<p>Anand signed up for a privilege card with Temasek Airlines where cardholders earn more reward points as they fly further with the airline. From the airlines' website, Anand identified \(8\) different destinations and recorded the flight distance, \(x\) miles, of each destination from Singapore. He also recorded the corresponding reward points, \(y\), credited into his card after each flight. The data are shown below.</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr><th style="text-align: left; --cell-justify: flex-start">Destination</th><th style="text-align: left; --cell-justify: flex-start">A</th><th style="text-align: left; --cell-justify: flex-start">B</th><th style="text-align: left; --cell-justify: flex-start">C</th><th style="text-align: left; --cell-justify: flex-start">D</th><th style="text-align: left; --cell-justify: flex-start">E</th><th style="text-align: left; --cell-justify: flex-start">F</th><th style="text-align: left; --cell-justify: flex-start">G</th><th style="text-align: left; --cell-justify: flex-start">H</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Flight distance, \(x\)</td><td style="text-align: left; --cell-justify: flex-start">\(200\)</td><td style="text-align: left; --cell-justify: flex-start">\(400\)</td><td style="text-align: left; --cell-justify: flex-start">\(600\)</td><td style="text-align: left; --cell-justify: flex-start">\(800\)</td><td style="text-align: left; --cell-justify: flex-start">\(1000\)</td><td style="text-align: left; --cell-justify: flex-start">\(1200\)</td><td style="text-align: left; --cell-justify: flex-start">\(1400\)</td><td style="text-align: left; --cell-justify: flex-start">\(1600\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Reward points, \(y\)</td><td style="text-align: left; --cell-justify: flex-start">\(110\)</td><td style="text-align: left; --cell-justify: flex-start">\(160\)</td><td style="text-align: left; --cell-justify: flex-start">\(190\)</td><td style="text-align: left; --cell-justify: flex-start">\(200\)</td><td style="text-align: left; --cell-justify: flex-start">\(210\)</td><td style="text-align: left; --cell-justify: flex-start">\(215\)</td><td style="text-align: left; --cell-justify: flex-start">\(220\)</td><td style="text-align: left; --cell-justify: flex-start">\(225\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch a scatter diagram of the data.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate the value of the product moment correlation coefficient, and explain why this value does not necessarily mean that a linear model is the best for the relationship between \(x\) and \(y\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Without calculating the product moment correlation coefficient, explain which of the following equations, where&nbsp;\(a\) and \(b\) are constants, and \(b &gt; 0\), can be used to model the relationship between \(x\) and \(y\).<br>(I) \(y = a + bx^2\)<br>(II) \(y = a + b \ln x\)<br>(III) \(y = a + be^{-x}\)<span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using the model identified in (c), find the equation of the corresponding regression line.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence estimate the reward points that can be earned for a destination \(700\) miles away from Singapore. Comment on the reliability of your estimation.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">It is given that&nbsp;\(1\) mile is approximately \(1.609\) kilometres. State what will happen to the product moment correlation coefficient found in (b) if the flight distance is measured in kilometres instead of miles.</span><span class="inline-marks">[1]</span></div></li></ol>

Anand signed up for a privilege card with Temasek Airlines where cardholders earn more reward points as they fly further with the airline. From the airlines' website, Anand identified \(8\) different destinations and recorded the flight distance, \(x\) miles, of each destination from Singapore. He also recorded the corresponding reward points, \(y\), credited into his card after each flight. The data are shown below.

Calculate the value of the product moment correlation coefficient, and explain why this value does not necessarily mean that a linear model is the best for the relationship between \(x\) and \(y\).

Without calculating the product moment correlation coefficient, explain which of the following equations, where \(a\) and \(b\) are constants, and \(b > 0\), can be used to model the relationship between \(x\) and \(y\).

Without calculating the product moment correlation coefficient, explain which of the following equations, where \(a\) and \(b\) are constants, and \(b > 0\), can be used to model the relationship between \(x\) and \(y\).
(I) \(y = a + bx^2\)
(II) \(y = a + b \ln x\)
(III) \(y = a + be^{-x}\)

Using the model identified in (c), find the equation of the corresponding regression line.

Hence estimate the reward points that can be earned for a destination \(700\) miles away from Singapore. Comment on the reliability of your estimation.

It is given that \(1\) mile is approximately \(1.609\) kilometres. State what will happen to the product moment correlation coefficient found in (b) if the flight distance is measured in kilometres instead of miles.

2025 TJC P2 Q10

2025 TJC P2 Q10