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  <question type="description">
    <name>
      <text>43.1.2.3 Eg1</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h4 class="HELM_example">Example</h4>
<p>A warehouse manager of a company dealing in large quantities of steel cable needs to be able to estimate how much cable is left on his partially used drums. A random sample of twelve partially used drums is taken and each drum is weighed and the corresponding length of cable measured. The results are given in the table below:</p>
<table border="1"><caption>&nbsp;</caption>
<thead>
<tr>
<th scope="col">Weight of drum and cable \( (x) \) kg.</th>
<th scope="col">Measured length of cable \( (y) \) m.</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: center;">30</td>
<td style="text-align: center;">70</td>
</tr>
<tr>
<td style="text-align: center;">40</td>
<td style="text-align: center;">90</td>
</tr>
<tr>
<td style="text-align: center;">40</td>
<td style="text-align: center;">100</td>
</tr>
<tr>
<td style="text-align: center;">50</td>
<td style="text-align: center;">120</td>
</tr>
<tr>
<td style="text-align: center;">50</td>
<td style="text-align: center;">130</td>
</tr>
<tr>
<td style="text-align: center;">50</td>
<td style="text-align: center;">150</td>
</tr>
<tr>
<td style="text-align: center;">60</td>
<td style="text-align: center;">160</td>
</tr>
<tr>
<td style="text-align: center;">70</td>
<td style="text-align: center;">190</td>
</tr>
<tr>
<td style="text-align: center;">70</td>
<td style="text-align: center;">200</td>
</tr>
<tr>
<td style="text-align: center;">80</td>
<td style="text-align: center;">200</td>
</tr>
<tr>
<td style="text-align: center;">80</td>
<td style="text-align: center;">220</td>
</tr>
<tr>
<td style="text-align: center;">80</td>
<td style="text-align: center;">230</td>
</tr>
</tbody>
</table>
<p><br><br></p>
Find the least squares regression line in the form \(y = mx+c\) and use it to predict the lengths of cable left on drums whose weights are:
<p></p>
<ol class="HELM_parts_inline">
<li>\(35\) <b>kg</b></li>
<li>\(85\) <b>kg</b></li>
<li>\(100\) <b>kg</b></li>
</ol>
In the latter case state any assumptions which you make in order to find the length
of cable left on the drum.
<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
<p>Excel calculations give
\( \sum x = 700\), \( \sum  x^2 = 44200 \), \( \sum y = 1860 \), \( \sum xy = 118600\),  so that
the formulae
\[ b = \dfrac{ \dfrac{ \sum xy}{n} - \dfrac{ \sum x}{n}\dfrac{ \sum y}{n}  }{ \dfrac{\sum x^2}{n} - \left( \dfrac{\sum x}{n} \right)^2 }, \text{   and   } a = \bar{y}  - b \bar{x} \]
give \(a  = -20\) and \(b = 3\). Our regression line is \(y = -20 + 3x\), so \( y = 3x - 20 \). 

Hence, the required predicted values are 
</p><li> \[ y_{35} = 3 \times 35 - 20 = 85 \] </li> <p></p>
<p></p><li> \[ y_{85} = 3 \times 85 - 20 = 235 \] </li> <p></p>
<p></p><li> \[ y_{100} = 3 \times 100 - 20 = 280 \] </li> <p></p>
all results being in metres.

To obtain the last result we have assumed that the linearity of the relationship continues beyond
the range of values actually taken.</ol>]]></text>
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