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<quiz>
  <question type="description">
    <name>
      <text>39.1.6.3 Applications of the Normal Distribution - Complex Examples</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h3>Applications</h3>
<hr>
<h4 class="HELM_example">Example</h4>
<p>Piston rings are mass-produced. The target internal diameter is \(45\) mm but
records show that the diameters are normally distributed with mean \(45\) mm and standard deviation \(0.05\) mm. An acceptable diameter is one within the range \(44.95\) mm to
\(45.05\) mm. What proportion of the output is unacceptable?</p> 
<h4 class="HELM_solution">Solution</h4>
<p>There are many words in the statement of the problem; we must read them
carefully to extract the necessary information. If \(X\) is the diameter of a piston ring, then  \(X\ \sim\ N (45,(0.05)^2)\).</p>
<p>The transformation is \(Z=\dfrac{X-\mu}{\sigma}=\dfrac{X-45}{0.05}.\)</p>
<p>The upper limit of acceptability is \(x_2=45.05\) so that \(z_2=(45.05 -45)/0.05=1.\)</p>
<p>The lower limit of acceptability is \(x_1=44.95\) so that \(z_1=(44.95-45)/0.05=-1\)</p>
<p>The range of `acceptable' \(Z\) values is therefore \(-1\) to \(1\), as shown in the Figure below.</p>
<center><img src="@@PLUGINFILE@@/39-1-fig-11-minus1and1.svg" alt="The Normal distribution curve with vertical lines drawn at z1=-1 and z2=1." width="229" height="78" class="img-responsive atto_image_button_text-bottom"><br></center><p></p>
<p>Using the symmetry of the curves, \(\text{P}(-1 &lt; Z &lt; 1) = 2\times \text{P}(0 &lt; Z &lt; 2) = 2 \times 0.3413 = 0.6826\).</p>
<p>Thus the proportion of unacceptable items is \(1-0.6826=0.3174\), or \(31.74\%\).</p>
<hr>

<h4 class="HELM_example">Example</h4>
<p>If the standard deviation is halved by improved production practices, what is now
the proportion of unacceptable items?</p> 
<h4 class="HELM_solution">Solution</h4>
<p>Now \(\sigma=0.025\) so that:</p>
<p>\(\qquad z_2=\dfrac{45.05-45}{0.025}=2\qquad\text{and}\qquad z_1=-2.\)</p>
<p>Then, \(\text{P}(-2 &lt; Z &lt; 2) = 2 \times \text{P}(0 &lt; Z &lt; 2)=2\times 0.4772=0.9544\). Hence the proportion of unacceptable items is reduced to \(1-0.9544=0.0456\) or \(4.56\%\).</p>
<p>We observe that less of the area under the curve now lies outside the interval \((44.95, 45.05)\).</p>
<p></p><center><img src="@@PLUGINFILE@@/39-1-fig-12-minus2and2.svg" alt="The normal distribution curve is shaded z=-2 and z=2." width="227" height="105" class="img-responsive atto_image_button_text-bottom"><br></center><p></p>]]></text>
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