the majority.</p>
<p class="p1">A number of students did not gain full marks in parts (bii) and (e), for not showing their process. In part (c), some candidates wrote things like “using GDC”, without showing relevant work, and so lost marks. Those who chose a formulaic approach to the conditional probability question in (dii) were often not as successful as those who could interpret the question in terms of the table values.</p>
<p class="p1">A large number of candidates could not find the mean value in (e). Some used the incorrect mid-interval values and others did not consider their use.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was well handled by most candidates. Except for miscalculations and incorrect readings from the cumulative frequency graph, the processes and concepts seemed to be well understood by the majority.</p>
<p class="p1">A number of students did not gain full marks in parts (bii) and (e), for not showing their process. In part (c), some candidates wrote things like “using GDC”, without showing relevant work, and so lost marks. Those who chose a formulaic approach to the conditional probability question in (dii) were often not as successful as those who could interpret the question in terms of the table values.</p>
<p class="p1">A large number of candidates could not find the mean value in (e). Some used the incorrect mid-interval values and others did not consider their use.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was well handled by most candidates. Except for miscalculations and incorrect readings from the cumulative frequency graph, the processes and concepts seemed to be well understood by the majority.</p>
<p class="p1">A number of students did not gain full marks in parts (bii) and (e), for not showing their process. In part (c), some candidates wrote things like “using GDC”, without showing relevant work, and so lost marks. Those who chose a formulaic approach to the conditional probability question in (dii) were often not as successful as those who could interpret the question in terms of the table values.</p>
<p class="p1">A large number of candidates could not find the mean value in (e). Some used the incorrect mid-interval values and others did not consider their use.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was well handled by most candidates. Except for miscalculations and incorrect readings from the cumulative frequency graph, the processes and concepts seemed to be well understood by the majority.</p>
<p class="p1">A number of students did not gain full marks in parts (bii) and (e), for not showing their process. In part (c), some candidates wrote things like “using GDC”, without showing relevant work, and so lost marks. Those who chose a formulaic approach to the conditional probability question in (dii) were often not as successful as those who could interpret the question in terms of the table values.</p>
<p class="p1">A large number of candidates could not find the mean value in (e). Some used the incorrect mid-interval values and others did not consider their use.</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following table gives the examination grades for 120 students.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/easley.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>p</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>q</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the mean grade.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the standard deviation.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(a) (i) evidence of appropriate approach <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(9 + 25 + 35\) , \(34 + 35\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(p = 69\) </span> <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(109 - \) <strong>their</strong> value of <em>p</em>, \(120 - (9 + 25 + 35 + 11)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(q = 40\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[4 marks]</strong></em></span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. substituting into </span><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{\sum {fx} }}{n}\)</span><span style="font-family: times new roman,times; font-size: medium;">, division by 120</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">mean \(= 3.16\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">1.09 <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates had little trouble finding the missing values in the frequency </span><span style="font-family: times new roman,times; font-size: medium;">distribution table.</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many did not seem comfortable calculating the mean and standard </span><span style="font-family: times new roman,times; font-size: medium;">deviation using their GDCs.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The correct mean was often found without the use of the statistical functions on the graphing </span><span style="font-family: times new roman,times; font-size: medium;">calculator, but a large number of candidates were unable to find the standard deviation.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many did not seem comfortable calculating the mean and standard </span><span style="font-family: times new roman,times; font-size: medium;">deviation using their GDCs.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The correct mean was often found without the use of the statistical functions on the graphing </span><span style="font-family: times new roman,times; font-size: medium;">calculator, but a large number of candidates were unable to find the standard deviation.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The masses of watermelons grown on a farm are normally distributed with a mean of \(10\) kg.</p>
<p class="p1">The watermelons are classified as small, medium or large.</p>
<p class="p2"><span class="s1">A watermelon is small if its mass is less than \(4\) kg</span>. Five percent of the watermelons are classified as small.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the standard deviation of the masses of the watermelons.</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 class="p1">The following table shows the percentages of small, medium and large watermelons grown on the farm.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_09.47.19.png" alt></p>
<p class="p1">A watermelon is large if its mass is greater than \(w\) <span class="s1">kg</span>.</p>
<p class="p1">Find the value of \(w\).</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 class="p1">All the medium and large watermelons are delivered to a grocer.</p>
<p class="p1">The grocer selects a watermelon at random from <strong>this </strong>delivery. Find the probability that it is medium.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">All the medium and large watermelons are delivered to a grocer.</p>
<p class="p1">The grocer sells all the medium watermelons for $1.75 each, and all the large watermelons for $3.00 each. His costs on this delivery are $300, and his total profit is $150<span class="s1">. Find the number of watermelons in the delivery.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>finding standardized value for 4 kg (seen anywhere) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;z = - 1.64485\)</p>
<p>attempt to standardize <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\sigma = \frac{{x - \mu }}{z},{\text{ }}\frac{{4 - 10}}{\sigma }\)</p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\; - 1.64 = \frac{{4 - 10}}{\sigma },{\text{ }}\frac{{4 - 10}}{{ - 1.64}}\)</p>
<p>\(\sigma = 3.64774\)</p>
<p>\(\sigma = 3.65\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;1 - p,{\text{ 0.62, }}\frac{{w - 10}}{{3.65}} = 0.305\)</p>
<p class="p1">\(w = 11.1143\)</p>
<p class="p1">\(w = 11.1\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to restrict melon population <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(95\% \) are delivered, \({\text{P}}({\text{medium}}|{\text{delivered}}),{\text{ }}57 + 38\)</p>
<p class="p1">correct probability for medium watermelons <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{0.57}}{{0.95}}\)</p>
<p class="p1">\(\frac{{57}}{{95}},{\text{ }}0.6,{\text{ }}60\% \) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">proportion of large watermelons (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{\text{P(large)}} = 0.4,{\text{ }}40\% \)</p>
<p class="p1">correct approach to find total sales (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;150 = {\text{sales}} - 300,{\text{ total sales}} = \$ 450\)</p>
<p class="p1">correct expression <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;1.75(0.6x) + 3(0.4x),{\text{ }}1.75(0.6) + 3(0.4)\)</p>
<p class="p1">evidence of correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;1.75(0.6x) + 3(0.4x) = 450,{\text{ }}2.25x = 450\)</p>
<p class="p1">200 watermelons in the delivery <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Notes: <span class="Apple-converted-space"> </span></strong>If candidate answers 0.57 in part (c), the <strong><em>FT </em></strong>values are \({\text{P(large)}} = 0.43\) and 197 watermelons. Award <strong><em>FT </em></strong>marks if working shown.</p>
<p class="p1">Award <strong><em>N0 </em></strong>for 197.</p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following table shows the amount of fuel (\(y\)<em> </em>litres) used by a car to travel certain distances (\(x\)<em> </em>km).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<table class="block_black_border" border="0" align="center">
<tbody>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Distance</strong> (<em>x</em> km)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">40</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">75</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">120</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">150</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">195</span></td>
</tr>
<tr>
<td><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Amount of fuel</strong> (<em>y</em> litres)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">3.6</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">6.5</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">9.9</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">13.1</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">16.2</span></td>
</tr>
</tbody>
</table>
<p> </p>
<p>This data can be modelled by the regression line with equation \(y = ax + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(a\) and of \(b\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Explain what the gradient \(a\) represents.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use the model to estimate the amount of fuel the car would use if it is driven \(110\) km.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 0.0823604{\text{, }}b = 0.306186\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 0.0824{\text{, }}b = 0.306\) <strong><em>A1A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct explanation with reference to number of litres</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">required for \(1\) km <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(a\) represents the (average) amount of fuel (litres) required to drive \(1\) km, (average) litres per kilometre, (average) rate of change in fuel used for each km travelled</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 marks]</em></strong></span></p>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(y = 0.0824(110) + 0.306\), sketch</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(9.36583\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(9.37\) (litres) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The random variable \(X\) is normally distributed with mean \(20\) and standard deviation \(5\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}(X \le 22.9)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Given that \({\rm{P}}(X < k) = 0.55\) , find the value of \(k\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(z = \frac{{22.9 - 20}}{5}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(z = 0.58\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \le 22.9) = 0.719\) <em><strong>A1 N3 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(z\)-score for \(0.55\) is \(0.12566…\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach (must be with \(z\)-values) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> using inverse normal, \(0.1257 = \frac{{k - 20}}{5}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 20.6\) <em><strong>A1 N3 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The normal distribution was handled better than in previous years with many candidates successful in both parts and very few blank responses. Some candidates used tables and \(z\)-scores while others used the GDC directly; the GDC approach earned full marks more often than the \(z\)-score approach.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The normal distribution was handled better than in previous years with many candidates successful in both parts and very few blank responses. Some candidates used tables and \(z\)-scores while others used the GDC directly; the GDC approach earned full marks more often than the \(z\)-score approach. A common error in part (b) was to set the expression for \(z\)-score equal to the probability. Many candidates had difficulty giving answers correct to three significant figures; this was particularly an issue if no working was shown.<br></span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows values of ln <em>x</em> and ln <em>y</em>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The relationship between ln <em>x</em> and ln <em>y</em> can be modelled by the regression equation ln <em>y</em> = <em>a</em> ln <em>x</em> + <em>b</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>a</em> and 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>Use the regression equation to estimate the value of <em>y</em> when<em> x</em> = 3.57.</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>The relationship between <em>x</em> and <em>y</em> can be modelled using the formula <em>y</em> = <em>kx<sup>n</sup></em>, where <em>k</em> ≠ 0 , <em>n</em> ≠ 0 , <em>n</em> ≠ 1.</p>
<p>By expressing ln <em>y</em> in terms of ln <em>x</em>, find the value of <em>n</em> and of <em>k</em>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em> one correct value</p>
<p>−0.453620, 6.14210</p>
<p><em>a</em> = −0.454, <em>b</em> = 6.14 <em><strong>A1A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution <em><strong> (A1)</strong></em></p>
<p><em>eg </em>−0.454 ln 3.57 + 6.14</p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em> ln <em>y</em> = 5.56484</p>
<p>261.083 (260.409 from 3 sf)</p>
<p><em>y</em> = 261, (<em>y</em> = 260 from 3sf) <em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> If no working shown, award <em><strong>N1</strong></em> for 5.56484.<br>If no working shown, award <em><strong>N2</strong> </em>for ln <em>y</em> = 5.56484.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>valid approach for expressing ln <em>y</em> in terms of ln <em>x</em> <em><strong>(M1)</strong></em></p>
<p><em>eg </em> \({\text{ln}}\,y = {\text{ln}}\,\left( {k{x^n}} \right),\,\,{\text{ln}}\,\left( {k{x^n}} \right) = a\,{\text{ln}}\,x + b\)</p>
<p>correct application of addition rule for logs <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,k + {\text{ln}}\,\left( {{x^n}} \right)\)</p>
<p>correct application of exponent rule for logs <em><strong>A1</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,k + n\,{\text{ln}}\,x\)</p>
<p>comparing one term with regression equation (check <em><strong>FT</strong></em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(n = a,\,\,b = {\text{ln}}\,k\)</p>
<p>correct working for <em>k</em> <strong>(A1)</strong></p>
<p><em>eg </em>\({\text{ln}}\,k = 6.14210,\,\,\,k = {e^{6.14210}}\)</p>
<p>465.030</p>
<p>\(n = - 0.454,\,\,k = 465\) (464 from 3sf) <em><strong>A1A1 N2N2</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\({e^{{\text{ln}}\,y}} = {e^{a\,{\text{ln}}\,x + b}}\)</p>
<p>correct use of exponent laws for \({e^{a\,{\text{ln}}\,x + b}}\) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\({e^{a\,{\text{ln}}\,x}} \times {e^b}\)</p>
<p>correct application of exponent rule for \(a\,{\text{ln}}\,x\) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,{x^a}\)</p>
<p>correct equation in<em> y</em> <em><strong>A1</strong></em></p>
<p><em>eg </em>\(y = {x^a} \times {e^b}\)</p>
<p>comparing one term with equation of model (check <em><strong>FT</strong></em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(k = {e^b},\,\,n = a\)</p>
<p>465.030</p>
<p>\(n = - 0.454,\,\,k = 465\) (464 from 3sf) <em><strong>A1A1 N2N2</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>valid approach for expressing ln <em>y</em> in terms of ln <em>x</em> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,y = {\text{ln}}\,\left( {k{x^n}} \right),\,\,{\text{ln}}\,\left( {k{x^n}} \right) = a\,{\text{ln}}\,x + b\)</p>
<p>correct application of exponent rule for logs (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,\left( {{x^a}} \right) + b\)</p>
<p>correct working for <em>b</em> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\(b = {\text{ln}}\,\left( {{e^b}} \right)\)</p>
<p>correct application of addition rule for logs <em><strong>A1</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,\left( {{e^b}{x^a}} \right)\)</p>
<p>comparing one term with equation of model (check <em><strong>FT</strong></em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(k = {e^b},\,\,n = a\)</p>
<p>465.030</p>
<p>\(n = - 0.454,\,\,k = 465\) (464 from 3sf) <em><strong>A1A1 N2N2</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">An environmental group records the numbers of coyotes and foxes in a wildlife reserve after \(t\) <span class="s1">years, starting on 1 January 1995</span>.</p>
<p class="p1">Let \(c\) be the number of coyotes in the reserve after \(t\) years. The following table shows the number of coyotes after \(t\) years.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_08.53.25.png" alt></p>
<p class="p1">The relationship between the variables can be modelled by the regression equation \(c = at + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\) and of \(b\).</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="p1">Use the regression equation to estimate the number of coyotes in the reserve when \(t = 7\).</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 class="p1">Let \(f\) be the number of foxes in the reserve after \(t\) years. The number of foxes can be modelled by the equation \(f = \frac{{2000}}{{1 + 99{{\text{e}}^{ - kt}}}}\), where \(k\) <span class="s1">is a constant.</span></p>
<p class="p2">Find the number of foxes in the reserve on 1 January 1995.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">After five years, there were 64 </span>foxes in the reserve. Find \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">During which year were the number of coyotes the same as the number of foxes?</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of setup <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)correct value for \(a\) or \(b\)</p>
<p class="p1">\(13.3823\), \(137.482\)</p>
<p class="p1">\(a{\rm{ }} = {\rm{ }}13.4\), \(b{\rm{ }} = {\rm{ }}137\) <strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong> </strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution into <strong>their </strong>regression equation</p>
<p class="p1"><em>eg</em>\(\;\;\;13.3823 \times 7 + 137.482\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">correct calculation</p>
<p class="p1">\(231.158\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(231\) (coyotes) (must be an integer) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing \(t = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f(0)\)</p>
<p class="p1">correct substitution into the model</p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{2000}}{{1 + 99{{\text{e}}^{ - k(0)}}}},{\text{ }}\frac{{2000}}{{100}}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(20\) (foxes) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing \((5,{\text{ }}64)\) satisfies the equation <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;f(5) = 64\)</p>
<p>correct substitution into the model</p>
<p><em>eg</em>\(\;\;\;64 = \frac{{2000}}{{1 + 99{{\text{e}}^{ - k(5)}}}},{\text{ }}64(1 + 99\(e\)^{ - 5k}}) = 2000\) <strong><em>(A1)</em></strong></p>
<p>\(0.237124\)</p>
<p>\(k = - \frac{1}{5}\ln \left( {\frac{{11}}{{36}}} \right){\text{ (exact), }}0.237{\text{ }}[0.237,{\text{ }}0.238]\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;c = f\), sketch of graphs</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{2000}}{{1 + 99{{\text{e}}^{ - 0.237124t}}}} = 13.382t + 137.482\), sketch of graphs, table of values</p>
<p class="p1">\(t = 12.0403\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(2007\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Exception to the <strong><em>FT </em></strong>rule. Award <strong><em>A1FT </em></strong>on their value of \(t\).</p>
<p class="p1"><em><strong>[4 marks]</strong></em></p>
<p class="p1"><em><strong>Total [16 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {{\text{e}}^{0.5x}} - 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For the graph of <em>f</em>:</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>write down the \(y\)-intercept;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>find the \(x\)-intercept;</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>write down the equation of the horizontal asymptote.</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 class="p1">On the following grid, sketch the graph of \(f\)<span class="s1">, for \( - 4 \leqslant x \leqslant 4\).</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2017-02-03_om_06.14.24.png" alt="M16/5/MATME/SP2/ENG/TZ2/03.b"></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(y = - 1\)</span> <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid attempt to find \(x\)-intercept <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f(x) = 0\)</p>
<p class="p2">1.38629 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p3">\(x = 2\ln 2{\text{ (exact), }}1.39\)</p>
<p class="p1">(iii) <span class="Apple-converted-space"> \(y = - 2\)</span> (must be equation) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-02-03_om_06.16.00.png" alt="M16/5/MATME/SP2/ENG/TZ2/03.b/M"> <strong><em>A1A1A1 N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part a), most candidates were successful at finding the intercepts with the x and y axis, though many failed to write the horizontal asymptote as an equation. Some candidates gave the answer for the horizontal asymptote as \(y \ne - 2\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">For part b), a considerable number of candidates could sketch the exponential function providing an approximately correct shape, although many of them did not use the correct domain, making it go beyond \(x = 4\). Others plotted an incorrect value of \(y\) at \(x = 4\), resulting in the loss of a mark.</p>
<p class="p1">Considering that all the question requires from students is to copy the graph off the GDC, it is important to stress which are the features that cannot be missed.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following table shows the probability distribution of a discrete random variable \(X\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_16.09.18.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(k\).</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="p1">Find \({\text{E}}(X)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of using \(\sum {{p_i}} = 1\) <strong><em>(M1)</em></strong></p>
<p>correct substitution <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;0.15 + k + 0.1 + 2k = 1,{\text{ }}3k + 0.25 = 1\)</p>
<p>\(k = 0.25\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;0 \times 0.15 + 1 \times 0.25 + 2 \times 0.1 + 3 \times 0.5\)</p>
<p class="p1">\({\text{E}}(X) = 1.95\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<p class="p1"><strong><em>Total [5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram is a box and whisker plot for a set of data.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sharpay.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The interquartile range is 20 and the range is 40.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the median value.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(a\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(b\) .<br></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">18 <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) 10 <em><strong>A2 N2 </strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) 44 <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to find the values for the median, lower quartile, and point b. A large majority answered this question correctly.<br></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to find the values for the median, lower quartile, and point b. A large majority answered this question correctly. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A jar contains <span class="s1">5 </span>red discs, <span class="s1">10 </span>blue discs and \(m\) green discs. A disc is selected at random and replaced. This process is performed four times.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the probability that the first disc selected is red.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Let \(X\) </span>be the number of red discs selected. Find the smallest value of \(m\) for which \({\text{Var}}(X{\text{ }}) < 0.6\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">\({\text{P(red)}} = \frac{5}{{15 + m}}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing binomial distribution <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(X \sim B(n,{\text{ }}p)\)</p>
<p class="p1">correct value for the complement of <strong>their</strong> \(p\) (seen anywhere) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(1 - \frac{5}{{15 + m}},{\text{ }}\frac{{10 + m}}{{15 + m}}\)</p>
<p class="p2">correct substitution into \({\text{Var}}(X) = np(1 - p)\) <span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(4\left( {\frac{5}{{15 + m}}} \right)\left( {\frac{{10 + m}}{{15 + m}}} \right),{\text{ }}\frac{{20(10 + m)}}{{{{(15 + m)}^2}}} < 0.6\)</p>
<p class="p2">\(m > 12.2075\) <span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p4"><span class="s2">\(m = 13\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p4"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two events <em>A</em> and <em>B</em> are such that P(<em>A</em>) = 0.62 and P\(\left( {A \cap B} \right)\) = 0.18.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find P(<em>A</em> ∩ <em>B′ </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>Given that P((<em>A</em> ∪ <em>B</em>)′<em> </em>) = 0.19, find P(<em>A </em>|<em> </em><em>B</em>′<em> </em>).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach</p>
<p><em>eg</em> Venn diagram, P(<em>A</em>) − P (<em>A</em> ∩ <em>B</em>), 0.62 − 0.18 <em><strong>(M1) </strong></em></p>
<p>P(<em>A</em> ∩ <em>B' </em>) = 0.44 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find either P(<em>B</em>′ ) or P(<em>B</em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em><img src="data:image/png;base64,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"> (seen anywhere), 1 − P(<em>A</em> ∩ <em>B</em>′<em> </em>) − P((<em>A</em> ∪ <em>B</em>)′<em> </em>)</p>
<p>correct calculation for P(<em>B</em>′ ) or P(<em>B</em>) <em><strong>(A1)</strong></em></p>
<p><em>eg </em> 0.44 + 0.19, 0.81 − 0.62 + 0.18</p>
<p>correct substitution into \(\frac{{{\text{P}}\left( {A \cap B'} \right)}}{{{\text{P}}\left( {B'} \right)}}\) <em><strong> (A1)</strong></em></p>
<p><em>eg</em> \(\frac{{0.44}}{{0.19 + 0.44}},\,\,\frac{{0.44}}{{1 - 0.37}}\)</p>
<p>0.698412</p>
<p>P(<em>A </em>|<em> </em><em>B</em>′<em> </em>) = \(\frac{{44}}{{63}}\) (exact), 0.698 <em><strong>A1 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The heights of adult males in a country are normally distributed with a mean of 180 cm and a standard deviation of \(\sigma {\text{ cm}}\). 17% of these men are shorter than 168 cm. 80% of them have heights between \((192 - h){\text{ cm}}\) and 192 cm.</p>
<p>Find the value of \(h\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>finding the \(z\)-value for 0.17 <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(z = - 0.95416\)</p>
<p>setting up equation to find \(\sigma \)<em>, <strong>(M1)</strong></em></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(z = \frac{{168 - 180}}{\sigma },{\text{ }} - 0.954 = \frac{{ - 12}}{\sigma }\)</p>
<p>\(\sigma = 12.5765\) <strong><em>(A1)</em></strong></p>
<p><strong>EITHER (Properties of the Normal curve)</strong></p>
<p>correct value (seen anywhere) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(X < 192) = 0.83,{\text{ P}}(X > 192) = 0.17\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(X < 192 - h) = 0.83 - 0.8,{\text{ P}}(X < 192 - h) = 1 - 0.8 - 0.17,\)</p>
<p>\({\text{P}}(X > 192 - h) = 0.8 + 0.17\)</p>
<p>correct equation in \(h\)</p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{(192 - h) - 180}}{{12.576}} = - 1.88079,{\text{ }}192 - h = 156.346\) <strong><em>(A1)</em></strong></p>
<p>35.6536</p>
<p>\(h = 35.7\) <strong><em>A1 N3</em></strong></p>
<p><strong>OR (Trial and error using different values of <em>h</em>)</strong></p>
<p><strong>two</strong> correct probabilities whose 2 sf will round up <strong>and</strong> down, respectively, to 0.8 <strong><em>A2</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P}}(192 - 35.6 < X < 192) = 0.799706,{\text{ P}}(157 < X < 192) = 0.796284,\)</p>
<p>\({\text{P}}(192 - 36 < X < 192) = 0.801824\)</p>
<p>\(h = 35.7\) <strong><em>A2</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The time taken for a student to complete a task is normally distributed with a mean of \(20\) minutes and a standard deviation of \(1.25\) minutes.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A student is selected at random. Find the probability that the student completes the task in less than \(21.8\) minutes.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The probability that a student takes between \(k\) and \(21.8\) minutes is \(0.3\). Find the value of \(k\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> There may be slight differences in answers, depending on whether candidates use tables or GDCs, or their 3 sf answers in subsequent parts. Do not penalise answers that are consistent with <strong>their </strong>working and check carefully for <strong><em>FT</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to standardize <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(z = \frac{{21.8 - 20}}{{1.25}},{\text{ 1.44}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(T < 21.8) = 0.925\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> There may be slight differences in answers, depending on whether candidates use tables or GDCs, or their 3 sf answers in subsequent parts. Do not penalise answers that are consistent with <strong>their </strong>working and check carefully for <strong><em>FT</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to subtract probabilities <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{P}}(T < 21.8) - {\text{P}}(T < k) = 0.3,{\text{ }}0.925 - 0.3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(T < k) = 0.625\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">finding the \(z\)-value for \(0.625\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(z = 0.3186\) (from tables), \(z = 0.3188\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to set up equation using <strong>their </strong>\(z\)-value <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.3186 = \frac{{k - 20}}{{1.25}},{\text{ }} - 0.524 \times 1.25 = k - 20\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 20.4\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 20.4\) <strong><em>A3 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In a large city, the time taken to travel to work is normally distributed with </span><span style="font-family: times new roman,times; font-size: medium;">mean \(\mu \) and standard deviation \(\sigma \) . It is found that \(4\% \) of the population take less </span><span style="font-family: times new roman,times; font-size: medium;">than 5 minutes to get to work, and \(70\% \) take less than 25 minutes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\mu \) and of \(\sigma \) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">correct <em>z</em>-values <em><strong>(A1)(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 1.750686 \ldots \) , \(0.524400 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up their equations, must involve <em>z</em>-values, not % <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. one correct equation </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">two correct equations <em><strong>A1A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\mu - 1.750686\sigma = 5\) , \(0.5244005 = \frac{{25 - \mu }}{\sigma }\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve their equations <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. substitution, matrices, one correct value </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\mu = 20.39006 \ldots \) , \(\sigma = 8.790874 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\mu = 20.4\) \([20.3{\text{, }}20.4]\), \(\sigma = 8.79\) \([8.79{\text{, }}8.80]\) <em><strong>A1A1 N4</strong> </em></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">A standard question for which well-prepared candidates frequently earned all eight marks. Common errors included the use of percentages rather than <em>z</em>-values and the inability to find the negative <em>z</em>-value. Others had correct equations but were not able to use their GDC to solve them and ultimately made errors in their algebra. </span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A box holds 240 eggs. The probability that an egg is brown is 0.05.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the expected number of brown eggs in the box.</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 style="font-family: times new roman,times; font-size: medium;">Find the probability that there are 15 brown eggs in the box.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that there are at least 10 brown eggs in the box.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution into formula for \({\rm{E}}(X)\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.05 \times 240\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}(X) = 12\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing binomial probability (may be seen in part (a)) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>{240}\\<br>{15}<br>\end{array}} \right){(0.05)^{15}}{(0.95)^{225}}\) </span><span style="font-family: times new roman,times; font-size: medium;">, </span><span style="font-family: times new roman,times; font-size: medium;">\(X \sim {\rm{B}}(240,0.05)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X = 15) = 0.0733\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \le 9) = 0.236\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. using complement, summing probabilities</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}(X \ge 10) = 0.764\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was answered correctly by most candidates. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In parts (b) and (c), many failed to recognize the binomial nature of this experiment and opted for incorrect techniques in simple probability. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In parts (b) and (c), many failed to recognize the binomial nature of this experiment and opted for incorrect techniques in simple probability. Although several candidates appreciated that (c) involved the idea of a complement, some resorted to elaborate probability addition suggesting they were unaware of the capabilities of their GDC. There was also a great deal of evidence to suggest that candidates did not understand the phrase "at least 10" as several candidates found either \(1 - {\rm{P}}(X \le 10)\) , \(1 - {\rm{P}}(X = 10)\) or \({\rm{P}}(X > 10)\) . </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The mass <em>M</em> of apples in grams is normally distributed with mean <em>μ</em>. The following table shows probabilities for values of <em>M</em>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The apples are packed in bags of ten.</p>
<p>Any apples with a mass less than 95 g are classified as small.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <em>k</em>.</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>Show that <em>μ</em> = 106.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>P</em>(M < 95) .</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a bag of apples selected at random contains at most one small apple.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the expected number of bags in this crate that contain at most one small apple.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that at least 48 bags in this crate contain at most one small apple.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of using \(\sum {{p_i}} = 1\) <em><strong>(M1)</strong></em></p>
<p><em>eg k</em> + 0.98 + 0.01 = 1</p>
<p><em>k</em> = 0.01 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing that 93 and 119 are symmetrical about <em>μ</em> <em><strong>(M1)</strong></em></p>
<p><em>eg μ</em> is midpoint of 93 and 119</p>
<p>correct working to find <em>μ</em> <em><strong>A1</strong></em></p>
<p>\(\frac{{119 + 93}}{2}\)</p>
<p><em>μ</em> = 106 <em><strong>AG N0</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finding standardized value for 93 or 119 <em><strong> (A1)</strong></em><br><em>eg</em> <em>z</em> = −2.32634, <em>z</em> = 2.32634</p>
<p>correct substitution using <strong>their</strong> <em>z</em> value <em><strong>(A1)</strong></em><br><em>eg </em> \(\frac{{93 - 106}}{\sigma } = - 2.32634,\,\,\frac{{119 - 106}}{{2.32634}} = \sigma \)</p>
<p>σ = 5.58815 <em><strong>(A1)</strong></em></p>
<p>0.024508</p>
<p>P(<em>X</em> < 95) = 0.0245 <em><strong> A2 N3</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of recognizing binomial <strong><em>(M1)<span class="Apple-converted-space"> </span></em></strong></p>
<p><em>eg </em>10, <em>anana</em><em>Cpqn</em>−=××and 0.024B(5,,)<em>pnp</em>=<span class="Apple-converted-space"> </span></p>
<p>valid approach <strong><em>(M1)<span class="Apple-converted-space"> </span></em></strong></p>
<p><em>eg </em>P(1),P(0)P(1)<em>XXX</em>≤=+=<span class="Apple-converted-space"> </span></p>
<p>0.976285<span class="Apple-converted-space"> </span></p>
<p>0.976 <strong><em>A1 N2<span class="Apple-converted-space"> </span></em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing <strong>new</strong> binomial probability <em><strong>(M1)</strong></em><br><em>eg </em> B(50, 0.976)</p>
<p>correct substitution <em><strong>(A1)</strong></em><br><em>eg</em> <em>E(X) = </em>50 (0.976285)</p>
<p>48.81425</p>
<p>48.8 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> P(X ≥ 48), 1 − P(X ≤ 47)</p>
<p>0.884688</p>
<p>0.885 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A Ferris wheel with diameter \(122\) metres rotates clockwise at a constant speed. </span><span style="font-family: times new roman,times; font-size: medium;">The wheel completes \(2.4\) rotations every hour. The bottom of the wheel is \(13\) metres </span><span style="font-family: times new roman,times; font-size: medium;">above the ground.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/open.png" alt></span></p>
<p> <span style="font-family: times new roman,times; font-size: medium;">A seat starts at the bottom of the wheel.</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">After <em><strong>t</strong> </em>minutes, the height \(h\) metres above the ground of the seat is given by\[h = 74 + a\cos bt {\rm{ .}}\]<br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the maximum height above the ground of the seat.</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 style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that the period of \(h\) is \(25\) minutes.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the <strong>exact</strong> value of \(b\) .</span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(b) (i) Show that the period of \(h\) is \(25\) minutes.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Write down the <strong>exact</strong> value of \(b\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) </span><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(a\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) </span><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(h\) , for \(0 \le t \le 50\) .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">bcd.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(a\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(h\) , for \(0 \le t \le 50\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In one rotation of the wheel, find the probability that a randomly selected seat is </span><span style="font-family: times new roman,times; font-size: medium;">at least \(105\) metres above the ground.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(13 + {\rm{diameter}}\) , \(13 + 122\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">maximum height \( = 135\) (m) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) period \( = \frac{{60}}{{2.4}}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">period \( = 25\) minutes <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(b = \frac{{2\pi }}{{25}}\) \(( = 0.08\pi )\) <strong><em>A1 N1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong><em>[2 marks]<br></em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) (i) period \( = \frac{{60}}{{2.4}}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">period \( = 25\) minutes <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(b = \frac{{2\pi }}{{25}}\) \(( = 0.08\pi )\) <strong><em>A1 N1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong><em>[2 marks]<br></em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) <strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{max}} - 74\) , \(\left| a \right| = \frac{{135 - 13}}{2}\) , \(74 - 13\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left| a \right| = 61\) (accept \(a = 61\) ) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = - 61\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1</span></em><span style="font-family: times new roman,times; font-size: medium;"><em> N2</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute valid point into equation for <em><strong>h</strong></em> <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(135 = 74 + a\cos \left( {\frac{{2\pi \times 12.5}}{{25}}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(135 = 74 + a\cos (\pi )\) , \(13 = 74 + a\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = - 61\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c)<br></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1A1 N4</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct domain, <em><strong>A1</strong></em> for approximately correct range,</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for approximately correct sinusoidal shape with \(2\) cycles. </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><strong> Only</strong> if this last <em><strong>A1</strong></em> awarded, award <em><strong>A1</strong></em> for max/min in approximately </span><span style="font-family: times new roman,times; font-size: medium;">correct positions. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [9 marks]<br></span></strong></em></p>
<div class="question_part_label">bcd.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{max}} - 74\) , \(\left| a \right| = \frac{{135 - 13}}{2}\) , \(74 - 13\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left| a \right| = 61\) (accept \(a = 61\) ) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = - 61\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1</span></em><span style="font-family: times new roman,times; font-size: medium;"><em> N2</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute valid point into equation for <em><strong>h</strong></em> <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(135 = 74 + a\cos \left( {\frac{{2\pi \times 12.5}}{{25}}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(135 = 74 + a\cos (\pi )\) , \(13 = 74 + a\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = - 61\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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" alt><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1A1 N4</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct domain, <em><strong>A1</strong></em> for approximately correct range,</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for approximately correct sinusoidal shape with \(2\) cycles. </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><strong> Only</strong> if this last <em><strong>A1</strong></em> awarded, award <em><strong>A1</strong></em> for max/min in approximately </span><span style="font-family: times new roman,times; font-size: medium;">correct positions. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">setting up inequality (accept equation) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(h > 105\) , \(105 = 74 + a\cos bt\) , sketch of graph with line \(y = 105\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">any <strong>two</strong> correct values for <em><strong>t</strong></em> (seen anywhere) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(t = 8.371 \ldots \) , \(t = 16.628 \ldots \) , \(t = 33.371 \ldots \) , \(t = 41.628 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{16.628 - 8.371}}{{25}}\) , \(\frac{{{t_1} - {t_2}}}{{25}}\) , \(\frac{{2 \times 8.257}}{{50}}\) , \(\frac{{2(12.5 - 8.371)}}{{25}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.330\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were successful with part (a).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A surprising number had difficulty producing enough work to show that the period was \(25\); writing down the exact value of \(b\) also overwhelmed a number of candidates. In part (c), candidates did not recognize that the seat on the Ferris wheel is a minimum at \(t = 0\) thereby making the value of a negative. Incorrect values of \(61\) were often seen with correct follow through obtained when sketching the graph in part (d). Graphs again frequently failed to show key features in approximately correct locations and candidates lost marks for incorrect domains and ranges.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A surprising number had difficulty producing enough work to show that the period was \(25\); writing down the exact value of \(b\) also overwhelmed a number of candidates. In part (c), candidates did not recognize that the seat on the Ferris wheel is a minimum at \(t = 0\) thereby making the value of a negative. Incorrect values of \(61\) were often seen with correct follow through obtained when sketching the graph in part (d). Graphs again frequently failed to show key features in approximately correct locations and candidates lost marks for incorrect domains and ranges.</p>
<div class="question_part_label">bcd.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A surprising number had difficulty producing enough work to show that the period was \(25\); writing down the exact value of \(b\) also overwhelmed a number of candidates. In part (c), candidates did not recognize that the seat on the Ferris wheel is a minimum at \(t = 0\) thereby making the value of a negative. Incorrect values of \(61\) were often seen with correct follow through obtained when sketching the graph in part (d). Graphs again frequently failed to show key features in approximately correct locations and candidates lost marks for incorrect domains and ranges.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A surprising number had difficulty producing enough work to show that the period was \(25\); writing down the exact value of \(b\) also overwhelmed a number of candidates. In part (c), candidates did not recognize that the seat on the Ferris wheel is a minimum at \(t = 0\) thereby making the value of a negative. Incorrect values of \(61\) were often seen with correct follow through obtained when sketching the graph in part (d). Graphs again frequently failed to show key features in approximately correct locations and candidates lost marks for incorrect domains and ranges.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) was very poorly done for those who attempted the question and most did not make the connection between height, time and probability. The idea of linking probability with a real-life scenario proved beyond most candidates. That said, there were a few novel approaches from the strongest of candidates using circles and angles to solve this part of question 10.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br>