<p>A car dealer is investigating how the value of a car depreciates over time. A random sample of eight cars of the same model is selected and the current resale value, $y$ thousand dollars of each car is recorded along with its age in $x$ years. The results are shown in the table.</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr><th style="text-align: left; --cell-justify: flex-start">Age of car ($x$ years)</th><th style="text-align: left; --cell-justify: flex-start">$0$</th><th style="text-align: left; --cell-justify: flex-start">$1$</th><th style="text-align: left; --cell-justify: flex-start">$2$</th><th style="text-align: left; --cell-justify: flex-start">$3$</th><th style="text-align: left; --cell-justify: flex-start">$4$</th><th style="text-align: left; --cell-justify: flex-start">$5$</th><th style="text-align: left; --cell-justify: flex-start">$6$</th><th style="text-align: left; --cell-justify: flex-start">$7$</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Resale Value ($y$ thousand dollars)</td><td style="text-align: left; --cell-justify: flex-start">$30.0$</td><td style="text-align: left; --cell-justify: flex-start">$25.8$</td><td style="text-align: left; --cell-justify: flex-start">$22.3$</td><td style="text-align: left; --cell-justify: flex-start">$19.2$</td><td style="text-align: left; --cell-justify: flex-start">$16.5$</td><td style="text-align: left; --cell-justify: flex-start">$14.4$</td><td style="text-align: left; --cell-justify: flex-start">$12.5$</td><td style="text-align: left; --cell-justify: flex-start">$10.9$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Draw the scatter diagram for these values, labelling the axes clearly.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>It is thought that the resale value of a car, $y$, can be modelled by one of the formulae \[y = a + bx \quad \text{or} \quad \ln y = c + dx\]where&nbsp;$a$, $b$, $c$ and $d$ are real constants.</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Find, correct to $5$ decimal places, the value of the product moment correlation coefficient between</span><span class="inline-marks">[2]</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$y$ and $x$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\ln y$ and $x$.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Use your answers to parts (a) and (b) to explain which of $y = a + bx$ and $\ln y = c + dx$ is the better model.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>It is required to estimate the age of a car with a resale value of $\$18000$.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the equation of a suitable regression line and use it to find the required estimate.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Without the use of a graphing calculator, re-write your equation from part (d) so that it can be used to estimate the resale value when the age is given in months.</span><span class="inline-marks">[2]</span></div></li></ol>

A car dealer is investigating how the value of a car depreciates over time. A random sample of eight cars of the same model is selected and the current resale value, $y$ thousand dollars of each car is recorded along with its age in $x$ years. The results are shown in the table.

Draw the scatter diagram for these values, labelling the axes clearly.

It is thought that the resale value of a car, $y$, can be modelled by one of the formulae \[y = a + bx \quad \text{or} \quad \ln y = c + dx\]where $a$, $b$, $c$ and $d$ are real constants.

Find, correct to $5$ decimal places, the value of the product moment correlation coefficient between

Use your answers to parts (a) and (b) to explain which of $y = a + bx$ and $\ln y = c + dx$ is the better model.

It is required to estimate the age of a car with a resale value of $\$18000$.

Find the equation of a suitable regression line and use it to find the required estimate.

Without the use of a graphing calculator, re-write your equation from part (d) so that it can be used to estimate the resale value when the age is given in months.

2025 TMJC P2 Q9

2025 TMJC P2 Q9