<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 style="width: 100%;"><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Age of car ($x$ years)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$0$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$7$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Resale Value ($y$ thousand dollars)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30.0$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$25.8$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$22.3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$16.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$14.4$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$12.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10.9$</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.

Correlation & Regressions

International Baccalaureate Higher Level Mathematics

2025 TMJC P2 Q9

2025 TMJC P2 Q9