A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

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

<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>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 

<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

Transforming Regression Equations for New Units

2023 DHS P2 Q11

<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>

Unbiased Estimates of Population Parameters

2023 EJC P2 Q10

Sampling

<p>An electrical appliance supplier is investigating the potential impact of monthly advertisement expenditure, denoted as&nbsp;$x$ (in thousands of dollars), on the popular social media platform SnapGram, on their monthly sales profits, denoted as&nbsp;$y$ (in thousands of dollars). The supplier has collected the following data over the course of the past year.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 635px;"><colgroup><col style="width: 95px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"><col style="width: 45px;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start">Month</th><th style="text-align: center; --cell-justify: center">Jan</th><th style="text-align: center; --cell-justify: center">Feb</th><th style="text-align: center; --cell-justify: center">Mar</th><th style="text-align: center; --cell-justify: center">Apr</th><th style="text-align: center; --cell-justify: center">May</th><th style="text-align: center; --cell-justify: center">Jun</th><th style="text-align: center; --cell-justify: center">Jul</th><th style="text-align: center; --cell-justify: center">Aug</th><th style="text-align: center; --cell-justify: center">Sep</th><th style="text-align: center; --cell-justify: center">Oct</th><th style="text-align: center; --cell-justify: center">Nov</th><th style="text-align: center; --cell-justify: center">Dec</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Advertisement Expenditure, $x$</td><td style="text-align: center; --cell-justify: center">$5$</td><td style="text-align: center; --cell-justify: center">$6.5$</td><td style="text-align: center; --cell-justify: center">$7$</td><td style="text-align: center; --cell-justify: center">$6$</td><td style="text-align: center; --cell-justify: center">$8$</td><td style="text-align: center; --cell-justify: center">$5.5$</td><td style="text-align: center; --cell-justify: center">$10$</td><td style="text-align: center; --cell-justify: center">$7.5$</td><td style="text-align: center; --cell-justify: center">$8.5$</td><td style="text-align: center; --cell-justify: center">$9$</td><td style="text-align: center; --cell-justify: center">$4$</td><td style="text-align: center; --cell-justify: center">$9.5$</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Sales Profits, $y$</td><td style="text-align: center; --cell-justify: center">$16$</td><td style="text-align: center; --cell-justify: center">$19$</td><td style="text-align: center; --cell-justify: center">$19.5$</td><td style="text-align: center; --cell-justify: center">$17$</td><td style="text-align: center; --cell-justify: center">$k$</td><td style="text-align: center; --cell-justify: center">$16.5$</td><td style="text-align: center; --cell-justify: center">$32.5$</td><td style="text-align: center; --cell-justify: center">$22$</td><td style="text-align: center; --cell-justify: center">$26$</td><td style="text-align: center; --cell-justify: center">$28$</td><td style="text-align: center; --cell-justify: center">$15.5$</td><td style="text-align: center; --cell-justify: center">$30$</td></tr></tbody></table></div></div>
<p>It is thought that this set of data can be modelled by one of the following equations:</p>
<p style="text-align: center">Model A: $y=mx+c$</p>
<p style="text-align: center">Model B: $\ln y=ax+b$</p>
<ol class="i-list"><li data-list-item-id="item-1760425312004-jjhhqdhzw" data-list-item-path="item-1760425312004-jjhhqdhzw"><div class="list-item-paragraph">Explain the meaning of the value of $m$ in the context of the data for Model A.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760425314186-9knkf91j8" data-list-item-path="item-1760425314186-9knkf91j8"><div class="list-item-paragraph">It is given that the least squares regression line for Model A is $y=0.3604+3.0194x$, correct to $4$ decimal places. Show that&nbsp;$k=23.5$, correct to $1$ decimal place.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760425315012-bd10m0uxl" data-list-item-path="item-1760425315012-bd10m0uxl"><div class="list-item-paragraph">Draw the scatter diagram for the data, labelling the axes clearly.<span class="inline-marks">[1]</span></div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760425363264-w0um7f5klr.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0MjUzNjMyNjQtdzB1bTdmNWtsci5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTU5MTQ0LCJleHAiOjE3OTM1NTk0NDR9.UkRK65Vm2EajqGPGNk8VCFZwZrNhvCDiO9Hi_4qnImI" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div></li><li data-list-item-id="item-1760425317982-h2kqturd5" data-list-item-path="item-1760425317982-h2kqturd5"><div class="list-item-paragraph">By calculating the product moment correlation coefficients and using the scatter diagram from part <strong>(iii)</strong>, explain whether Model A or Model B is a more appropriate model for the given set of data.<span class="inline-marks">[3]</span></div></li></ol>
<p>Use the more appropriate model selected in part <strong>(iv)</strong> for the rest of this question.</p>
<ol class="i-list" start="5"><li data-list-item-id="item-1760425391004-z2oxh5nde" data-list-item-path="item-1760425391004-z2oxh5nde"><div class="list-item-paragraph">Use a regression line to obtain the estimate for the sales profit if the advertisement expenditure is $\$11,000$ for a particular month.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760425405634-ofp1b7u3k" data-list-item-path="item-1760425405634-ofp1b7u3k"><div class="list-item-paragraph">Comment on the reliability of the estimate obtained in <strong>(v)</strong>.<span class="inline-marks">[1]</span></div></li></ol>

An electrical appliance supplier is investigating the potential impact of monthly advertisement expenditure, denoted as $x$ (in thousands of dollars), on the popular social media platform SnapGram, on their monthly sales profits, denoted as $y$ (in thousands of dollars). The supplier has collected the following data over the course of the past year.

It is thought that this set of data can be modelled by one of the following equations:

Use the more appropriate model selected in part **(iv)** for the rest of this question.

Finding Missing Data from Regression Information

2023 ACJC P2 Q7

<ol class="abc-list" start="2"><li><div class="list-item-paragraph">May travels regularly from home to the hospital for treatment. She leaves home at $x$ minutes after 8 am and takes $y$ minutes to arrive at the hospital. Ten pairs of data are recorded in the table below.</div><div class="table-container"><table style="width: 100%;"><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">$5$</th><th style="text-align: center; --cell-justify: center">$10$</th><th style="text-align: center; --cell-justify: center">$15$</th><th style="text-align: center; --cell-justify: center">$20$</th><th style="text-align: center; --cell-justify: center">$25$</th><th style="text-align: center; --cell-justify: center">$30$</th><th style="text-align: center; --cell-justify: center">$35$</th><th style="text-align: center; --cell-justify: center">$40$</th><th style="text-align: center; --cell-justify: center">$45$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$</td><td style="text-align: center; --cell-justify: center">$31$</td><td style="text-align: center; --cell-justify: center">$42$</td><td style="text-align: center; --cell-justify: center">$33$</td><td style="text-align: center; --cell-justify: center">$48$</td><td style="text-align: center; --cell-justify: center">$47$</td><td style="text-align: center; --cell-justify: center">$53$</td><td style="text-align: center; --cell-justify: center">$68$</td><td style="text-align: center; --cell-justify: center">$65$</td><td style="text-align: center; --cell-justify: center">$71$</td><td style="text-align: center; --cell-justify: center">$70$</td></tr></tbody></table></div><ol class="i-list"><li><div class="list-item-paragraph">Draw a scatter diagram to illustrate the data, labelling the axes clearly.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Calculate the equation of the least squares regression line of $y$ on $x$ and the value of the linear product moment correlation coefficient for the data.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Interpret the regression coefficient in the context of the question.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Draw the regression line of $y$ on $x$ on the scatter diagram.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">May has a hospital treatment appointment at 8.45 am on a particular day. If she leaves home at 8.10 am, explain whether she is likely to be on time for her hospital treatment. Comment on the reliability of your answer.<span class="inline-marks">[3]</span></div></li></ol></li></ol>

2013 TJC P2 Q11

<p>A large cohort of students sat for a mathematics examination. Based on selected data of the examination results, the following table shows $y$, the proportions of students who scored $x$ marks.</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: center; --cell-justify: center"><p>$x$</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>30</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>40</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>60</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>70</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>80</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>90</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>0.00029</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>0.00174</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>0.00663</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>0.0161</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>0.0252</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>0.0252</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>0.0161</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>0.00663</p></td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Draw a scatter diagram for these values, labelling the axes.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Explain why, in this context, a linear model is not appropriate.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is decided to fit a model of the form $\ln y=a{{(x-m)}^{2}}+b$, where and is a suitable constant, to the data. The product moment correlation coefficient between ${{\left( x-m \right)}^{2}}$ and $\ln y$ is denoted by $r$. The table below gives values of $r$ for some possible values of $m$.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>m</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>62.5</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>65</p></th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>67.5</p></th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>r</p></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>-0.9899292</p></td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center"><p>-0.9938968</p></td></tr></tbody></table></div></div>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Calculate the value of $r$ for&nbsp;$m=65$, giving your answer correct to $7$ decimal places.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Use the table and your answer in part <strong>(iii)</strong> to suggest with a reason which of $62.5$, $65$ or $67.5$ is the most appropriate value for&nbsp;$m$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Using the value of found in part <strong>(iv)</strong>, calculate the values of $a$ and $b$, and use them to predict the proportion of students who scored $45$ marks. Comment on the reliability of your prediction.<span class="inline-marks">[5]</span></div></li></ol>

A large cohort of students sat for a mathematics examination. Based on selected data of the examination results, the following table shows $y$, the proportions of students who scored $x$ marks.

It is decided to fit a model of the form $\ln y=a{{(x-m)}^{2}}+b$, where and is a suitable constant, to the data. The product moment correlation coefficient between ${{\left( x-m \right)}^{2}}$ and $\ln y$ is denoted by $r$. The table below gives values of $r$ for some possible values of $m$.

2017 HCI P2 Q10

<p>The data below shows the average height, measured in feet, of cherry trees from age $1$ year to age $11$ years.</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: center; --cell-justify: center">Age ($x$)</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><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$8$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$11$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Average height ($y$)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$13$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$15$</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">$17.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$18.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.7$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.8$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate $r$, the value of the product moment correlation coefficient between $x$ and $y$. State, with a reason, whether the value of $r$ would be different if the average height is measured in meters instead.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Give a sketch of the scatter diagram for the data.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">It is desired to predict the average height of cheery trees beyond age $11$ years. Explain why neither a linear nor a quadratic model is likely to be appropriate.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is suggested that the average height $y$ can be modelled by the formula $y=a+b\ln x$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the equation of the regression line for the suggested model, and the product moment correlation coefficient between $\ln x$ and $y$. Estimate the average height of cherry trees age $5.5$ years, and comment on the reliability of your answer.<span class="inline-marks">[4]</span></div></li></ol>

The data below shows the average height, measured in feet, of cherry trees from age $1$ year to age $11$ years.

It is suggested that the average height $y$ can be modelled by the formula $y=a+b\ln x$.

2013 RI P2 Q9

<p>John is training for National Shooting Championship and he practices shooting each week. His percentage score, $y\%$ in week $x$, are as follows.</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">$1$</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">$6$</th><th style="text-align: center; --cell-justify: center">$7$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$</td><td style="text-align: center; --cell-justify: center">$40$</td><td style="text-align: center; --cell-justify: center">$57$</td><td style="text-align: center; --cell-justify: center">$68$</td><td style="text-align: center; --cell-justify: center">$74$</td><td style="text-align: center; --cell-justify: center">$80$</td><td style="text-align: center; --cell-justify: center">$85$</td><td style="text-align: center; --cell-justify: center">$88$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Draw a scatter diagram to illustrate the data.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Calculate the value of the product moment correlation coefficient, and explain why its value does not necessarily mean that a linear model is an appropriate model for the relationship between $x$ and $y$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is decided to fit a model of the form $\mathrm{ln}\left( 95-y \right)=a+bx$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Find, correct to $4$ decimal places, the value of the product moment correlation coefficient between $\mathrm{ln}\left( 95-y \right)$ and $x$ and comment on its value.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Calculate the values of $a$ and $b$ correct to $4$ decimal places. Use them to predict, to the nearest whole number, the percentage score John will obtain for week $8$. Comment on the reliability of the estimate.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Give an interpretation, in context, of the value $95$ in the model $\mathrm{ln}\left( 95-y \right)=a+bx$.<span class="inline-marks">[1]</span></div></li></ol>

John is training for National Shooting Championship and he practices shooting each week. His percentage score, $y\%$ in week $x$, are as follows.

It is decided to fit a model of the form $\mathrm{ln}\left( 95-y \right)=a+bx$.

2013 IJC P2 Q9

<p>A swim school takes in both male and female primary school students for competitive swimming lessons. The school assesses its students’ progress each year by recording the time, $t$ seconds, each student takes to swim a $50$-metre lap in breaststroke, and the number of months, $m$, that he or she has been at the school. The records for&nbsp;$8$ randomly chosen students are shown in the following table.</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">$m$</th><th style="text-align: center; --cell-justify: center">$6$</th><th style="text-align: center; --cell-justify: center">$7$</th><th style="text-align: center; --cell-justify: center">$10$</th><th style="text-align: center; --cell-justify: center">$12$</th><th style="text-align: center; --cell-justify: center">$15$</th><th style="text-align: center; --cell-justify: center">$19$</th><th style="text-align: center; --cell-justify: center">$21$</th><th style="text-align: center; --cell-justify: center">$24$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$t$</td><td style="text-align: center; --cell-justify: center">$92.32$</td><td style="text-align: center; --cell-justify: center">$87.11$</td><td style="text-align: center; --cell-justify: center">$66.12$</td><td style="text-align: center; --cell-justify: center">$59.41$</td><td style="text-align: center; --cell-justify: center">$53.94$</td><td style="text-align: center; --cell-justify: center">$43.82$</td><td style="text-align: center; --cell-justify: center">$42.07$</td><td style="text-align: center; --cell-justify: center">$41.45$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Labelling the axes clearly, draw a scatter diagram for the data and explain, in context, why a linear model would not be suitable to predict the time taken by a student to swim a lap of breaststroke given the number of months that he or she has been at the school.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is desired to fit a model of the form $\ln \left( t-C \right)=a+bm$, where&nbsp;$C$ is a suitable constant. The product moment correlation coefficient $r$ between $m$ and $\ln \left( t-C \right)$ for some possible values of $C$ are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$C$</th><th style="text-align: center; --cell-justify: center">$36$</th><th style="text-align: center; --cell-justify: center">$37$</th><th style="text-align: center; --cell-justify: center">$38$</th><th style="text-align: center; --cell-justify: center">$39$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$r$</td><td style="text-align: center; --cell-justify: center">$-0.992114$</td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: center; --cell-justify: center">$-0.992681$</td><td style="text-align: center; --cell-justify: center">$-0.992192$</td></tr></tbody></table></div></div>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Calculate the value of $r$ for&nbsp;$C=37$, giving your answer correct to 6 decimal places.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Use the table and your answer to <strong>(ii)</strong> to choose the most appropriate value for&nbsp;$C$. Explain your choice.<span class="inline-marks">[2]</span></div></li></ol>
<p>For the remainder of this question, use the value of $C$ that you have chosen in <strong>(iii)</strong>.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the equation of the least squares regression line of $\ln \left( t-C \right)$ on $m$. Give an interpretation of $C$ in the context of the question.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Another student who has been swimming at the school for&nbsp;$9$ months clocked a time of $60.33$ seconds for a lap of breaststroke. Using your regression line, comment on the student’s swimming ability.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Suggest an improvement to the data collection process so that the results could provide a fairer gauge of the expected outcome for the students in the first $2$ years of lessons.<span class="inline-marks">[1]</span></div></li></ol>

A swim school takes in both male and female primary school students for competitive swimming lessons. The school assesses its students’ progress each year by recording the time, $t$ seconds, each student takes to swim a $50$-metre lap in breaststroke, and the number of months, $m$, that he or she has been at the school. The records for $8$ randomly chosen students are shown in the following table.

It is desired to fit a model of the form $\ln \left( t-C \right)=a+bm$, where $C$ is a suitable constant. The product moment correlation coefficient $r$ between $m$ and $\ln \left( t-C \right)$ for some possible values of $C$ are shown in the table below.

For the remainder of this question, use the value of $C$ that you have chosen in **(iii)**.

Math Question Bank