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      <text>39.1.8.1 Probability Intervals - General Normal Distribution</text>
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      <text><![CDATA[<h3>Probability Intervals - General Normal Distribution</h3>

<p>We saw in a previous subsection that \(95\%\) of the area under the standard normal curve lay between \(z_1=-1.96\) and \(z_2=1.96\). Using the formula \(Z=\dfrac{X-\mu}{\sigma}\) in the re-arrangement \(X=\mu+Z\sigma\), we can see that \(95\%\) of the area under the general normal curve lies between \(x_1=\mu-1.96\sigma\) and \(x_2=\mu+1.96\sigma\). This is shown in the Figure below.</p><p></p><center><img src="@@PLUGINFILE@@/39-1-fig-14-normal95confinterval.svg" alt="The Normal distribution curve with 95% of the area shaded in between x-values, x1 = mu - 1.96 times sigma, and x2 = mu + 1.96 times sigma." width="244" height="107" class="img-responsive atto_image_button_text-bottom"></center>
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<h3 class="HELM_example">Example</h3>
<p>Suppose that the internal diameters of mass-produced pipes are normally distributed with mean \(50\) mm and standard deviation \(2\) mm. What are the \(95\%\) probability limits on the internal diameter of a single pipe?</p> 
<h3 class="HELM_solution">Solution</h3>
<p>Here \(\mu=50\) and \(\sigma=2\) so that the \(95\%\) probability limits are</p>
<p>\(\qquad 50\pm 1.96\times 2 = 50\pm 3.92\) mm,</p>
<p>i.e. \(46.08\) mm and \(53.92\) mm.</p>
<p>Hence the probability interval is \((46.08, 53.92)\).</p>]]></text>
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