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<p>An estimate is reliable when it is made under conditions that are well supported by the data. In regression, we should first check whether the value being used is within the range of the original data; this is called interpolation and is usually more reliable. </p>
<p>If the value is outside the data range, it is extrapolation, which is less reliable because the same pattern may not continue beyond the observed data. We should also check whether there is a strong linear correlation between the two variables, because regression estimates are only dependable when the data follows a clear linear trend. </p>
<p>So, a reliable estimate usually requires both conditions: the value is within the data range and the variables have a strong linear correlation.</p>
<p><strong>Interpolation</strong> - The process of carrying out estimation within the given data range.<br><strong>Extrapolation</strong> – The process of carrying out estimation beyond the given data range.</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/pasted/1775711500945-gfbv3ul3z96.png?token=eyJraWQiOiJzdG9yYWdlLXVybC1zaWduaW5nLWtleV9iZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU3MTE1MDA5NDUtZ2ZidjN1bDN6OTYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4NTcwOTU4NiwiZXhwIjoxNzg1NzA5ODg2fQ.h7NJoq6XDSHwXtTNqDYmtkjnKgRHTmbubcqL_0e1mR8" style="max-width: min(600px, 100%); height: auto; display: block;" alt="Flowchart showing reliability of estimates based on interpolation and linear correlation" class="aligncenter custom-width" /></figure></div>

An estimate is reliable when it is made under conditions that are well supported by the data. In regression, we should first check whether the value being used is within the range of the original data; this is called interpolation and is usually more reliable. 

If the value is outside the data range, it is extrapolation, which is less reliable because the same pattern may not continue beyond the observed data. We should also check whether there is a strong linear correlation between the two variables, because regression estimates are only dependable when the data follows a clear linear trend. 

So, a reliable estimate usually requires both conditions: the value is within the data range and the variables have a strong linear correlation.

**Interpolation** - The process of carrying out estimation within the given data range.
**Extrapolation** – The process of carrying out estimation beyond the given data range.

Reliability of Estimates

Correlation & Regressions

<p>We need to choose the correct regression line because each line is designed to minimise a different type of prediction error. When we use $x$ to estimate $y$, the regression line of $y$ on $x$ is chosen because it makes the errors in the predicted $y$-values as small as possible. <br></p>
<p>On the other hand, when we use $y$ to estimate $x$, the regression line of $x$ on $y$ is used because it minimises the errors in the predicted $x$-values. Therefore, the choice of regression line depends on which variable we are trying to predict. Using the wrong line may not give the smallest possible errors for the variable of interest.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center">Scenario</th><th style="text-align: center; --cell-justify: center">Given&nbsp;$x$ to estimate $y$</th><th style="text-align: center; --cell-justify: center">Given&nbsp;$y$ to estimate $x$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$ depends on $x$</td><td colspan="2" style="text-align: center; --cell-justify: center">Use the $y$ on $x$ line equation.</td></tr><tr><td style="text-align: center; --cell-justify: center">$x$ depends on $y$</td><td colspan="2" style="text-align: center; --cell-justify: center">Use the $x$ on $y$ line equation.</td></tr><tr><td style="text-align: center; --cell-justify: center">No observable dependence</td><td style="text-align: center; --cell-justify: center">Use the $y$ on $x$ line equation.</td><td style="text-align: center; --cell-justify: center">Use the $x$ on $y$ line equation.</td></tr></tbody></table></div></div>

We need to choose the correct regression line because each line is designed to minimise a different type of prediction error. When we use $x$ to estimate $y$, the regression line of $y$ on $x$ is chosen because it makes the errors in the predicted $y$-values as small as possible. 


On the other hand, when we use $y$ to estimate $x$, the regression line of $x$ on $y$ is used because it minimises the errors in the predicted $x$-values. Therefore, the choice of regression line depends on which variable we are trying to predict. Using the wrong line may not give the smallest possible errors for the variable of interest.

Choice of Regression Lines

Scatter Diagrams and Correlation

<p>An energy company is studying the relationship between the average daily temperature, $x$ (in degrees Celsius), and the amount of electricity consumed, $y$ (in megawatt-hours MWh). For a random sample of ten days, the records are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Average daily temperature ($x{}^\circ$C)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$20$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$22$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$24$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$26$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$28$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$32$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$34$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$36$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$38$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Electricity consumption ($y$ MWh)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$12.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$12.8$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$13.2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$14.0$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$14.7$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$16.2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$16.2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.9$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$22.3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$29.5$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a-2" data-list-item-path="item-a-2"><div class="list-item-content"><span class="list-item-text">Draw a scatter diagram for these values, labelling the axes clearly.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>It is thought that the electricity consumption and average daily temperature can be modelled by one of the formulae\[y = a + bx^2, \quad y = c + dx,\]where&nbsp;$a$, $b$, $c$ and $d$ are constants.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b-2" data-list-item-path="item-b-2"><div class="list-item-content"><span class="list-item-text">Find the value of the product moment correlation coefficient between</span><span class="inline-marks">[2]</span></div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b-2/item-b-i"><div class="list-item-content"><span class="list-item-text">$x$ and $y$,</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b-2/item-b-ii"><div class="list-item-content"><span class="list-item-text">$x^2$ and $y$.</span></div></li></ol></li><li data-list-item-id="item-c-2" data-list-item-path="item-c-2"><div class="list-item-paragraph">Using your answers in parts <strong>(a)</strong> and <strong>(b)</strong>, explain which of $y = a + bx^2$ and $y = c + dx$ is the better model and find the equation of a suitable regression line for this model.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d-2" data-list-item-path="item-d-2"><div class="list-item-content"><span class="list-item-text">Use the equation of your regression line to estimate the electricity consumption on a day with an average temperature of $37{}^\circ$C. Comment on the reliability of your estimate.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>In some regions, temperature is measured in degrees Fahrenheit instead of degrees Celsius. A temperature of $C$ degrees Celsius is equivalent to $F$ degrees Fahrenheit, where&nbsp;$F = \frac{9}{5}C + 32$.</p>
<ol class="abc-list" start="5"><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Rewrite the equation of your regression line found in part <strong>(c)</strong> in terms of $y$ and $F$, where&nbsp;$F$ is the average daily temperature in degrees Fahrenheit.<span class="inline-marks">[1]</span></div></li></ol>

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

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

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

Find the value of the product moment correlation coefficient between

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

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

2025 VJC P2 Q11

<p>A car dealer is investigating how the value of a car depreciates over time. A random sample of eight cars of the same model is selected and the current resale value, $y$ thousand dollars of each car is recorded along with its age in $x$ years. The results are shown in the table.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Age of car ($x$ years)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$0$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$7$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start">Resale Value ($y$ thousand dollars)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30.0$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$25.8$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$22.3$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.2$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$16.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$14.4$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$12.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10.9$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Draw the scatter diagram for these values, labelling the axes clearly.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>It is thought that the resale value of a car, $y$, can be modelled by one of the formulae \[y = a + bx \quad \text{or} \quad \ln y = c + dx\]where&nbsp;$a$, $b$, $c$ and $d$ are real constants.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Find, correct to $5$ decimal places, the value of the product moment correlation coefficient between</span><span class="inline-marks">[2]</span></div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-content"><span class="list-item-text">$y$ and $x$,</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-content"><span class="list-item-text">$\ln y$ and $x$.</span></div></li></ol></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Use your answers to parts <strong>(a)</strong> and <strong>(b)</strong> to explain which of $y = a + bx$ and $\ln y = c + dx$ is the better model.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is required to estimate the age of a car with a resale value of $\$18000$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-content"><span class="list-item-text">Find the equation of a suitable regression line and use it to find the required estimate.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">Without the use of a graphing calculator, re-write your equation from part <strong>(d)</strong> so that it can be used to estimate the resale value when the age is given in months.<span class="inline-marks">[2]</span></div></li></ol>

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

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

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

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

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

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

2025 TMJC P2 Q9

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

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

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

Sketch a scatter diagram for these values. Explain, using your diagram, which of the following equations provides the most accurate model of the relationship between $x$ and $y$.\n\n\[(A) \quad y = a - bx^2,\]\n\[(B) \quad y = a + be^{-x},\]\n\[(C) \quad y = a + b \ln(10 - x),\]\n\nwhere $a$ and $b$ are constants such that $a > 0$ and $b > 0$.

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

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

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

2025 RVHS P2 Q10

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

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

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

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

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

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

2025 RI P2 Q8

<p>The director of an art gallery wishes to find out the relationship between the time spent, $t$ minutes, in the art gallery and the age, $a$ (in years), of the visitors. Ten visitors were randomly selected and the data are summarised as follows:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 580px;"><colgroup><col style="width: 80px;"><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;"><col style="width: 50px;"></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-content"><span class="list-item-text">Sketch a scatter diagram of $t$ against $a$ for the data given in the table and draw the line given in (a) on the same diagram.</span><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-content"><span class="list-item-text">Use the least squares regression line of $t$ on $a$ in (a) to estimate the time spent in the art gallery by a $60$-year old visitor.</span><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 $k = 40$.

Sketch a scatter diagram of $t$ against $a$ for the data given in the table and draw the line given in (a) on the same diagram.

Use the least squares regression line of $t$ on $a$ in (a) to estimate the time spent in the art gallery by a $60$-year old visitor.

A regression line $t = ma + c$ where $m$ and $c$ are constants is used to model the above set of data. The residual for a data point $(a_i, t_i)$ where $i = 1, 2, \dots, 10$ which is plotted on a scatter diagram is defined to be the number\n\[t_i - (ma_i + c).\]\nThe sum of the squares of these residuals, denoted by $S$, is given by\n\[S = \sum_{i=1}^{10} [t_i - (ma_i + c)]^2.\]\nFind an inequality that is satisfied by $S$.

2025 NYJC P2 Q10

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

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

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

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

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

2025 NJC P2 Q8

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Draw separate scatter diagrams, each with 7 points, all in the first quadrant, which represent the situation where the product moment correlation coefficient between variables $x$ and $y$ is</span></div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-content"><span class="list-item-text">0,</span><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-content"><span class="list-item-text">approximately 0.8.</span><span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Eight cities in a certain country are linked by rail to the capital city. The table below shows the distance of each city from the capital and the rail price from the city to the capital.</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start">City</th><th style="text-align: center; --cell-justify: center">A</th><th style="text-align: center; --cell-justify: center">B</th><th style="text-align: center; --cell-justify: center">C</th><th style="text-align: center; --cell-justify: center">D</th><th style="text-align: center; --cell-justify: center">E</th><th style="text-align: center; --cell-justify: center">F</th><th style="text-align: center; --cell-justify: center">G</th><th style="text-align: center; --cell-justify: center">H</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Distance, $d$ km</td><td style="text-align: center; --cell-justify: center">124</td><td style="text-align: center; --cell-justify: center">44</td><td style="text-align: center; --cell-justify: center">76</td><td style="text-align: center; --cell-justify: center">148</td><td style="text-align: center; --cell-justify: center">16</td><td style="text-align: center; --cell-justify: center">180</td><td style="text-align: center; --cell-justify: center">104</td><td style="text-align: center; --cell-justify: center">195</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Rail price, $p$</td><td style="text-align: center; --cell-justify: center">156</td><td style="text-align: center; --cell-justify: center">53</td><td style="text-align: center; --cell-justify: center">99</td><td style="text-align: center; --cell-justify: center">169</td><td style="text-align: center; --cell-justify: center">23</td><td style="text-align: center; --cell-justify: center">177</td><td style="text-align: center; --cell-justify: center">138</td><td style="text-align: center; --cell-justify: center">178</td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-content"><span class="list-item-text">Draw a scatter diagram of the data.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-content"><span class="list-item-text">For a line of best fit $p = f(d)$, the residual for a point $(a, b)$ plotted on the scatter diagram is the vertical distances between $(a, f(a))$ and $(a, b)$. Explain why, in general, the sum of the squares of the residuals rather than the sum of the residuals is used.</span><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Using the scatter diagram in <strong>(i)</strong>, explain if a linear relationship between the distance and the rail price is appropriate.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-content"><span class="list-item-text">Suppose that the relationship between $d$ and $p$ are modelled by an equation of the form $p = s + t \ln d$, where&nbsp;$s$ and $t$ are constants. Find the equation of this regression line and use it to estimate the rail price for a city which is 210 km from the capital.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-v" data-list-item-path="item-v"><div class="list-item-paragraph">Comment on the reliability of the estimate in <strong>(iv)</strong>.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-vi" data-list-item-path="item-vi"><div class="list-item-paragraph">Following an adjustment to the rail prices, all rail prices are increased by 10 dollars. Without any further calculations, state any change you would expect in the values of the constants $s$ and $t$ found in <strong>(iv)</strong>.<span class="inline-marks">[2]</span></div></li></ol>

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

Eight cities in a certain country are linked by rail to the capital city. The table below shows the distance of each city from the capital and the rail price from the city to the capital.

For a line of best fit $p = f(d)$, the residual for a point $(a, b)$ plotted on the scatter diagram is the vertical distances between $(a, f(a))$ and $(a, b)$. Explain why, in general, the sum of the squares of the residuals rather than the sum of the residuals is used.

Suppose that the relationship between $d$ and $p$ are modelled by an equation of the form $p = s + t \ln d$, where $s$ and $t$ are constants. Find the equation of this regression line and use it to estimate the rail price for a city which is 210 km from the capital.

2025 JPJC P2 Q11

<p>An ice-cream seller records the monthly ice cream sales, $s$ thousands dollars for different temperature, $t$ degrees Celsius during the winter season. The recorded values are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$t$</th><th style="text-align: center; --cell-justify: center">$1$</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><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">$s$</td><td style="text-align: center; --cell-justify: center">$14$</td><td style="text-align: center; --cell-justify: center">$15$</td><td style="text-align: center; --cell-justify: center">$15$</td><td style="text-align: center; --cell-justify: center">$16$</td><td style="text-align: center; --cell-justify: center">$18$</td><td style="text-align: center; --cell-justify: center">$21$</td><td style="text-align: center; --cell-justify: center">$23$</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a-5" data-list-item-path="item-a-5"><div class="list-item-content"><span class="list-item-text">It is given that the regression line of $s$ on $t$ is $s = 1.125t + 11$. Using this regression line, find the sum of the squares of the residuals.</span><span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b-5" data-list-item-path="item-b-5"><div class="list-item-paragraph">State the coordinates of an additional data point such that, with all $8$ data points, the regression line remains the same as in part <strong>(a)</strong>.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c-5" data-list-item-path="item-c-5"><div class="list-item-content"><span class="list-item-text">Sketch a scatter diagram of $s$ against $t$ for the data given in the table.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The following three models are proposed, where&nbsp;$a$, $b$, $c$, $d$, $f$ and $h$ are positive constants.</p>
<ol class="ABC-list" style="margin-left: 8rem"><li data-list-item-id="model-a" data-list-item-path="model-a"><div class="list-item-content"><span class="list-item-text">$s = at^2 + b$</span></div></li><li data-list-item-id="model-b" data-list-item-path="model-b"><div class="list-item-content"><span class="list-item-text">$s = -ce^t + d$</span></div></li><li data-list-item-id="model-c" data-list-item-path="model-c"><div class="list-item-content"><span class="list-item-text">$s = f \ln(t + h)$</span></div></li></ol>
<ol class="abc-list" start="4"><li data-list-item-id="item-d-5" data-list-item-path="item-d-5"><div class="list-item-content"><span class="list-item-text">Explain which of these models give the best fit to the data. State the values of the constants for the chosen model.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>A temperature of $F$ degrees Fahrenheit is equivalent to a temperature of $C$ degrees Celsius, where&nbsp;$F = \frac{9}{5}C + 32$.</p>
<ol class="abc-list" start="5"><li data-list-item-id="item-e-5" data-list-item-path="item-e-5"><div class="list-item-paragraph">Using the model you chose in part <strong>(d)</strong>, re-write the equation so that it can be used to estimate the monthly sales when the temperature, $T$, is given in degrees Fahrenheit.<span class="inline-marks">[2]</span></div></li></ol>

An ice-cream seller records the monthly ice cream sales, $s$ thousands dollars for different temperature, $t$ degrees Celsius during the winter season. The recorded values are shown in the table below.

It is given that the regression line of $s$ on $t$ is $s = 1.125t + 11$. Using this regression line, find the sum of the squares of the residuals.

Sketch a scatter diagram of $s$ against $t$ for the data given in the table.

The following three models are proposed, where $a$, $b$, $c$, $d$, $f$ and $h$ are positive constants.

Explain which of these models give the best fit to the data. State the values of the constants for the chosen model.

A temperature of $F$ degrees Fahrenheit is equivalent to a temperature of $C$ degrees Celsius, where $F = \frac{9}{5}C + 32$.

2025 HCI P2 Q7

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

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

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

2015 YIJC P2 Q10

<p>Linearize the following equations where $a$ and $b$ are constants.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$y=a x^{2}-\frac{b}{x}$</div></li><li><div class="list-item-paragraph">$\frac{1}{y}=a x^{3}-\frac{b}{x^{2}}$</div></li><li><div class="list-item-paragraph">$y=3 b^{x}$</div></li><li><div class="list-item-paragraph">$y=b^{2x+1}-1$</div></li></ol>

Linearize the following equations where $a$ and $b$ are constants.

Linearization of Equations

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

<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 $k$.

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

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

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

Finding The Equations of The Regression Lines

Estimated Product moment correlation coefficient:  

$r=\frac{\sum{\left( x-\bar{x} \right)\left( y-\bar{y} \right)}}{\sqrt{\left\{ \sum{{{\left( x-\bar{x} \right)}^{2}}} \right\}\left\{ \sum{{{\left( y-\bar{y} \right)}^{2}}} \right\}}}=\frac{\sum{xy}-\frac{\sum{x}\sum{y}}{n}}{\sqrt{\left( \sum{{{x}^{2}}}-\frac{{{\left( \sum{x} \right)}^{2}}}{n} \right)\left( \sum{{{y}^{2}}}-\frac{{{\left( \sum{y} \right)}^{2}}}{n} \right)}}$ (From MF 26)

$y-\bar{y}=b\left( x-\bar{x} \right)$, where $b=\frac{\sum{\left( x-\bar{x} \right)\left( y-\bar{y} \right)}}{\sum{{{\left( x-\bar{x} \right)}^{2}}}}$ . (From MF 26)

The value of $r$ gives us an indication of:

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