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

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

Sketch a scatter plot of the above data.	

One of the values of $h$ appears to be incorrect. Indicate the corresponding point on your diagram by labelling it $P$.

Omit $P$ for the remainder of this question.

Calculate the product moment correlation coefficient for this set of data. Use the equation of an appropriate regression line to predict the value of $s$ when $h=100$, justifying your choice of regression line. 

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

Determine which of the two models is a better choice, giving a reason for your answer.	

Suppose a new data pair $\left( \overline{s},\overline{h} \right)$ is added to the table above, where $\overline{s}$ and $\overline{h}$  are the patient’s sample mean walking speed (in km/h) and his sample mean heart-beat rate (in bpm) respectively, based on the data above. Without any calculations, explain whether the equation of the regression line you have obtained in part **(iii) ** would change.

Hypothesis Testing

International Baccalaureate Higher Level Mathematics

2015 NJC P2 Q12

2015 NJC P2 Q12