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PSLE

PSLE Foundation Mathematics

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

Reliability of Regression Estimates

Linearisation of Nonlinear Models

Effects of Linear Transformations on Correlation

Finding Regression Line Equations

Interpreting Regression Models and Coefficients

Comparing Regression Models

Prediction Using Regression Models

Scatter Diagrams and Association

Product Moment Correlation Coefficient

Transforming Regression Equations for New Units

2023 DHS P2 Q11

Correlation & Regressions

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA0Mjg2MzUzNzctdDFqcnVjMTE4ZGkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzU1OTE0OCwiZXhwIjoxNzkzNTU5NDQ4fQ.7dq1yXFqksECVPvPZlBUh1EQogaYopjouMMdQSlL1cM" 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.

Finding Missing Data from Regression Information

2023 JPJC P2 Q9

<p>Tom records the maximum speeds of an athlete running in a temperature controlled environment. The table below shows the maximum speed, $y$ metres per second, recorded for various specific temperatures, $x$ degree Celsius.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 480px;"><colgroup><col style="width: 80px;"><col style="width: 80px;"><col style="width: 80px;"><col style="width: 80px;"><col style="width: 80px;"><col style="width: 80px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center">$x$</th><th style="text-align: center; --cell-justify: center">$6$</th><th style="text-align: center; --cell-justify: center">$11$</th><th style="text-align: center; --cell-justify: center">$16$</th><th style="text-align: center; --cell-justify: center">$21$</th><th style="text-align: center; --cell-justify: center">$26$</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$y$</td><td style="text-align: center; --cell-justify: center">$2.96$</td><td style="text-align: center; --cell-justify: center">$2.94$</td><td style="text-align: center; --cell-justify: center">$2.85$</td><td style="text-align: center; --cell-justify: center">$2.71$</td><td style="text-align: center; --cell-justify: center">$2.52$</td></tr></tbody></table></div></div>
<ol class="i-list"><li data-list-item-id="item-1760419959192-gfj0qfkfa" data-list-item-path="item-1760419959192-gfj0qfkfa"><div class="list-item-paragraph">Tom decides to model the data using the line $y=-0.02x+3.1$.</div><ol class="abc-list"><li data-list-item-id="subitem-1760419970104-kf8nb0wk0" data-list-item-path="item-1760419959192-gfj0qfkfa/subitem-1760419970104-kf8nb0wk0"><div class="list-item-paragraph">On the grid opposite</div><div class="list-item-paragraph">• draw a scatter diagram of the data, and</div><div class="list-item-paragraph">• draw the line $y=-0.02x+3.1$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="subitem-1760419997359-gc9phj22i" data-list-item-path="item-1760419959192-gfj0qfkfa/subitem-1760419997359-gc9phj22i"><div class="list-item-paragraph">For a line of best fit $y=\mathrm{f}\left( x \right)$, the residual for a point $\left( a,b \right)$ plotted on the scatter diagram is the vertical distance between $\left( a,\mathrm{f}\left( a \right) \right)$ and $\left( a,b \right)$. Mark the residual for each point on your diagram.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1760420009427-b9ah2jr8l" data-list-item-path="item-1760419959192-gfj0qfkfa/subitem-1760420009427-b9ah2jr8l"><div class="list-item-paragraph">Calculate the sum of the squares of the residuals for Tom’s line.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1760420026512-053doarb0" data-list-item-path="item-1760419959192-gfj0qfkfa/subitem-1760420026512-053doarb0"><div class="list-item-paragraph">Explain why, in general, the sum of the squares of the residuals rather than the sum of the residuals is used.<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-1760420039096-16bzy6mx2" data-list-item-path="item-1760420039096-16bzy6mx2"><div class="list-item-paragraph">State the value of the product moment correlation coefficient.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760420049454-12egjlgon" data-list-item-path="item-1760420049454-12egjlgon"><div class="list-item-paragraph">Use your calculator to find the equation of a suitable regression line to estimate the temperature for a given maximum speed. Hence estimate the temperature when the maximum speed is $2.48$ metres/second. Explain whether you would expect this value to be reliable.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1760420057126-ipbg46x7m" data-list-item-path="item-1760420057126-ipbg46x7m"><div class="list-item-paragraph">Explain whether the product moment correlation coefficient would differ if Tom converted the speed to kilometres per hour.<span class="inline-marks">[1]</span></div></li></ol>

Tom records the maximum speeds of an athlete running in a temperature controlled environment. The table below shows the maximum speed, $y$ metres per second, recorded for various specific temperatures, $x$ degree Celsius.

Least-Squares Regression and Residuals

Choosing the Appropriate Regression Line

2023 SAJC P2 Q10

<p>The data below shows the average height, measured in feet, of cherry trees from age $1$ year to age $11$ years.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Age ($x$)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$1$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$2$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$3$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$5$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$7$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$8$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$10$</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$11$</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Average height ($y$)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$6$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$13$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$15$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$16.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$17.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$18.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.5$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.7$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$19.8$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate $r$, the value of the product moment correlation coefficient between $x$ and $y$. State, with a reason, whether the value of $r$ would be different if the average height is measured in meters instead.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Give a sketch of the scatter diagram for the data.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">It is desired to predict the average height of cheery trees beyond age $11$ years. Explain why neither a linear nor a quadratic model is likely to be appropriate.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is suggested that the average height $y$ can be modelled by the formula $y=a+b\ln x$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Find the equation of the regression line for the suggested model, and the product moment correlation coefficient between $\ln x$ and $y$. Estimate the average height of cherry trees age $5.5$ years, and comment on the reliability of your answer.<span class="inline-marks">[4]</span></div></li></ol>

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

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

2013 RI P2 Q9

Math Question Bank