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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The table gives the record time, $t$ seconds, for a particular $100$-metre freestyle swimming long course event for women for the number of years, $y$, after Year $2000$.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$y$</th><th style="text-align: center; --cell-justify: center">$4$</th><th style="text-align: center; --cell-justify: center">$6$</th><th style="text-align: center; --cell-justify: center">$8$</th><th style="text-align: center; --cell-justify: center">$12$</th><th style="text-align: center; --cell-justify: center">$16$</th><th style="text-align: center; --cell-justify: center">$18$</th><th style="text-align: center; --cell-justify: center">$19$</th><th style="text-align: center; --cell-justify: center">$20$</th><th style="text-align: center; --cell-justify: center">$21$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$t$</td><td style="text-align: center; --cell-justify: center">$53.52$</td><td style="text-align: center; --cell-justify: center">$53.11$</td><td style="text-align: center; --cell-justify: center">$52.88$</td><td style="text-align: center; --cell-justify: center">$52.25$</td><td style="text-align: center; --cell-justify: center">$52.06$</td><td style="text-align: center; --cell-justify: center">$52.03$</td><td style="text-align: center; --cell-justify: center">$52.03$</td><td style="text-align: center; --cell-justify: center">$52.9$</td><td style="text-align: center; --cell-justify: center">$51.96$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph">Draw the scatter diagram for these values, labelling the axes clearly. Circle the point on the scatter diagram that seems to have the value of $t$ recorded wrongly and label it as&nbsp;$P$.<span class="inline-marks">[3]</span></div></li></ol>
<p><strong>Omit point P identified in part (a) for the rest of this question.</strong></p>
<ol class="abc-list" start="2"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">It is desired to predict the record time in future years. Explain why, in this context, a linear model is likely to be inappropriate.<span class="inline-marks">[1]</span></div></li></ol>
<p>The following models are proposed to describe the relationship between $t$ and $y$:</p>
<p>For&nbsp;$a &gt; 0$ and $b &gt; 0$,</p>
<p style="text-align: center">Model A: $t = a - b y^2$; $\qquad$ Model B: $t = a + \frac{b}{y}$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="imported-item-block-8-item-0" data-list-item-path="imported-item-block-8-item-0"><div class="list-item-paragraph">Explain, without further calculation, why Model B provides a more accurate model of the relationship between $t$ and $y$ than Model A.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-8-item-1" data-list-item-path="imported-item-block-8-item-1"><div class="list-item-paragraph">For Model B, find the product moment correlation coefficient between $t$ and $\frac{1}{y}$. State the values of $a$ and of $b$ correct to $5$ significant figures.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-8-item-2" data-list-item-path="imported-item-block-8-item-2"><div class="list-item-paragraph">Use your equation to estimate the record time in $2025$, correct to $3$ significant figures. Explain whether your estimate is reliable.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-8-item-3" data-list-item-path="imported-item-block-8-item-3"><div class="list-item-paragraph">It is required to estimate the value of $y$ for which $t = 52.45$. Use a suitable equation to find the estimated value of $y$, correct to the nearest integer. Explain why neither the regression line of $y^2$ on $t$ nor the regression line of $\frac{1}{y}$ on $t$ should be used.<span class="inline-marks">[2]</span></div></li></ol>

The table gives the record time, $t$ seconds, for a particular $100$-metre freestyle swimming long course event for women for the number of years, $y$, after Year $2000$.

Draw the scatter diagram for these values, labelling the axes clearly. Circle the point on the scatter diagram that seems to have the value of $t$ recorded wrongly and label it as $P$.

**Omit point P identified in part (a) for the rest of this question.**

It is desired to predict the record time in future years. Explain why, in this context, a linear model is likely to be inappropriate.

The following models are proposed to describe the relationship between $t$ and $y$:

Model A: $t = a - b y^2$; $\qquad$ Model B: $t = a + \frac{b}{y}$.

Explain, without further calculation, why Model B provides a more accurate model of the relationship between $t$ and $y$ than Model A.

For Model B, find the product moment correlation coefficient between $t$ and $\frac{1}{y}$. State the values of $a$ and of $b$ correct to $5$ significant figures.

Use your equation to estimate the record time in $2025$, correct to $3$ significant figures. Explain whether your estimate is reliable.

It is required to estimate the value of $y$ for which $t = 52.45$. Use a suitable equation to find the estimated value of $y$, correct to the nearest integer. Explain why neither the regression line of $y^2$ on $t$ nor the regression line of $\frac{1}{y}$ on $t$ should be used.

Reliability of Regression Estimates

Linearisation of Nonlinear Models

Finding Regression Line Equations

Interpreting Regression Models and Coefficients

Comparing Regression Models

Prediction Using Regression Models

Scatter Diagrams and Association

Choosing the Appropriate Regression Line

Product Moment Correlation Coefficient

2023 HCI P2 Q12

Correlation & Regressions

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph"><p>It is given that&nbsp;$X$ is the number of times a student is late for school in a year and $Y$ is the student's performance in the Mathematics Examination. The product moment correlation coefficient of a bivariate sample $(x_i, y_i)$, $i = 1, 2, \dots, n$, for&nbsp;$n$ students is $r$.</p><p>State, giving a reason, whether each of the following statements is true or false.</p></div><ol class="i-list"><li data-list-item-id="imported-item-block-0-item-0-sub-0" data-list-item-path="imported-item-block-0-item-0/imported-item-block-0-item-0-sub-0"><div class="list-item-paragraph">When the value of $r$ is zero, it can be implied that the variables $X$ and $Y$ are not related.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-0-item-0-sub-1" data-list-item-path="imported-item-block-0-item-0/imported-item-block-0-item-0-sub-1"><div class="list-item-paragraph">When the value of $r$ is $-1$, it can be implied that late-coming causes poor performance in the subject.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="imported-item-block-0-item-1" data-list-item-path="imported-item-block-0-item-1"><div class="list-item-paragraph">The table below shows the daily sales of cups of iced coffee in a week by a shop and the maximum daily temperature.</div><div class="table-container"><table style="table-layout: fixed; width: 490px;"><colgroup><col style="width: 28.5714%;"><col style="width: 10.2041%;"><col style="width: 10.2041%;"><col style="width: 10.2041%;"><col style="width: 10.2041%;"><col style="width: 10.2041%;"><col style="width: 10.2041%;"><col style="width: 10.2041%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"></th><th style="text-align: center; --cell-justify: center">Mon</th><th style="text-align: center; --cell-justify: center">Tue</th><th style="text-align: center; --cell-justify: center">Wed</th><th style="text-align: center; --cell-justify: center">Thu<br></th><th style="text-align: center; --cell-justify: center">Fri</th><th style="text-align: center; --cell-justify: center">Sat</th><th style="text-align: center; --cell-justify: center">Sun</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Temperature ($^\circ\mathrm{C}$), $t$</td><td style="text-align: center; --cell-justify: center">$30.5$</td><td style="text-align: center; --cell-justify: center">$31.3$</td><td style="text-align: center; --cell-justify: center">$31.9$</td><td style="text-align: center; --cell-justify: center">$34.8$</td><td style="text-align: center; --cell-justify: center">$25.9$</td><td style="text-align: center; --cell-justify: center">$28.5$</td><td style="text-align: center; --cell-justify: center">$29.2$</td></tr><tr><td style="text-align: center; --cell-justify: center">Daily sales, $y$</td><td style="text-align: center; --cell-justify: center">$95$</td><td style="text-align: center; --cell-justify: center">$101$</td><td style="text-align: center; --cell-justify: center">$115$</td><td style="text-align: center; --cell-justify: center">$79$</td><td style="text-align: center; --cell-justify: center">$81$</td><td style="text-align: center; --cell-justify: center">$86$</td><td style="text-align: center; --cell-justify: center">$88$</td></tr></tbody></table></div><ol class="i-list"><li data-list-item-id="subitem-1785052937038-aak3vr5wi" data-list-item-path="imported-item-block-0-item-1/subitem-1785052937038-aak3vr5wi"><div class="list-item-paragraph"><p>Sketch a scatter diagram of $y$ against $t$, labelling the axes clearly.</p><span class="inline-marks">[2]</span></div></li></ol><div class="list-item-after-content"><p>One of the values of $y$ appears to be incorrect.</p></div><div class="list-item-after-content"><ol class="i-list" start="2"><li data-list-item-id="item-1785052993309-9pcovoox0" data-list-item-path="item-1785052993309-9pcovoox0"><div class="list-item-paragraph"><p>Indicate the corresponding point on your diagram by labelling it $P$. Omitting $P$, find the equation of the least squares regression line of $\frac{1}{y}$ on $t$, and the value of the product moment correlation coefficient between $\frac{1}{y}$ and $t$. Comment on this value.</p><span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1785053005257-5g27wrlec" data-list-item-path="item-1785053005257-5g27wrlec"><div class="list-item-paragraph"><p>Use an appropriate regression line to give an estimate of the daily sales when the temperature is $20.4^\circ\mathrm{C}$. State, with a reason, whether the estimate is reliable.</p><span class="inline-marks">[2]</span></div></li></ol></div></li></ol>

Correlation and Causation

2024 RI P2 Q8

<p>An experiment was carried out to investigate the growth rate of a particular species of plant. The following table gives the height of the plant specimen, $h$ centimeters, at the start of the $n^{\text{th}}$ month.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 350px;"><colgroup><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$n$</th><th style="text-align: center; --cell-justify: center">$1$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$4$</th><th style="text-align: center; --cell-justify: center">$6$</th><th style="text-align: center; --cell-justify: center">$8$</th><th style="text-align: center; --cell-justify: center">$10$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$h$</td><td style="text-align: center; --cell-justify: center">$6.22$</td><td style="text-align: center; --cell-justify: center">$9.06$</td><td style="text-align: center; --cell-justify: center">$13.62$</td><td style="text-align: center; --cell-justify: center">$16.62$</td><td style="text-align: center; --cell-justify: center">$18.46$</td><td style="text-align: center; --cell-justify: center">$19.72$</td></tr></tbody></table></div></div>
<p>A possible model for the growth rate is given to be</p>
<p style="text-align: center">\[h = \frac{an}{b + n},\]</p>
<p>where&nbsp;$a$ and $b$ are constants.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-5-item-0" data-list-item-path="imported-item-block-5-item-0"><div class="list-item-paragraph">By writing the above equation in a form that is linear in $\frac{1}{h}$ and $\frac{1}{n}$, calculate the equation of the least squares regression line of $\frac{1}{h}$ on $\frac{1}{n}$. Hence find estimates for the values of $a$ and $b$, correct to $3$ decimal places.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="imported-item-block-5-item-1" data-list-item-path="imported-item-block-5-item-1"><div class="list-item-paragraph"><p>Sketch a scatter diagram for&nbsp;$\frac{1}{h}$ against $\frac{1}{n}$ and include the least squares regression line found in part <strong>(a)</strong>.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-5-item-2" data-list-item-path="imported-item-block-5-item-2"><div class="list-item-paragraph">Explain in the context of the question, the significance of the value of $a$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-5-item-3" data-list-item-path="imported-item-block-5-item-3"><div class="list-item-paragraph"><p>Use the least squares regression line in part <strong>(a)</strong> to find the least integer value of $n$ required for the plant to reach a height of $18$ centimeters. Explain whether you would expect this estimate to be reliable.</p><span class="inline-marks">[3]</span></div></li></ol>
<p>For a line of best fit $y = \mathrm{f}(x)$, the residual for a point $(p, q)$ plotted on the scatter diagram is the vertical distance between $(p, \mathrm{f}(p))$ and $(p, q)$.</p>
<ol class="abc-list" start="5"><li data-list-item-id="imported-item-block-7-item-0" data-list-item-path="imported-item-block-7-item-0"><div class="list-item-paragraph"><p>Mark the residual for each point on the scatter diagram in part <strong>(b)</strong>.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-7-item-1" data-list-item-path="imported-item-block-7-item-1"><div class="list-item-paragraph">Find the sum of squares of the residuals for the least squares regression line of $\frac{1}{h}$ on $\frac{1}{n}$, giving your answer correct to $5$ significant figures.<span class="inline-marks">[1]</span></div></li></ol>

An experiment was carried out to investigate the growth rate of a particular species of plant. The following table gives the height of the plant specimen, $h$ centimeters, at the start of the $n^{\text{th}}$ month.

A possible model for the growth rate is given to be

By writing the above equation in a form that is linear in $\frac{1}{h}$ and $\frac{1}{n}$, calculate the equation of the least squares regression line of $\frac{1}{h}$ on $\frac{1}{n}$. Hence find estimates for the values of $a$ and $b$, correct to $3$ decimal places.

Explain in the context of the question, the significance of the value of $a$.

For a line of best fit $y = \mathrm{f}(x)$, the residual for a point $(p, q)$ plotted on the scatter diagram is the vertical distance between $(p, \mathrm{f}(p))$ and $(p, q)$.

Find the sum of squares of the residuals for the least squares regression line of $\frac{1}{h}$ on $\frac{1}{n}$, giving your answer correct to $5$ significant figures.

Least-Squares Regression and Residuals

2024 HCI P2 Q11

<p>An energy company is studying the relationship between the average daily temperature, $x$ (in degrees Celsius), and the amount of electricity consumed, $y$ (in megawatt-hours MWh). For a random sample of ten days, the records are shown in the table below.</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">Average daily temperature ($x{}^\circ$C)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$20$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$22$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$24$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$26$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$28$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$32$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$34$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$36$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$38$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Electricity consumption ($y$ MWh)</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">$12.8$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$13.2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$14.0$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$14.7$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$16.2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$16.2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.9$</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">$29.5$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a-2" data-list-item-path="item-a-2"><div class="list-item-content"><span class="list-item-text">Draw a 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 electricity consumption and average daily temperature can be modelled by one of the formulae\[y = a + bx^2, \quad y = c + dx,\]where&nbsp;$a$, $b$, $c$ and $d$ are constants.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b-2" data-list-item-path="item-b-2"><div class="list-item-content"><span class="list-item-text">Find the value of the product moment correlation coefficient between</span><span class="inline-marks">[2]</span></div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b-2/item-b-i"><div class="list-item-content"><span class="list-item-text">$x$ and $y$,</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b-2/item-b-ii"><div class="list-item-content"><span class="list-item-text">$x^2$ and $y$.</span></div></li></ol></li><li data-list-item-id="item-c-2" data-list-item-path="item-c-2"><div class="list-item-paragraph">Using your answers in parts <strong>(a)</strong> and <strong>(b)</strong>, explain which of $y = a + bx^2$ and $y = c + dx$ is the better model and find the equation of a suitable regression line for this model.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d-2" data-list-item-path="item-d-2"><div class="list-item-content"><span class="list-item-text">Use the equation of your regression line to estimate the electricity consumption on a day with an average temperature of $37{}^\circ$C. Comment on the reliability of your estimate.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>In some regions, temperature is measured in degrees Fahrenheit instead of degrees Celsius. A temperature of $C$ degrees Celsius is equivalent to $F$ degrees Fahrenheit, where&nbsp;$F = \frac{9}{5}C + 32$.</p>
<ol class="abc-list" start="5"><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Rewrite the equation of your regression line found in part <strong>(c)</strong> in terms of $y$ and $F$, where&nbsp;$F$ is the average daily temperature in degrees Fahrenheit.<span class="inline-marks">[1]</span></div></li></ol>

An energy company is studying the relationship between the average daily temperature, $x$ (in degrees Celsius), and the amount of electricity consumed, $y$ (in megawatt-hours MWh). For a random sample of ten days, the records are shown in the table below.

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

It is thought that the electricity consumption and average daily temperature can be modelled by one of the formulae\[y = a + bx^2, \quad y = c + dx,\]where $a$, $b$, $c$ and $d$ are constants.

Find the value of the product moment correlation coefficient between

Use the equation of your regression line to estimate the electricity consumption on a day with an average temperature of $37{}^\circ$C. Comment on the reliability of your estimate.

In some regions, temperature is measured in degrees Fahrenheit instead of degrees Celsius. A temperature of $C$ degrees Celsius is equivalent to $F$ degrees Fahrenheit, where $F = \frac{9}{5}C + 32$.

Transforming Regression Equations for New Units

2025 VJC P2 Q11

<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 data-list-item-id="item-a" data-list-item-path="item-a"><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 data-list-item-id="item-b" data-list-item-path="item-b"><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 data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-content"><span class="list-item-text">$y$ and $x$,</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-content"><span class="list-item-text">$\ln y$ and $x$.</span></div></li></ol></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Use your answers to parts <strong>(a)</strong> and <strong>(b)</strong> to explain which of $y = a + bx$ and $\ln y = c + dx$ is the better model.<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 data-list-item-id="item-d" data-list-item-path="item-d"><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 data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Without the use of a graphing calculator, re-write your equation from part <strong>(d)</strong> so that it can be used to estimate the resale value when the age is given in months.<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

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.

2025 TMJC P2 Q9

<p>As part of the sound calibration for the National Day Parade at the Padang, sound engineers test how sound intensity changes with distance from the main stage speakers during rehearsals. This is helpful for calibrating the speaker system to avoid sound distortion or echo across seating zones.</p>
<p>At a particular rehearsal, the engineers record the sound intensity, $y$, measured in decibels, at various distances, $x$, measured in metres from the stage speakers and recorded the data in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 350px;"><colgroup><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$0$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$4$</th><th style="text-align: center; --cell-justify: center">$6$</th><th style="text-align: center; --cell-justify: center">$8$</th><th style="text-align: center; --cell-justify: center">$10$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$</td><td style="text-align: center; --cell-justify: center">$100$</td><td style="text-align: center; --cell-justify: center">$96$</td><td style="text-align: center; --cell-justify: center">$84$</td><td style="text-align: center; --cell-justify: center">$65$</td><td style="text-align: center; --cell-justify: center">$36$</td><td style="text-align: center; --cell-justify: center">$1$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph"><p>Sketch a scatter diagram for these values. Explain, using your diagram, which of the following equations provides the most accurate model of the relationship between $x$ and $y$.</p></div><div class="list-item-paragraph" style="margin-left: 2.5rem"><p>$(A) \quad y = a - bx^2,$</p><p>$(B) \quad y = a + be^{-x},$</p><p>$(C) \quad y = a + b \ln(10 - x),$</p></div><div class="list-item-paragraph"><p>where&nbsp;$a$ and $b$ are constants such that&nbsp;$a &gt; 0$ and $b &gt; 0$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Using the model you chose in part <strong>(a)</strong>, find the equation of the least squares regression line. State the corresponding value of the product moment correlation coefficient, correct to $5$ decimal places.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">At a particular spectator seat in the Padang, the sound intensity was recorded to be $20$ decibels. Estimate the distance of this spectator seat from the stage speakers. Explain whether your estimate is reliable.<span class="inline-marks">[2]</span></div></li></ol>
<p>As another part of sound calibration for the National Day Parade, engineers also measured the time that it takes for sound to reach the spectator stands, $t$, measured in milliseconds, at various distances, $x$, measured in metres from the stage speakers.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 350px;"><colgroup><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"><col style="width: 14.2857%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$10$</th><th style="text-align: center; --cell-justify: center">$20$</th><th style="text-align: center; --cell-justify: center">$30$</th><th style="text-align: center; --cell-justify: center">$40$</th><th style="text-align: center; --cell-justify: center">$50$</th><th style="text-align: center; --cell-justify: center">$60$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$t$</td><td style="text-align: center; --cell-justify: center">$29.1$</td><td style="text-align: center; --cell-justify: center">$58.3$</td><td style="text-align: center; --cell-justify: center">$87.5$</td><td style="text-align: center; --cell-justify: center">$116.6$</td><td style="text-align: center; --cell-justify: center">$145.8$</td><td style="text-align: center; --cell-justify: center">$174.9$</td></tr></tbody></table></div></div>
<ol class="abc-list" start="4"><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">The engineers would like to investigate if the model $t = 2.916x$ is a good fit of the above data. For a model $t = \mathrm{f}(x)$, the residual for a point $(p, q)$ is $q - \mathrm{f}(p)$.</div><ol class="i-list"><li data-list-item-id="item-d-i" data-list-item-path="item-d/item-d-i"><div class="list-item-paragraph">Explain why they should use the sum of the squares of the residuals rather than the sum of the residuals when assessing the fit of the model.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-d-ii" data-list-item-path="item-d/item-d-ii"><div class="list-item-paragraph">Calculate the sum of the squares of the residuals for the line $t = 2.916x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d-iii" data-list-item-path="item-d/item-d-iii"><div class="list-item-paragraph">It is further found that another line $t = 2.916x + D$ where&nbsp;$D \neq 0$ is a better fit for the above data. Explain how the sum of the squares of the residuals for the line $t = 2.916x + D$ differs from the sum of the squares of the residuals of $t = 2.916x$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

As part of the sound calibration for the National Day Parade at the Padang, sound engineers test how sound intensity changes with distance from the main stage speakers during rehearsals. This is helpful for calibrating the speaker system to avoid sound distortion or echo across seating zones.

At a particular rehearsal, the engineers record the sound intensity, $y$, measured in decibels, at various distances, $x$, measured in metres from the stage speakers and recorded the data in the table below.

At a particular spectator seat in the Padang, the sound intensity was recorded to be $20$ decibels. Estimate the distance of this spectator seat from the stage speakers. Explain whether your estimate is reliable.

As another part of sound calibration for the National Day Parade, engineers also measured the time that it takes for sound to reach the spectator stands, $t$, measured in milliseconds, at various distances, $x$, measured in metres from the stage speakers.

The engineers would like to investigate if the model $t = 2.916x$ is a good fit of the above data. For a model $t = \mathrm{f}(x)$, the residual for a point $(p, q)$ is $q - \mathrm{f}(p)$.

2025 RVHS P2 Q10

<p>A small cafe sells one type of coffee. The selling price of a cup of coffee is reviewed and adjusted at the beginning of every year depending on market conditions. Based on sales figures collected over $7$ years, the cafe owner, Mr Tay, studied the effect of the selling price of a cup of coffee, , on the average number of cups, $y$ cups, sold per day within the year. The data is shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$2.0$</th><th style="text-align: center; --cell-justify: center">$2.2$</th><th style="text-align: center; --cell-justify: center">$2.5$</th><th style="text-align: center; --cell-justify: center">$3.0$</th><th style="text-align: center; --cell-justify: center">$4.0$</th><th style="text-align: center; --cell-justify: center">$4.4$</th><th style="text-align: center; --cell-justify: center">$4.5$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$</td><td style="text-align: center; --cell-justify: center">$280$</td><td style="text-align: center; --cell-justify: center">$250$</td><td style="text-align: center; --cell-justify: center">$190$</td><td style="text-align: center; --cell-justify: center">$150$</td><td style="text-align: center; --cell-justify: center">$90$</td><td style="text-align: center; --cell-justify: center">$100$</td><td style="text-align: center; --cell-justify: center">$70$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a-3" data-list-item-path="item-a-3"><div class="list-item-content"><span class="list-item-text">On the grid in the Printed Answer Booklet, draw a scatter diagram of the data.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Mr Tay realised that one of the values of $y$ has been wrongly stated in the data table.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b-3" data-list-item-path="item-b-3"><div class="list-item-content"><span class="list-item-text">Indicate the corresponding point on your diagram by labelling it $P$. Explain why the scatter diagram for the remaining points may be consistent with a model of the form $y=a+\frac{b}{x}$, where&nbsp;$a$ and $b$ are constants.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c-3" data-list-item-path="item-c-3"><div class="list-item-content"><span class="list-item-text">Omitting $P$, calculate the product moment correlation coefficient and the least squares estimates of $a$ and $b$ for the model $y=a+\frac{b}{x}$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d-3" data-list-item-path="item-d-3"><div class="list-item-paragraph">Use the model $y=a+\frac{b}{x}$ with the values of $a$ and $b$ found in part <strong>(c) </strong>to estimate the value of $y$ that has been wrongly stated in the data table. Give two reasons why you would expect this estimate to be reliable.<span class="inline-marks">[3]</span></div></li></ol>

A small cafe sells one type of coffee. The selling price of a cup of coffee is reviewed and adjusted at the beginning of every year depending on market conditions. Based on sales figures collected over $7$ years, the cafe owner, Mr Tay, studied the effect of the selling price of a cup of coffee, , on the average number of cups, $y$ cups, sold per day within the year. The data is shown in the table below.

On the grid in the Printed Answer Booklet, draw a scatter diagram of the data.

Mr Tay realised that one of the values of $y$ has been wrongly stated in the data table.

Indicate the corresponding point on your diagram by labelling it $P$. Explain why the scatter diagram for the remaining points may be consistent with a model of the form $y=a+\frac{b}{x}$, where $a$ and $b$ are constants.

Omitting $P$, calculate the product moment correlation coefficient and the least squares estimates of $a$ and $b$ for the model $y=a+\frac{b}{x}$.

2025 RI P2 Q8

<p>The director of an art gallery wishes to find out the relationship between the time spent, $t$ minutes, in the art gallery and the age, $a$ (in years), of the visitors. Ten visitors were randomly selected and the data are summarised as follows:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 580px;"><colgroup><col style="width: 13.7931%;"><col style="width: 8.6207%;"><col style="width: 8.6207%;"><col style="width: 8.6207%;"><col style="width: 8.6207%;"><col style="width: 8.6207%;"><col style="width: 8.6207%;"><col style="width: 8.6207%;"><col style="width: 8.6207%;"><col style="width: 8.6207%;"><col style="width: 8.6207%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$t$ (minutes)</th><th style="text-align: center; --cell-justify: center">$20$</th><th style="text-align: center; --cell-justify: center">$70$</th><th style="text-align: center; --cell-justify: center">$56$</th><th style="text-align: center; --cell-justify: center">$75$</th><th style="text-align: center; --cell-justify: center">$k$</th><th style="text-align: center; --cell-justify: center">$42$</th><th style="text-align: center; --cell-justify: center">$50$</th><th style="text-align: center; --cell-justify: center">$52$</th><th style="text-align: center; --cell-justify: center">$60$</th><th style="text-align: center; --cell-justify: center">$60$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$a$ (years)</td><td style="text-align: center; --cell-justify: center">$21$</td><td style="text-align: center; --cell-justify: center">$56$</td><td style="text-align: center; --cell-justify: center">$42$</td><td style="text-align: center; --cell-justify: center">$77$</td><td style="text-align: center; --cell-justify: center">$22$</td><td style="text-align: center; --cell-justify: center">$34$</td><td style="text-align: center; --cell-justify: center">$40$</td><td style="text-align: center; --cell-justify: center">$41$</td><td style="text-align: center; --cell-justify: center">$48$</td><td style="text-align: center; --cell-justify: center">$47$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-6a" data-list-item-path="item-6a"><div class="list-item-paragraph">A linear model is proposed for the relationship between $t$ and $a$. Given that the least squares regression line of $t$ on $a$ is \[t = 0.8957914a + 14.16013,\] show that&nbsp;$k = 40$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-6b" data-list-item-path="item-6b"><div class="list-item-paragraph">Sketch a scatter diagram of $t$ against $a$ for the data given in the table and draw the line given in <strong>(a)</strong> on the same diagram.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-6c" data-list-item-path="item-6c"><div class="list-item-paragraph">Use the least squares regression line of $t$ on $a$ in <strong>(a)</strong> to estimate the time spent in the art gallery by a $60$-year old visitor.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-6d" data-list-item-path="item-6d"><div class="list-item-paragraph">A regression line $t = ma + c$ where&nbsp;$m$ and $c$ are constants is used to model the above set of data. The residual for a data point $(a_i, t_i)$ where&nbsp;$i = 1, 2, \dots, 10$ which is plotted on a scatter diagram is defined to be the number \[t_i - (ma_i + c).\]The sum of the squares of these residuals, denoted by $S$, is given by\[S = \sum_{i=1}^{10} [t_i - (ma_i + c)]^2.\]Find an inequality that is satisfied by $S$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-6e" data-list-item-path="item-6e"><div class="list-item-paragraph">The director wishes to explore the following two models which represent the relationship between $t$ and $a$, where&nbsp;$p$ and $q$ are constants.<br> \[(A) \quad t = p + \frac{q}{a} ,\qquad (B) \quad t = pe^{qa}. \] <br>For each model, find the product moment correlation coefficient and hence explain with a reason which model is more appropriate.<span class="inline-marks">[3]</span></div></li></ol>

The director of an art gallery wishes to find out the relationship between the time spent, $t$ minutes, in the art gallery and the age, $a$ (in years), of the visitors. Ten visitors were randomly selected and the data are summarised as follows:

A linear model is proposed for the relationship between $t$ and $a$. Given that the least squares regression line of $t$ on $a$ is\n\[t = 0.8957914a + 14.16013,\]\nshow that $k = 40$.

Finding Missing Data from Regression Information

2025 NYJC P2 Q10

<p>The Ministry of Health declared "War on Diabetes" in $2016$ to rally a whole-of-nation effort to tackle diabetes. A study was conducted to investigate the effects of diabetes on one’s health, specifically to determine if the blood pressure of an adult aged between $50$ and $60$ years old is dependent on one’s haemoglobin level. Data from six patients in this age group from a hospital was collected. Their haemoglobin level, $s$ mmol/mol, and blood pressure, $p$ mmHg, are shown in the table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 350px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr style="height: 30px;"><th style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$s$</th><th style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$32$</th><th style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$37$</th><th style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$41$</th><th style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$46$</th><th style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$50$</th><th style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$54$</th></tr></thead><tbody><tr style="height: 30px;"><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$p$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$120$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$150$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$168$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$172$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$179$</td><td style="height: 30px; min-height: 30px; text-align: center; --cell-justify: center">$183$</td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-content"><span class="list-item-text">Draw a scatter diagram of these data.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>It is thought that blood pressure $p$ can be modelled by one of the formulae \[p = a + bs^2 \quad \text{or} \quad p = c + \frac{\mathrm{d}}{s}\]where&nbsp;$a$, $b$ and $c$ are positive real numbers and $d$ is a negative real number.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-content"><span class="list-item-text">By using your diagram, explain which of $p = a + bs^2$ or $p = c + \frac{\mathrm{d}}{s}$ is the better model.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Using the model you chose in part <strong>(ii)</strong>, write down the equation for the relationship between $s$ and $p$, giving the numerical values of the coefficients. State the product moment correlation coefficient for this model.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-content"><span class="list-item-text">Estimate the value of $p$ when&nbsp;$s = 45$. Determine whether this estimate is reliable.</span><span class="inline-marks">[3]</span></div></li></ol>

The Ministry of Health declared "War on Diabetes" in $2016$ to rally a whole-of-nation effort to tackle diabetes. A study was conducted to investigate the effects of diabetes on one’s health, specifically to determine if the blood pressure of an adult aged between $50$ and $60$ years old is dependent on one’s haemoglobin level. Data from six patients in this age group from a hospital was collected. Their haemoglobin level, $s$ mmol/mol, and blood pressure, $p$ mmHg, are shown in the table.

It is thought that blood pressure $p$ can be modelled by one of the formulae \[p = a + bs^2 \quad \text{or} \quad p = c + \frac{\mathrm{d}}{s}\]where $a$, $b$ and $c$ are positive real numbers and $d$ is a negative real number.

By using your diagram, explain which of $p = a + bs^2$ or $p = c + \frac{\mathrm{d}}{s}$ is the better model.

Estimate the value of $p$ when $s = 45$. Determine whether this estimate is reliable.

2025 NJC P2 Q8

<ol class="abc-list"><li data-list-item-id="imported-item-block-0-item-0" data-list-item-path="imported-item-block-0-item-0"><div class="list-item-paragraph">Draw separate scatter diagrams, each with $7$ points, all in the first quadrant, which represent the situation where the product moment correlation coefficient between variables $x$ and $y$ is</div><ol class="i-list"><li data-list-item-id="imported-item-block-0-item-0-sub-0" data-list-item-path="imported-item-block-0-item-0/imported-item-block-0-item-0-sub-0"><div class="list-item-paragraph">$0$,<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-0-item-0-sub-1" data-list-item-path="imported-item-block-0-item-0/imported-item-block-0-item-0-sub-1"><div class="list-item-paragraph">approximately $0.8$.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="imported-item-block-0-item-1" data-list-item-path="imported-item-block-0-item-1"><div class="list-item-paragraph">Eight cities in a certain country are linked by rail to the capital city. The table below shows the distance of each city from the capital and the rail price from the city to the capital.</div><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 20.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"><col style="width: 10.0000%;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start">City</th><th style="text-align: center; --cell-justify: center">$A$</th><th style="text-align: center; --cell-justify: center">$B$</th><th style="text-align: center; --cell-justify: center">$C$</th><th style="text-align: center; --cell-justify: center">$D$</th><th style="text-align: center; --cell-justify: center">$E$</th><th style="text-align: center; --cell-justify: center">$F$</th><th style="text-align: center; --cell-justify: center">$G$</th><th style="text-align: center; --cell-justify: center">$H$</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Distance, $d\text{ km}$</td><td style="text-align: center; --cell-justify: center">$124$</td><td style="text-align: center; --cell-justify: center">$44$</td><td style="text-align: center; --cell-justify: center">$76$</td><td style="text-align: center; --cell-justify: center">$148$</td><td style="text-align: center; --cell-justify: center">$16$</td><td style="text-align: center; --cell-justify: center">$180$</td><td style="text-align: center; --cell-justify: center">$104$</td><td style="text-align: center; --cell-justify: center">$195$</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Rail price, $\$p$</td><td style="text-align: center; --cell-justify: center">$156$</td><td style="text-align: center; --cell-justify: center">$53$</td><td style="text-align: center; --cell-justify: center">$99$</td><td style="text-align: center; --cell-justify: center">$169$</td><td style="text-align: center; --cell-justify: center">$23$</td><td style="text-align: center; --cell-justify: center">$177$</td><td style="text-align: center; --cell-justify: center">$138$</td><td style="text-align: center; --cell-justify: center">$178$</td></tr></tbody></table></div><ol class="i-list"><li data-list-item-id="subitem-1786780602637-3kv3z5vly" data-list-item-path="imported-item-block-0-item-1/subitem-1786780602637-3kv3z5vly"><div class="list-item-paragraph"><p>Draw a scatter diagram of the data.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1786780605525-to8v571uo" data-list-item-path="imported-item-block-0-item-1/subitem-1786780605525-to8v571uo"><div class="list-item-paragraph"><p>For a line of best fit $p = \mathrm{f}(d)$, the residual for a point $(a, b)$ plotted on the scatter diagram is the vertical distances between $(a, \mathrm{f}(a))$ and $(a, b)$. Explain why, in general, the sum of the squares of the residuals rather than the sum of the residuals is used.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1786780605926-sdi2g5tll" data-list-item-path="imported-item-block-0-item-1/subitem-1786780605926-sdi2g5tll"><div class="list-item-paragraph"><p>Using the scatter diagram in <strong>(i)</strong>, explain if a linear relationship between the distance and the rail price is appropriate.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1786780606344-4g4f1r1ag" data-list-item-path="imported-item-block-0-item-1/subitem-1786780606344-4g4f1r1ag"><div class="list-item-paragraph"><p>Suppose that the relationship between $d$ and $p$ are modelled by an equation of the form $p = s + t \ln d$, where&nbsp;$s$ and $t$ are constants. Find the equation of this regression line and use it to estimate the rail price for a city which is $210\text{ km}$ from the capital.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1786780607007-vrbxkm8vh" data-list-item-path="imported-item-block-0-item-1/subitem-1786780607007-vrbxkm8vh"><div class="list-item-paragraph"><p>Comment on the reliability of the estimate in <strong>(iv)</strong>.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1786780607467-s3tb6tph0" data-list-item-path="imported-item-block-0-item-1/subitem-1786780607467-s3tb6tph0"><div class="list-item-paragraph"><p>Following an adjustment to the rail prices, all rail prices are increased by $10$ dollars. Without any further calculations, state any change you would expect in the values of the constants $s$ and $t$ found in <strong>(iv)</strong>.</p><span class="inline-marks">[2]</span></div></li></ol></li></ol>

Draw separate scatter diagrams, each with $7$ points, all in the first quadrant, which represent the situation where the product moment correlation coefficient between variables $x$ and $y$ is

2025 JPJC P2 Q11

<p>A study was conducted on a group of pre-diabetic seniors to examine whether the number of hours of exercise per week ($t$) has any effect on their fasting blood sugar levels ($S$, measured in mg/dL). The results were as follows.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 450px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$t$</th><th style="text-align: center; --cell-justify: center">$1.5$</th><th style="text-align: center; --cell-justify: center">$1.8$</th><th style="text-align: center; --cell-justify: center">$2.1$</th><th style="text-align: center; --cell-justify: center">$2.4$</th><th style="text-align: center; --cell-justify: center">$2.5$</th><th style="text-align: center; --cell-justify: center">$3.1$</th><th style="text-align: center; --cell-justify: center">$3.5$</th><th style="text-align: center; --cell-justify: center">$4.2$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$S$</td><td style="text-align: center; --cell-justify: center">$119$</td><td style="text-align: center; --cell-justify: center">$108$</td><td style="text-align: center; --cell-justify: center">$100$</td><td style="text-align: center; --cell-justify: center">$95$</td><td style="text-align: center; --cell-justify: center">$93$</td><td style="text-align: center; --cell-justify: center">$88$</td><td style="text-align: center; --cell-justify: center">$86$</td><td style="text-align: center; --cell-justify: center">$85$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Draw the scatter diagram for these values, labelling the axes.</span><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Comment on whether a linear model would be appropriate, referring both to the scatter diagram and the context of the question.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">By considering the shape of the following graphs, explain which model is the most appropriate for the data collected, where&nbsp;$a$ and $b$ are positive constants.</span><span class="inline-marks">[1]</span></div></li></ol>
<p style="text-align: center">$\mathbf{A}: S=a+bt^2, \quad \mathbf{B}: S=a+\frac{b}{t}, \quad \mathbf{C}: S=a+b \ln t$</p>
<p>Using the appropriate model selected above, find</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-content"><span class="list-item-text">the value of the product moment correlation coefficient,</span><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-content"><span class="list-item-text">the equation of the regression line of $S$ on $t$, and hence find an estimate of the exercise time required for the fasting blood sugar to be $99$ mg/dL. Comment on the reliability of your answer.</span><span class="inline-marks">[3]</span></div></li></ol>

A study was conducted on a group of pre-diabetic seniors to examine whether the number of hours of exercise per week ($t$) has any effect on their fasting blood sugar levels ($S$, measured in mg/dL). The results were as follows.

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

Comment on whether a linear model would be appropriate, referring both to the scatter diagram and the context of the question.

By considering the shape of the following graphs, explain which model is the most appropriate for the data collected, where $a$ and $b$ are positive constants.

$\mathbf{A}: S=a+bt^2, \quad \mathbf{B}: S=a+\frac{b}{t}, \quad \mathbf{C}: S=a+b \ln t$

Using the appropriate model selected above, find

the value of the product moment correlation coefficient,

the equation of the regression line of $S$ on $t$, and hence find an estimate of the exercise time required for the fasting blood sugar to be $99$ mg/dL. Comment on the reliability of your answer.

2025 ASRJC P2 Q8

<p>Scientists are studying the growth of a particular species of bamboo. They want to understand how the height of the bamboo stalk changes over time. The table below shows the height, $h$ inches, of a bamboo stalk at age, $t$ weeks, after sprouting.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 450px;"><colgroup><col style="width: 11.1111%;"><col style="width: 11.1111%;"><col style="width: 11.1111%;"><col style="width: 11.1111%;"><col style="width: 11.1111%;"><col style="width: 11.1111%;"><col style="width: 11.1111%;"><col style="width: 11.1111%;"><col style="width: 11.1111%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$t$</th><th style="text-align: center; --cell-justify: center">$1$</th><th style="text-align: center; --cell-justify: center">$2$</th><th style="text-align: center; --cell-justify: center">$3$</th><th style="text-align: center; --cell-justify: center">$4$</th><th style="text-align: center; --cell-justify: center">$5$</th><th style="text-align: center; --cell-justify: center">$7$</th><th style="text-align: center; --cell-justify: center">$8$</th><th style="text-align: center; --cell-justify: center">$9$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$h$</td><td style="text-align: center; --cell-justify: center">$3.6$</td><td style="text-align: center; --cell-justify: center">$7.4$</td><td style="text-align: center; --cell-justify: center">$10.2$</td><td style="text-align: center; --cell-justify: center">$11.7$</td><td style="text-align: center; --cell-justify: center">$12.6$</td><td style="text-align: center; --cell-justify: center">$14.1$</td><td style="text-align: center; --cell-justify: center">$14.5$</td><td style="text-align: center; --cell-justify: center">$15.5$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Draw a scatter diagram of these data, and explain using the diagram why the relationship between $h$ and $t$ is not well-modelled by an equation of the form $h=a+bt$ or $h=a+bt^2$, where&nbsp;$b&gt;0$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph"><p>Consider the two models below:</p></div><div class="list-item-paragraph" style="margin-left: 11rem"><p style="text-align: left;">Model $A$: $h=c+d\sqrt{t}$,</p><p style="text-align: left;">Model $B$: $h=c+d\ln t$,</p></div><div class="list-item-after-content"><p>where&nbsp;$c$ and $d$ are constants. Explain which of Model $A$ or Model $B$ is a better fit to the data. State the equation of the least squares regression line in this case, giving your answer to $3$ decimal places.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph"><p>Use the equation of your regression line in<strong> (b)</strong> to estimate the length of a $6$-week-old bamboo stalk. Comment on the reliability of this estimate.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Comment on the suitability of using this model in the long run.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph"><p>Let&nbsp;$H$ denote the height of the bamboo stalk calculated for each value of $t$ in the table using the least squares regression line obtained in <strong>(b)</strong>. Find the value of $S$, where&nbsp;$S=\sum(h-H)^2$ giving your answer to $2$ decimal places.</p><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-f" data-list-item-path="item-f"><div class="list-item-paragraph">For each of the eight sample values of $t$, $H'$ is given by $H'=p+q\ln t$, where&nbsp;$p$ and $q$ are constants. Would $\sum(h-H')^2$ be greater or less than $S$ and why?<span class="inline-marks">[1]</span></div></li></ol>

Scientists are studying the growth of a particular species of bamboo. They want to understand how the height of the bamboo stalk changes over time. The table below shows the height, $h$ inches, of a bamboo stalk at age, $t$ weeks, after sprouting.

Draw a scatter diagram of these data, and explain using the diagram why the relationship between $h$ and $t$ is not well-modelled by an equation of the form $h=a+bt$ or $h=a+bt^2$, where $b>0$.

Comment on the suitability of using this model in the long run.

For each of the eight sample values of $t$, $H'$ is given by $H'=p+q\ln t$, where $p$ and $q$ are constants. Would $\sum(h-H')^2$ be greater or less than $S$ and why?

2025 ACJC P2 Q10 

<p>The following table shows the weight ($w$) of a foetus, in grams, taken at various weeks ($t$).</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 320px;"><colgroup><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"><col style="width: 40px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$t$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$12$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$16$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$20$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$24$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$28$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$32$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$36$</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$w$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$15$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$100$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$300$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$600$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1005$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1700$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$2622$</p></td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-1767014745226-drj420vsn" data-list-item-path="item-1767014745226-drj420vsn"><div class="list-item-paragraph">Draw a scatter diagram to illustrate the data, labelling the axes clearly.         <span class="inline-marks">[2]</span></div></li></ol>
<p>It is thought that the weight of the foetus at week $t$ can be modelled by one of the formulae</p>
<p style="text-align: center">$w=a+b\ln t$	or	$w=c{{t}^{\text{}\hspace{0.5em}d}}$,</p>
<p>where&nbsp;$a$, $b$, $c$ and $d$ are positive constants.     </p>
<ol class="i-list" start="2"><li data-list-item-id="item-1767014775616-59uxsn40o" data-list-item-path="item-1767014775616-59uxsn40o"><div class="list-item-paragraph">Using your scatter diagram, explain which of $w=a+b\ln t$ or $w=c{{t}^{\text{}\hspace{0.5em}d}}$ is the better model.	<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1767014788552-hyl4t74d1" data-list-item-path="item-1767014788552-hyl4t74d1"><div class="list-item-paragraph">It is required to estimate the value of $w$ for which $t=30$. Find the equation of a suitable regression line, and use it to find the required estimate, giving your answer to the nearest grams.	<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1767014798956-pet6la0br" data-list-item-path="item-1767014798956-pet6la0br"><div class="list-item-paragraph">It is also required to estimate the value of $t$ for which $w=2000$. Explain whether the regression line of $t$ on $w$ or $\ln t$ on $\ln w$ could be used to find this estimate.  	<span class="inline-marks">[1]</span></div></li></ol>
<p></p>

The following table shows the weight ($w$) of a foetus, in grams, taken at various weeks ($t$).

It is thought that the weight of the foetus at week $t$ can be modelled by one of the formulae

where $a$, $b$, $c$ and $d$ are positive constants.     

2015 RI P2 Q7

Hypothesis Testing

<p>A weight is suspended in mid-air by means of a string of length $x$ cm. In an experiment, the weight was displaced slightly and the time, $y$ s, taken to complete $10$ oscillations was recorded. The results are shown in the table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 350px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$x$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$10$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$30$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$50$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$100$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$150$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$200$</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$y$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$7.05$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$9.85$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$15.53$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$18.86$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$25.91$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$28.31$</p></td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-1767014126919-u6eeyguga" data-list-item-path="item-1767014126919-u6eeyguga"><div class="list-item-paragraph">Draw a scatter diagram to illustrate the data.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is thought that the time taken $y$ can be modelled by one of the formulae $y=a+bx$ or $y=c+d\sqrt{x}$ where&nbsp;$a$, $b$, $c$ and $d$ are constants.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1767014150110-tbnzws5bi" data-list-item-path="item-1767014150110-tbnzws5bi"><div class="list-item-paragraph">Find the value of the product moment correlation coefficient between</div><ol class="abc-list"><li data-list-item-id="subitem-1767014154692-w16oish9q" data-list-item-path="item-1767014150110-tbnzws5bi/subitem-1767014154692-w16oish9q"><div class="list-item-paragraph">$y$ and $x$,</div></li><li data-list-item-id="subitem-1767014162372-w4zzdydsc" data-list-item-path="item-1767014150110-tbnzws5bi/subitem-1767014162372-w4zzdydsc"><div class="list-item-paragraph">$y$ and $\sqrt{x}$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1767014184062-itpvqa4wh" data-list-item-path="item-1767014184062-itpvqa4wh"><div class="list-item-paragraph">Use your answers to <strong>(i)</strong> and <strong>(ii)</strong> to explain which formula is the better model.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1767014203651-c7yva1u4g" data-list-item-path="item-1767014203651-c7yva1u4g"><div class="list-item-paragraph">Find the equation of a suitable regression line and use it to estimate the value of $y$ when&nbsp;$x=45$. Comment on the reliability of the estimate.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1767014211422-exfh0vhuz" data-list-item-path="item-1767014211422-exfh0vhuz"><div class="list-item-paragraph">The addition of a data point will not change the equation of the regression line found in <strong>(iv)</strong>. State the coordinates of such a point.<span class="inline-marks">[1]</span></div></li></ol>

A weight is suspended in mid-air by means of a string of length $x$ cm. In an experiment, the weight was displaced slightly and the time, $y$ s, taken to complete $10$ oscillations was recorded. The results are shown in the table.

It is thought that the time taken $y$ can be modelled by one of the formulae $y=a+bx$ or $y=c+d\sqrt{x}$ where $a$, $b$, $c$ and $d$ are constants.

2015 YIJC P2 Q10

<p style="margin-bottom: 1rem">A medical officer wishes to investigate a patient’s walking $s$ speed km/h  and his heart-beat rate $h$ beats per minute (bpm). The data is shown below:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 550px;"><colgroup><col style="width: 18.1818%;"><col style="width: 9.0909%;"><col style="width: 9.0909%;"><col style="width: 9.0909%;"><col style="width: 9.0909%;"><col style="width: 9.0909%;"><col style="width: 9.0909%;"><col style="width: 9.0909%;"><col style="width: 9.0909%;"><col style="width: 9.0909%;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$s$ (km/h)</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$1.5$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$2$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$2.5$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$3$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$3.5$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$4$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$4.5$</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$5$</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$h$ (bpm)</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$60$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$63$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$66$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$75$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$86$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$99$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$150$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$110$</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>$130$</p></td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-1767002821691-53dx8a1qv" data-list-item-path="item-1767002821691-53dx8a1qv"><div class="list-item-paragraph">Sketch a scatter plot of the above data.	<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1767002837508-rs6m37n9z" data-list-item-path="item-1767002837508-rs6m37n9z"><div class="list-item-paragraph">One of the values of $h$ appears to be incorrect. Indicate the corresponding point on your diagram by labelling it $P$.<span class="inline-marks">[1]</span></div></li></ol>
<p>Omit $P$ for the remainder of this question.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-1767002856971-w8ze67gun" data-list-item-path="item-1767002856971-w8ze67gun"><div class="list-item-paragraph">Calculate the product moment correlation coefficient for this set of data. Use the equation of an appropriate regression line to predict the value of $s$ when&nbsp;$h=100$, justifying your choice of regression line. <span class="inline-marks">[4]</span></div></li></ol>
<p>It is suggested to use one of the following two models instead:</p>
<p style="text-align: center">Model (I): $h=a+b{{s}^{2}}$, </p>
<p style="text-align: center">Model (II): $h=a+b{{e}^{s}}$,</p>
<p>where&nbsp;$a$ and $b$ are real constants. </p>
<ol class="i-list" start="4"><li data-list-item-id="item-1767002902916-rt1qk7ei2" data-list-item-path="item-1767002902916-rt1qk7ei2"><div class="list-item-paragraph">Determine which of the two models is a better choice, giving a reason for your answer.	<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1767002903538-38a3l60au" data-list-item-path="item-1767002903538-38a3l60au"><div class="list-item-paragraph">Suppose a new data pair $\left( \overline{s},\overline{h} \right)$ is added to the table above, where&nbsp;$\overline{s}$ and $\overline{h}$  are the patient’s sample mean walking speed (in km/h) and his sample mean heart-beat rate (in bpm) respectively, based on the data above. Without any calculations, explain whether the equation of the regression line you have obtained in part <strong>(iii) </strong> would change.<span class="inline-marks">[2]</span></div></li></ol>

A medical officer wishes to investigate a patient’s walking $s$ speed km/h  and his heart-beat rate $h$ beats per minute (bpm). The data is shown below:

Omit $P$ for the remainder of this question.

It is suggested to use one of the following two models instead:

2015 NJC P2 Q12

<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 550px;"><colgroup><col style="width: 100px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">Test 1 mark, $x$</th><th style="text-align: center; --cell-justify: center">$17$</th><th style="text-align: center; --cell-justify: center">$19$</th><th style="text-align: center; --cell-justify: center">$10$</th><th style="text-align: center; --cell-justify: center">$25$</th><th style="text-align: center; --cell-justify: center">$28$</th><th style="text-align: center; --cell-justify: center">$30$</th><th style="text-align: center; --cell-justify: center">$22$</th><th style="text-align: center; --cell-justify: center">$27$</th><th style="text-align: center; --cell-justify: center">$12$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Test 2 mark, $y$</td><td style="text-align: center; --cell-justify: center">$18$</td><td style="text-align: center; --cell-justify: center">$19$</td><td style="text-align: center; --cell-justify: center">$12$</td><td style="text-align: center; --cell-justify: center">$23$</td><td style="text-align: center; --cell-justify: center">$27$</td><td style="text-align: center; --cell-justify: center">$26$</td><td style="text-align: center; --cell-justify: center">$19$</td><td style="text-align: center; --cell-justify: center">$30$</td><td style="text-align: center; --cell-justify: center">$9$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Find the equation of the regression line of</div><ol class="abc-list"><li><div class="list-item-paragraph">$y$ on $x$</div></li><li><div class="list-item-paragraph">$x$ on $y$</div></li></ol></li><li><div class="list-item-paragraph">Tarzan scored $23$ in test $2$ but missed test $1$. Use the appropriate line to estimate what his mark for test 1 should be. Is this estimated value reliable? Why?</div></li></ol>

Reliability of Estimates

<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 490px;"><colgroup><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"><col style="width: 70px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$23$</th><th style="text-align: center; --cell-justify: center">$51$</th><th style="text-align: center; --cell-justify: center">$61$</th><th style="text-align: center; --cell-justify: center">$79$</th><th style="text-align: center; --cell-justify: center">$91$</th><th style="text-align: center; --cell-justify: center">$103$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$</td><td style="text-align: center; --cell-justify: center">$104$</td><td style="text-align: center; --cell-justify: center">$92$</td><td style="text-align: center; --cell-justify: center">$53$</td><td style="text-align: center; --cell-justify: center">$45$</td><td style="text-align: center; --cell-justify: center">$11$</td><td style="text-align: center; --cell-justify: center">$1$</td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-1763365784529-3k5uvtq5c" data-list-item-path="item-1763365784529-3k5uvtq5c"><div class="list-item-paragraph">Find the equations of the regression lines of $y$ on $x$ and $x$ on $y$, and sketch both lines in a single scatter diagram.</div></li></ol>
<p>Suppose the variables $x$ and $y$ are such that neither depends on the other.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-1782993755566-xdbv5t2zb" data-list-item-path="item-1782993755566-xdbv5t2zb"><div class="list-item-paragraph">Using a suitable regression line, estimate the value of</div><ol class="abc-list"><li data-list-item-id="item-1782993885281-jdh5rrlkl" data-list-item-path="item-1782993885281-jdh5rrlkl"><div class="list-item-paragraph">$y$, when&nbsp;$x=81$,</div></li><li data-list-item-id="item-1782993891688-x53dyfoss" data-list-item-path="item-1782993891688-x53dyfoss"><div class="list-item-paragraph">$x$, when&nbsp;$y=27$.</div></li></ol></li></ol>
<p>Justify your choice of regression line for each of parts <strong>(ii)(a)</strong> and <strong>(ii)(b)</strong>.</p>

Suppose the variables $x$ and $y$ are such that neither depends on the other.

Justify your choice of regression line for each of parts **(ii)(a)** and **(ii)(b)**.

Finding The Equations of The Regression Lines

<ol class="abc-list"><li data-list-item-id="item-1760687362296-pd7ssmaff" data-list-item-path="item-1760687362296-pd7ssmaff"><div class="list-item-paragraph">Sketch a scatter diagram that might be expected when&nbsp;$x$ and $y$ are related approximately as given in each of the cases (A) and (B) below. In each case, your diagram should include $6$ points, approximately equally spaced with respective to $x$, and with all $x$- and $y$-values positive. The letters $a$, $b$, $c$ and $d$ represent constants.</div><ol class="ABC-list"><li data-list-item-id="item-1783357164726-6lu7wwqq9" data-list-item-path="item-1783357164726-6lu7wwqq9"><div class="list-item-paragraph">$y=a+b\sqrt{x}$, where&nbsp;$a$ is positive and $b$ is negative</div></li><li data-list-item-id="item-1783357186334-rn13b3ffd" data-list-item-path="item-1783357186334-rn13b3ffd"><div class="list-item-paragraph">$y=c+\frac{\mathrm{d}}{x}$, where&nbsp;$c$ is positive and $d$ is negative.</div></li></ol></li><li data-list-item-id="item-1760687363001-2qb82dpah" data-list-item-path="item-1760687363001-2qb82dpah"><div class="list-item-paragraph">The data below shows the diameter $h$, in cm, of coconuts of $9$ different ages, in $k$ months.</div><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start">Age in month ($k$)</th><th style="text-align: center; --cell-justify: center">1</th><th style="text-align: center; --cell-justify: center">2</th><th style="text-align: center; --cell-justify: center">3.5</th><th style="text-align: center; --cell-justify: center">4.5</th><th style="text-align: center; --cell-justify: center">5</th><th style="text-align: center; --cell-justify: center">7.5</th><th style="text-align: center; --cell-justify: center">9</th><th style="text-align: center; --cell-justify: center">10.5</th><th style="text-align: left; --cell-justify: flex-start">12</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Diameter in cm ($h$)</td><td style="text-align: center; --cell-justify: center">6</td><td style="text-align: center; --cell-justify: center">9.5</td><td style="text-align: center; --cell-justify: center">13</td><td style="text-align: center; --cell-justify: center">15</td><td style="text-align: center; --cell-justify: center">16.5</td><td style="text-align: center; --cell-justify: center">17.5</td><td style="text-align: center; --cell-justify: center">18.5</td><td style="text-align: center; --cell-justify: center">19</td><td style="text-align: left; --cell-justify: flex-start">19.5</td></tr></tbody></table></div><ol class="i-list"><li data-list-item-id="subitem-1760687410686-fbacsilh9" data-list-item-path="item-1760687363001-2qb82dpah/subitem-1760687410686-fbacsilh9"><div class="list-item-paragraph">Calculate the value of the product moment correlation coefficient $r$ between $k$ and $h$. State, with a reason, whether the value of $r$ would be different if the diameter is measured in inches instead.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760687411519-sli6zy4ir" data-list-item-path="item-1760687363001-2qb82dpah/subitem-1760687411519-sli6zy4ir"><div class="list-item-paragraph">Give a sketch of the scatter diagram for the data.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1760687413461-ij0v1b3yx" data-list-item-path="item-1760687363001-2qb82dpah/subitem-1760687413461-ij0v1b3yx"><div class="list-item-paragraph">It is desired to predict the diameter of coconuts in the long run. Explain why neither a linear nor a quadratic model is likely to be appropriate.<span class="inline-marks">[2]</span></div><div class="list-item-paragraph">It is suggested that diameter $h$ can be modelled by the formula $h=m+n\ln k$.</div></li><li data-list-item-id="subitem-1760687418434-ov8sebi1r" data-list-item-path="item-1760687363001-2qb82dpah/subitem-1760687418434-ov8sebi1r"><div class="list-item-paragraph">Find the equation of the regression line for the suggested model, and the value of the product moment correlation coefficient between $\ln k$ and $h$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760687418989-tlgjsea7m" data-list-item-path="item-1760687363001-2qb82dpah/subitem-1760687418989-tlgjsea7m"><div class="list-item-paragraph">Estimate, to $1$ decimal place, the age of a coconut with a diameter of $13.7$ cm, and comment on the reliability of your answer.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1760687419544-cbzh0zlw3" data-list-item-path="item-1760687363001-2qb82dpah/subitem-1760687419544-cbzh0zlw3"><div class="list-item-paragraph">Given that&nbsp;$1$ inch $=2.54$cm, state the equation of regression line if the diameter is measured in inches instead of cm.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

Effects of Linear Transformations on Correlation

2023 DHS P2 Q11

<p>The sales manager of a company that sells air conditioning systems presented the data of average daily temperature, $t\left( ^{\circ }\mathrm{C} \right)$ and sales, $s$ (in hundred units) at a meeting.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 550px;"><colgroup><col style="width: 150px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr style="height: 60px;"><th style="height: 60px; min-height: 60px; text-align: left; --cell-justify: flex-start">Average daily temperature, $t$ (${}^\circ\mathrm{C}$)</th><th style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$18$</th><th style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$20$</th><th style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$23$</th><th style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$26$</th><th style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$30$</th><th style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$32$</th><th style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$33$</th><th style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$34$</th></tr></thead><tbody><tr style="height: 60px;"><td style="height: 60px; min-height: 60px; text-align: left; --cell-justify: flex-start">Sales, $s$ (in hundred units)</td><td style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$307$</td><td style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$366$</td><td style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$497$</td><td style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$523$</td><td style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$565$</td><td style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$580$</td><td style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$588$</td><td style="height: 60px; min-height: 60px; text-align: center; --cell-justify: center">$596$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1760685288548-7wqbef6yh" data-list-item-path="item-1760685288548-7wqbef6yh"><div class="list-item-paragraph">Draw a scatter diagram of these data and calculate the value of the product moment correlation coefficient between $s$ and $t$. Comment on whether a linear model would be appropriate, referring both to the scatter diagram and the value of the product moment correlation coefficient found.<span class="inline-marks">[3]</span></div></li></ol>
<p>The marketing director proposes that the data should be modelled instead by the regression equation $s=a\ln t+b$, where&nbsp;$a$ and $b$ are constants.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760685304389-ktwa7ooaj" data-list-item-path="item-1760685304389-ktwa7ooaj"><div class="list-item-paragraph">Find the values of $a$ and $b$, giving your answers to $3$ decimal places.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760685305049-zcc2oddip" data-list-item-path="item-1760685305049-zcc2oddip"><div class="list-item-paragraph">Calculate the product moment correlation coefficient between $s$ and $\ln t$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760685306286-mf6xvtvbo" data-list-item-path="item-1760685306286-mf6xvtvbo"><div class="list-item-paragraph">Using parts <strong>(a)</strong> and <strong>(c)</strong>, explain which is a better model.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760685307922-hyexpd2vk" data-list-item-path="item-1760685307922-hyexpd2vk"><div class="list-item-paragraph">Use the model proposed by the marketing director to estimate the number of units sold when the average daily temperature is ${{38}^{\circ }}\mathrm{C}$ and comment on its reliability.<span class="inline-marks">[2]</span></div></li></ol>

The sales manager of a company that sells air conditioning systems presented the data of average daily temperature, $t\left( ^{\circ }\mathrm{C} \right)$ and sales, $s$ (in hundred units) at a meeting.

The marketing director proposes that the data should be modelled instead by the regression equation $s=a\ln t+b$, where $a$ and $b$ are constants.

2023 CJC P2 Q6

<p>A batch of plants is grown and then studied to examine how well they are growing. For each plant, data on two indicators are collected: the average root hair length $x$ mm, and the amount of potassium found in the roots $y$ mg. The data from a random sample of $8$ plants are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 550px;"><colgroup><col style="width: 150px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>Average root hair</p><p>length</p><p>($x$ mm)</p></th><th style="text-align: center; --cell-justify: center">$3.3$</th><th style="text-align: center; --cell-justify: center">$3.9$</th><th style="text-align: center; --cell-justify: center">$7.5$</th><th style="text-align: center; --cell-justify: center">$8.1$</th><th style="text-align: center; --cell-justify: center">$13.3$</th><th style="text-align: center; --cell-justify: center">$22.1$</th><th style="text-align: center; --cell-justify: center">$32.2$</th><th style="text-align: center; --cell-justify: center">$36.1$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>Amount of</p><p>potassium</p><p>($y$ mg)</p></td><td style="text-align: center; --cell-justify: center">$9$</td><td style="text-align: center; --cell-justify: center">$t$</td><td style="text-align: center; --cell-justify: center">$25$</td><td style="text-align: center; --cell-justify: center">$27$</td><td style="text-align: center; --cell-justify: center">$33$</td><td style="text-align: center; --cell-justify: center">$37$</td><td style="text-align: center; --cell-justify: center">$41$</td><td style="text-align: center; --cell-justify: center">$44$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1760684977906-zb7cr839b" data-list-item-path="item-1760684977906-zb7cr839b"><ol class="i-list"><li data-list-item-id="subitem-1760684979039-asbzxqa2h" data-list-item-path="item-1760684977906-zb7cr839b/subitem-1760684979039-asbzxqa2h"><div class="list-item-paragraph">Calculate an unbiased estimate of the population variance of the average root hair lengths.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1760684981116-ydlypavut" data-list-item-path="item-1760684977906-zb7cr839b/subitem-1760684981116-ydlypavut"><div class="list-item-paragraph">Find an expression for the unbiased estimate of the population variance of the amount of potassium in the roots, in the form</div><div class="list-item-paragraph" style="text-align: center">$\frac{1}{A}\left[ \left( B+{{t}^{2}} \right)-\frac{{{\left( C+t \right)}^{2}}}{D} \right]$,</div><div class="list-item-paragraph">where&nbsp;$A$, $B$, $C$ and $D$ are positive integers to be determined.<span class="inline-marks">[2]</span></div></li></ol></li></ol>
<p>You are now given that&nbsp;$t=17$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760685021729-g1c4o20at" data-list-item-path="item-1760685021729-g1c4o20at"><div class="list-item-paragraph">Sketch a scatter diagram of $y$ against $x$. State the value of the product moment correlation coefficient between $y$ and $x$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760685023679-xhu4or4fu" data-list-item-path="item-1760685023679-xhu4or4fu"><div class="list-item-paragraph">With reference to your answers in part <strong>(b)</strong>, explain whether an equation of the form $y=mx+k$, where&nbsp;$m$ and $k$ are constants, provides an accurate model of the relationship between $x$ and $y$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760685024871-qhilt32ha" data-list-item-path="item-1760685024871-qhilt32ha"><div class="list-item-paragraph">The data is instead modelled by the regression equation $y=p\ln x+q$. Find the values of $p$ and $q$, and the value of the corresponding product moment correlation coefficient.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760685025652-iwd9vofk4" data-list-item-path="item-1760685025652-iwd9vofk4"><div class="list-item-paragraph">Give a contextual interpretation for the value of $p$ found in part <strong>(d)</strong>.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760685027609-8akzuj8dc" data-list-item-path="item-1760685027609-8akzuj8dc"><div class="list-item-paragraph">Hence find the estimated amount of potassium in the roots of a plant with average root hair length $30.0$mm. Explain whether your estimate is reliable.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760685028824-kuknrm5pn" data-list-item-path="item-1760685028824-kuknrm5pn"><div class="list-item-paragraph">It is now required to estimate the average root hair length for a plant with $28$ mg of potassium in its roots. Explain whether it is appropriate to use the equation in part <strong>(d)</strong> to obtain this estimate.<span class="inline-marks">[1]</span></div></li></ol>

A batch of plants is grown and then studied to examine how well they are growing. For each plant, data on two indicators are collected: the average root hair length $x$ mm, and the amount of potassium found in the roots $y$ mg. The data from a random sample of $8$ plants are shown in the table below.

<p>Average root hair</p><p>length</p><p>($x$ mm)</p>

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