<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><thead><tr><th style="text-align: left; --cell-justify: flex-start">\(x\)</th><th style="text-align: left; --cell-justify: flex-start">\(2.0\)</th><th style="text-align: left; --cell-justify: flex-start">\(2.2\)</th><th style="text-align: left; --cell-justify: flex-start">\(2.5\)</th><th style="text-align: left; --cell-justify: flex-start">\(3.0\)</th><th style="text-align: left; --cell-justify: flex-start">\(4.0\)</th><th style="text-align: left; --cell-justify: flex-start">\(4.4\)</th><th style="text-align: left; --cell-justify: flex-start">\(4.5\)</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">\(y\)</td><td style="text-align: left; --cell-justify: flex-start">\(280\)</td><td style="text-align: left; --cell-justify: flex-start">\(250\)</td><td style="text-align: left; --cell-justify: flex-start">\(190\)</td><td style="text-align: left; --cell-justify: flex-start">\(150\)</td><td style="text-align: left; --cell-justify: flex-start">\(90\)</td><td style="text-align: left; --cell-justify: flex-start">\(100\)</td><td style="text-align: left; --cell-justify: flex-start">\(70\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><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><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 \(a\) and \(b\) are constants.</span><span class="inline-marks">[2]</span></div></li><li><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><div class="list-item-content"><span class="list-item-text">Use the model \(y=a+\frac{b}{x}\) with the values of \(a\) and \(b\) found in part (c) 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><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 \(y=a+\frac{b}{x}\).

Use the model \(y=a+\frac{b}{x}\) with the values of \(a\) and \(b\) found in part (c) 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.

2025 RI P2 Q8

2025 RI P2 Q8