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 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

Linearisation of Nonlinear Models

Comparing Regression Models

Least-Squares Regression and Residuals

Prediction Using Regression Models

Scatter Diagrams and Association

Product Moment Correlation Coefficient

2025 NYJC P2 Q10

Correlation & Regressions

<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">$17$</th><th style="text-align: center; --cell-justify: center">$11$</th><th style="text-align: center; --cell-justify: center">$19$</th><th style="text-align: center; --cell-justify: center">$23$</th><th style="text-align: center; --cell-justify: center">$29$</th><th style="text-align: center; --cell-justify: center">$35$</th><th style="text-align: center; --cell-justify: center">$41$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$</td><td style="text-align: center; --cell-justify: center">$25$</td><td style="text-align: center; --cell-justify: center">$k$</td><td style="text-align: center; --cell-justify: center">$29$</td><td style="text-align: center; --cell-justify: center">$41$</td><td style="text-align: center; --cell-justify: center">$47$</td><td style="text-align: center; --cell-justify: center">$58$</td><td style="text-align: center; --cell-justify: center">$71$</td></tr></tbody></table></div></div>
<p>The least square regression line of $x$ on $y$ has equation $x=0.53806y+3.01639$. Find the value of $k$.</p>

The least square regression line of $x$ on $y$ has equation $x=0.53806y+3.01639$. Find the value of $k$.

Finding Unknown Values in Tables

<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">$x$</th><th style="text-align: center; --cell-justify: center">$220$</th><th style="text-align: center; --cell-justify: center">$111$</th><th style="text-align: center; --cell-justify: center">$120$</th><th style="text-align: center; --cell-justify: center">$124$</th><th style="text-align: center; --cell-justify: center">$85$</th><th style="text-align: center; --cell-justify: center">$80$</th><th style="text-align: center; --cell-justify: center">$71$</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">$162$</td><td style="text-align: center; --cell-justify: center">$123$</td><td style="text-align: center; --cell-justify: center">$113$</td><td style="text-align: center; --cell-justify: center">$32$</td><td style="text-align: center; --cell-justify: center">$k$</td><td style="text-align: center; --cell-justify: center">$53$</td><td style="text-align: center; --cell-justify: center">$40$</td><td style="text-align: center; --cell-justify: center">$11$</td></tr></tbody></table></div></div>
<p>The least square regression line of $y$ on $x$ has equation $y=0.79985x-9.8347$. Find the value of $k$.</p>

The least square regression line of $y$ on $x$ has equation $y=0.79985x-9.8347$. Find the value of $k$.

<table><thead><tr><th>$x$</th><th>220</th><th>111</th><th>120</th><th>124</th><th>85</th><th>80</th><th>71</th><th>45</th></tr></thead><tbody><tr><td>$y$</td><td>162</td><td>123</td><td>113</td><td>32</td><td>$k$</td><td>53</td><td>40</td><td>11</td></tr></tbody></table>
<p>The least square regression line of $y$ on $x$ has equation $y=0.79985x-9.8347$. Find the value of $k$.</p>

Worked Example 3

<p>The shear strength of soil is the ability of the soil to resist tensional stresses and sustain loads. The water content of soil is the ratio of the weight of water to the weight of the solids in a mass of soil. A soil investigation laboratory collected soil from a reclaimed site to study the effect of water content on shear strength. The information gathered about the shear strength $P$, measured in KiloPascal (KPa), at different water content $w$, measured in percentage is presented in the table.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760428635377-t1jruc118di.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0Mjg2MzUzNzctdDFqcnVjMTE4ZGkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU2NDgwMywiZXhwIjoxNzkzNTY1MTAzfQ.JHaqP8JhVCgGEVEuFYeERIX9sSizk5hzJiGv4sbVbuc" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">Draw a scatter diagram of these data. Use your diagram to explain whether the relationship between $P$ and $w$ is likely to be well modelled by an equation of the form $P=aw+b$, where $a$ and $b$ are onstants.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">By calculating the relevant product moment correlation coefficients, determine whether the relationship between $P$ and $w$ is modelled better by $P=aw+b$ or $P=a\sqrt{w}+b$. Explain how you decide which model is better, and state the equation in this case.</div><p style="text-align: right;">[5]</p></li><li><div class="list-item-paragraph">Use your equation to estimate the shear strength when the water content is $40%$. Explain whether your estimate is reliable.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Subsequently, it is known that one data point is not presented in the information table and that the inclusion of this data point does not affect the appropriate equation in <strong>(ii)</strong>.</div><div class="list-item-paragraph">State the shear strength for the missing data point, giving your answer correct to the nearest integer.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Given that $1$ MegaPascal $=1000$ KiloPascal, re-write your equation in <strong>(ii)</strong> so that it can be used when the shear strength is measured in MegaPascal. Explain whether the corresponding product moment correlation coefficient would differ from that calculated in <strong>(ii)</strong>.</div><p style="text-align: right;">[2]</p></li></ol>

The shear strength of soil is the ability of the soil to resist tensional stresses and sustain loads. The water content of soil is the ratio of the weight of water to the weight of the solids in a mass of soil. A soil investigation laboratory collected soil from a reclaimed site to study the effect of water content on shear strength. The information gathered about the shear strength $P$, measured in KiloPascal (KPa), at different water content $w$, measured in percentage is presented in the table.

Reliability of Regression Estimates

Effects of Linear Transformations on Correlation

Finding Regression Line Equations

Transforming Regression Equations for New Units

2023 JPJC P2 Q9

<p>Various studies suggest that plants exposed to music tend to grow taller compared to those not exposed to music. A junior biologist wants to investigate the effects of music on the growth of different plants. In one experiment, vines of a particular species were exposed to classical music. He recorded the average length of the vines, $h$ centimetres, at age $t$ days, in the table below.</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"><p>$t$</p></th><th style="text-align: center; --cell-justify: center"><p>$7$</p></th><th style="text-align: center; --cell-justify: center"><p>$10$</p></th><th style="text-align: center; --cell-justify: center"><p>$13$</p></th><th style="text-align: center; --cell-justify: center"><p>$16$</p></th><th style="text-align: center; --cell-justify: center"><p>$19$</p></th><th style="text-align: center; --cell-justify: center"><p>$22$</p></th><th style="text-align: center; --cell-justify: center"><p>$25$</p></th><th style="text-align: center; --cell-justify: center"><p>$28$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$h$</p></td><td style="text-align: center; --cell-justify: center"><p>$19.2$</p></td><td style="text-align: center; --cell-justify: center"><p>$25.4$</p></td><td style="text-align: center; --cell-justify: center"><p>$36.5$</p></td><td style="text-align: center; --cell-justify: center"><p>$41.4$</p></td><td style="text-align: center; --cell-justify: center"><p>$45$</p></td><td style="text-align: center; --cell-justify: center"><p>$k$</p></td><td style="text-align: center; --cell-justify: center"><p>$51.9$</p></td><td style="text-align: center; --cell-justify: center"><p>$53.8$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1760425580653-jsucf73eg" data-list-item-path="item-1760425580653-jsucf73eg"><div class="list-item-paragraph">Using the data, the junior biologist calculated the least squares regression line of $h$ on $t$ to be $h=1.6393t+11.475$. Show that the value of $k$ is $48.1$, correct to 1 decimal place.<span class="inline-marks">[2]</span></div></li></ol>
<p>The junior biologist realised that the relationship between $h$ and $t$ is not linear and can be modelled by either</p>
<p style="text-align: center">$h=a\sqrt[3]{t}+b$ or $h=c\sqrt{t}+d$,</p>
<p>where&nbsp;$a$, $b$, $c$ and $d$ are constants.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-1760425641021-4hwtvdjzt" data-list-item-path="item-1760425641021-4hwtvdjzt"><div class="list-item-paragraph">By calculating the relevant product moment correlation coefficients, explain why $h=a\sqrt[3]{t}+b$ is a better model than $h=c\sqrt{t}+d$. Find the equation of the better model.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760425644583-7yeq71vjw" data-list-item-path="item-1760425644583-7yeq71vjw"><div class="list-item-paragraph">An estimate for the average length of the vines when they are $2$ months old is obtained using the better model. Explain whether the estimate is reliable.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760425646009-kvcesv596" data-list-item-path="item-1760425646009-kvcesv596"><div class="list-item-paragraph">Sketch a scatter diagram for&nbsp;$h$ against $\sqrt[3]{t}$. Draw the line of best fit on your scatter diagram.<span class="inline-marks">[2]</span></div></li></ol>
<p>For a line of best fit $y=\mathrm{f}\left( x \right)$, the residual is the difference between the observed and predicted value. The difference between an observed value $\left( p,q \right)$ plotted on the scatter diagram and the predicted value $\left( p,\mathrm{f}\left( p \right) \right)$ is defined as&nbsp;$q-\mathrm{f}\left( p \right)$.</p>
<ol class="abc-list" start="5"><li data-list-item-id="item-1760425678694-jldz47l9f" data-list-item-path="item-1760425678694-jldz47l9f"><div class="list-item-paragraph">Find the value of the residual corresponding to $t=28$, for the line of best fit found in part <strong>(b)</strong>.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760425679593-18dkop0d0" data-list-item-path="item-1760425679593-18dkop0d0"><div class="list-item-paragraph">Explain why the line of best fit is referred to as the ‘least squares’ regression line in relation to the residuals.<span class="inline-marks">[2]</span></div></li></ol>

Various studies suggest that plants exposed to music tend to grow taller compared to those not exposed to music. A junior biologist wants to investigate the effects of music on the growth of different plants. In one experiment, vines of a particular species were exposed to classical music. He recorded the average length of the vines, $h$ centimetres, at age $t$ days, in the table below.

The junior biologist realised that the relationship between $h$ and $t$ is not linear and can be modelled by either

$h=a\sqrt[3]{t}+b$ or $h=c\sqrt{t}+d$,

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

For a line of best fit $y=\mathrm{f}\left( x \right)$, the residual is the difference between the observed and predicted value. The difference between an observed value $\left( p,q \right)$ plotted on the scatter diagram and the predicted value $\left( p,\mathrm{f}\left( p \right) \right)$ is defined as $q-\mathrm{f}\left( p \right)$.

2024 NYJC P2 Q11

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0MjUzNjMyNjQtdzB1bTdmNWtsci5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTY0ODAzLCJleHAiOjE3OTM1NjUxMDN9.b_1nTF6gRS36_aRrQE__C-sMsNEY3F-UIfBdn0XYedE" 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.

Interpreting Regression Models and Coefficients

2023 ACJC P2 Q7

<p>A random sample of eight pairs of values of $x$ and $y$ is used to obtain the following equations of the regression lines of $y$ on $x$ and of $x$ on $y$ respectively.</p>
<p style="text-align: center">$y=-\frac{7}{10}x+\frac{151}{10}$, $x=-\frac{7}{6}y+20$</p>
<p>Seven of the pairs of data are given in the table.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760342420985-6566o67scvm.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjAzNDI0MjA5ODUtNjU2Nm82N3Njdm0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU2NDgwMywiZXhwIjoxNzkzNTY1MTAzfQ.hitFx5emxG8BjmiFGYpfTWe0_ELOpWKY0XifG3nd_Kg" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find the eighth pair of values of $x$ and $y$. <span class="inline-marks">[5]</span></p>
<p>Determine the value of the product moment correlation coefficient, and say what it leads you to expect about the scatter diagram for this sample. <span class="inline-marks">[4]</span></p>
<p>Let&nbsp;$Y$ be the value obtained by substituting a sample value of $x$ into the equation of the regression line of $y$ on $x$. Evaluate $Y$ for each of the eight values of $x$ and verify that&nbsp;${\sum (y - Y)^2 = 8.8}$.<span class="inline-marks">[3]</span></p>
<p>For each of the eight sample values of $x$, ${Y}'$ is given by ${Y}'=a+bx$, where&nbsp;$a$ and $b$ are any constants. What can you say about the value of $\sum{{{\left( y-{Y}' \right)}^{2}}}$?<span class="inline-marks">[2]</span></p>


A random sample of eight pairs of values of $x$ and $y$ is used to obtain the following equations of the regression lines of $y$ on $x$ and of $x$ on $y$ respectively.

$y=-\frac{7}{10}x+\frac{151}{10}$, $x=-\frac{7}{6}y+20$

Seven of the pairs of data are given in the table.

Find the eighth pair of values of $x$ and $y$. 

Determine the value of the product moment correlation coefficient, and say what it leads you to expect about the scatter diagram for this sample. 

Let $Y$ be the value obtained by substituting a sample value of $x$ into the equation of the regression line of $y$ on $x$. Evaluate $Y$ for each of the eight values of $x$ and verify that ${\sum (y - Y)^2 = 8.8}$.

For each of the eight sample values of $x$, ${Y}'$ is given by ${Y}'=a+bx$, where $a$ and $b$ are any constants. What can you say about the value of $\sum{{{\left( y-{Y}' \right)}^{2}}}$?

N2003 P2 Q11

<p>In a particular study, a medical student recorded the age and blood pressure of 8 men. The regression line of $y$ on $x$ is determined to be $y=1.144x+79.914$. Unfortunately, the student then lost the record of one of the men. He urgently needs the missing data, $b$ in his report.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 520px;"><colgroup><col style="width: 120px;"><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: left; --cell-justify: flex-start">Age ($x$)</th><th style="text-align: center; --cell-justify: center">$49$</th><th style="text-align: center; --cell-justify: center">$58$</th><th style="text-align: center; --cell-justify: center">$44$</th><th style="text-align: center; --cell-justify: center">$75$</th><th style="text-align: center; --cell-justify: center">$37$</th><th style="text-align: center; --cell-justify: center">$57$</th><th style="text-align: center; --cell-justify: center">$63$</th><th style="text-align: center; --cell-justify: center">$66$</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Blood Pressure ($y$)</td><td style="text-align: center; --cell-justify: center">$133$</td><td style="text-align: center; --cell-justify: center">$148$</td><td style="text-align: center; --cell-justify: center">$b$</td><td style="text-align: center; --cell-justify: center">$166$</td><td style="text-align: center; --cell-justify: center">$123$</td><td style="text-align: center; --cell-justify: center">$147$</td><td style="text-align: center; --cell-justify: center">$154$</td><td style="text-align: center; --cell-justify: center">$152$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$b$ is 130.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Draw a scatter diagram for the data in the table.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the linear product moment correlation coefficient between $y$ and $x$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Use the given regression line to estimate the age of a man when his blood pressure is $150$, giving your answer to the nearest integer.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Explain whether this choice of the regression line is appropriate.<span class="inline-marks">[1]</span></div></li></ol>

In a particular study, a medical student recorded the age and blood pressure of 8 men. The regression line of $y$ on $x$ is determined to be $y=1.144x+79.914$. Unfortunately, the student then lost the record of one of the men. He urgently needs the missing data, $b$ in his report.

Choosing the Appropriate Regression Line

2010 MI P2 Q8

Graphing Techniques

<p>A retail manager of a large electrical appliances store wants to investigate the relationship between the monthly advertising expenditure, $x$ hundred dollars, and the monthly sales of their refrigerators, $y$ thousand dollars. The table shows the results of the investigation.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1759749686024-ijmiyyregq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTk3NDk2ODYwMjQtaWptaXl5cmVncS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTY0ODAzLCJleHAiOjE3OTM1NjUxMDN9.xrlT-I2cblIUrv1_S9ghwWFdrb67iqAdxHL51VGzOQ8" width="400" class="aligncenter" /></figure>
<ol class="i-list"><li><div class="list-item-paragraph">The manager concludes that an increase in monthly advertising expenditure will result in an increase in the monthly sales of refrigerators. State, with a reason, whether you agree with this conclusion.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Draw a scatter diagram to illustrate the above data. Explain why a linear model is not likely to be appropriate.</div><p style="text-align: right;">[2]</p></li></ol>
<p>It is thought that the monthly sales $y$ thousand dollars can be modelled by one of the formulae</p>
<p style="text-align: center">$y=a+b\,{{\mathrm{e}}^{\sqrt{x}}}$ or $y=a+b\,{{x}^{2}}$</p>
<p>where $a$ and $b$ are constants.</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</div><div class="list-item-paragraph"><strong>(A)</strong> ${{\mathrm{e}}^{\sqrt{x}}}$ and $y$, </div><div class="list-item-paragraph"><strong>(B)</strong> ${{x}^{2}}$ and $y$.</div><div class="list-item-paragraph">Explain which of $y=a+b\,{{\mathrm{e}}^{\sqrt{x}}}$ or $y=a+b\,{{x}^{2}}$ is the better model.</div><p style="text-align: right;">[2]</p></li></ol>
<p>Assume that the better model in part <strong>(iii)</strong> holds for part <strong>(iv)</strong>.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">The manager forgot to record the monthly advertising expenditure when the monthly sales of refrigerators was $\$11300$. Combining this with the above data set, it is found that $a=10.876$ and $b=0.09906$ for the model. Find the monthly advertising expenditure that the manager forgot to record, leaving your answer to the nearest hundred.</div><p style="text-align: right;">[3]</p></li></ol>

A retail manager of a large electrical appliances store wants to investigate the relationship between the monthly advertising expenditure, $x$ hundred dollars, and the monthly sales of their refrigerators, $y$ thousand dollars. The table shows the results of the investigation.

It is thought that the monthly sales $y$ thousand dollars can be modelled by one of the formulae

$y=a+b\,{{\mathrm{e}}^{\sqrt{x}}}$ or $y=a+b\,{{x}^{2}}$

Assume that the better model in part **(iii)** holds for part **(iv)**.

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