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<div class="section">
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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.stat_tut.weg.f_eg"></a><a class="link" href="f_eg.html" title="F Distribution Examples">F Distribution Examples</a>
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</h4></div></div></div>
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<p>
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Imagine that you want to compare the standard deviations of two sample
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to determine if they differ in any significant way, in this situation you
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use the F distribution and perform an F-test. This situation commonly occurs
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when conducting a process change comparison: "is a new process more
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consistent that the old one?".
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</p>
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<p>
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In this example we'll be using the data for ceramic strength from <a href="http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm" target="_top">http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm</a>.
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The data for this case study were collected by Said Jahanmir of the NIST
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Ceramics Division in 1996 in connection with a NIST/industry ceramics consortium
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for strength optimization of ceramic strength.
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</p>
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<p>
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The example program is <a href="../../../../../example/f_test.cpp" target="_top">f_test.cpp</a>,
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program output has been deliberately made as similar as possible to the
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DATAPLOT output in the corresponding <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm" target="_top">NIST
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EngineeringStatistics Handbook example</a>.
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</p>
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<p>
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We'll begin by defining the procedure to conduct the test:
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</p>
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<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">f_test</span><span class="special">(</span>
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<span class="keyword">double</span> <span class="identifier">sd1</span><span class="special">,</span> <span class="comment">// Sample 1 std deviation</span>
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<span class="keyword">double</span> <span class="identifier">sd2</span><span class="special">,</span> <span class="comment">// Sample 2 std deviation</span>
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<span class="keyword">double</span> <span class="identifier">N1</span><span class="special">,</span> <span class="comment">// Sample 1 size</span>
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<span class="keyword">double</span> <span class="identifier">N2</span><span class="special">,</span> <span class="comment">// Sample 2 size</span>
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<span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">)</span> <span class="comment">// Significance level</span>
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<span class="special">{</span>
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</pre>
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<p>
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The procedure begins by printing out a summary of our input data:
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</p>
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<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
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<span class="comment">// Print header:</span>
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<span class="identifier">cout</span> <span class="special"><<</span>
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<span class="string">"____________________________________\n"</span>
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<span class="string">"F test for equal standard deviations\n"</span>
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<span class="string">"____________________________________\n\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Sample 1:\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Number of Observations"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">N1</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Sample Standard Deviation"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">sd1</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Sample 2:\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Number of Observations"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">N2</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Sample Standard Deviation"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">sd2</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span>
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</pre>
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<p>
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The test statistic for an F-test is simply the ratio of the square of the
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two standard deviations:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">F = s<sub>1</sub><sup>2</sup> / s<sub>2</sub><sup>2</sup></span>
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</p></blockquote></div>
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<p>
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where s<sub>1</sub> is the standard deviation of the first sample and s<sub>2</sub>
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is the standard
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deviation of the second sample. Or in code:
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</p>
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<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">F</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">sd1</span> <span class="special">/</span> <span class="identifier">sd2</span><span class="special">);</span>
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<span class="identifier">F</span> <span class="special">*=</span> <span class="identifier">F</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Test Statistic"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">F</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span>
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</pre>
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<p>
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At this point a word of caution: the F distribution is asymmetric, so we
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have to be careful how we compute the tests, the following table summarises
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the options available:
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Hypothesis
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</p>
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</th>
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<th>
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<p>
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Test
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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The null-hypothesis: there is no difference in standard deviations
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(two sided test)
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</p>
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</td>
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<td>
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<p>
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Reject if F <= F<sub>(1-alpha/2; N1-1, N2-1)</sub> or F >= F<sub>(alpha/2;
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N1-1, N2-1)</sub>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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The alternative hypothesis: there is a difference in means (two
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sided test)
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</p>
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</td>
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<td>
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<p>
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Reject if F<sub>(1-alpha/2; N1-1, N2-1)</sub> <= F <= F<sub>(alpha/2; N1-1,
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N2-1)</sub>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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The alternative hypothesis: Standard deviation of sample 1 is
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greater than that of sample 2
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</p>
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</td>
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<td>
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<p>
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Reject if F < F<sub>(alpha; N1-1, N2-1)</sub>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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The alternative hypothesis: Standard deviation of sample 1 is
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less than that of sample 2
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</p>
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</td>
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<td>
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<p>
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Reject if F > F<sub>(1-alpha; N1-1, N2-1)</sub>
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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<p>
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Where F<sub>(1-alpha; N1-1, N2-1)</sub> is the lower critical value of the F distribution
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with degrees of freedom N1-1 and N2-1, and F<sub>(alpha; N1-1, N2-1)</sub> is the upper
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critical value of the F distribution with degrees of freedom N1-1 and N2-1.
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</p>
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<p>
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The upper and lower critical values can be computed using the quantile
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function:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">F<sub>(1-alpha; N1-1, N2-1)</sub> = <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">fisher_f</span><span class="special">(</span><span class="identifier">N1</span><span class="special">-</span><span class="number">1</span><span class="special">,</span>
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<span class="identifier">N2</span><span class="special">-</span><span class="number">1</span><span class="special">),</span> <span class="identifier">alpha</span><span class="special">)</span></code></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">F<sub>(alpha; N1-1, N2-1)</sub> = <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">fisher_f</span><span class="special">(</span><span class="identifier">N1</span><span class="special">-</span><span class="number">1</span><span class="special">,</span>
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<span class="identifier">N2</span><span class="special">-</span><span class="number">1</span><span class="special">),</span> <span class="identifier">alpha</span><span class="special">))</span></code></span>
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</p></blockquote></div>
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<p>
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In our example program we need both upper and lower critical values for
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alpha and for alpha/2:
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</p>
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<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">ucv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">));</span>
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<span class="keyword">double</span> <span class="identifier">ucv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">));</span>
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<span class="keyword">double</span> <span class="identifier">lcv</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">);</span>
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<span class="keyword">double</span> <span class="identifier">lcv2</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">alpha</span> <span class="special">/</span> <span class="number">2</span><span class="special">);</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Upper Critical Value at alpha: "</span> <span class="special"><<</span> <span class="string">"= "</span>
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<span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">scientific</span> <span class="special"><<</span> <span class="identifier">ucv</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Upper Critical Value at alpha/2: "</span> <span class="special"><<</span> <span class="string">"= "</span>
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<span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">scientific</span> <span class="special"><<</span> <span class="identifier">ucv2</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Lower Critical Value at alpha: "</span> <span class="special"><<</span> <span class="string">"= "</span>
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<span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">scientific</span> <span class="special"><<</span> <span class="identifier">lcv</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Lower Critical Value at alpha/2: "</span> <span class="special"><<</span> <span class="string">"= "</span>
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<span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">scientific</span> <span class="special"><<</span> <span class="identifier">lcv2</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span>
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</pre>
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<p>
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The final step is to perform the comparisons given above, and print out
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|
whether the hypothesis is rejected or not:
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</p>
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<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">55</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span>
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|
<span class="string">"Results for Alternative Hypothesis and alpha"</span> <span class="special"><<</span> <span class="string">"= "</span>
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<span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">alpha</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Alternative Hypothesis Conclusion\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard deviations are unequal (two sided test) "</span><span class="special">;</span>
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<span class="keyword">if</span><span class="special">((</span><span class="identifier">ucv2</span> <span class="special"><</span> <span class="identifier">F</span><span class="special">)</span> <span class="special">||</span> <span class="special">(</span><span class="identifier">lcv2</span> <span class="special">></span> <span class="identifier">F</span><span class="special">))</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
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<span class="keyword">else</span>
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|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard deviation 1 is less than standard deviation 2 "</span><span class="special">;</span>
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<span class="keyword">if</span><span class="special">(</span><span class="identifier">lcv</span> <span class="special">></span> <span class="identifier">F</span><span class="special">)</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
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<span class="keyword">else</span>
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|
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Standard deviation 1 is greater than standard deviation 2 "</span><span class="special">;</span>
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<span class="keyword">if</span><span class="special">(</span><span class="identifier">ucv</span> <span class="special"><</span> <span class="identifier">F</span><span class="special">)</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"ACCEPTED\n"</span><span class="special">;</span>
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<span class="keyword">else</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"REJECTED\n"</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
|
|
</pre>
|
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<p>
|
|
Using the ceramic strength data as an example we get the following output:
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|
</p>
|
|
<pre class="programlisting">F test for equal standard deviations
|
|
____________________________________
|
|
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|
Sample 1:
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|
Number of Observations = 240
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|
Sample Standard Deviation = 65.549
|
|
|
|
Sample 2:
|
|
Number of Observations = 240
|
|
Sample Standard Deviation = 61.854
|
|
|
|
Test Statistic = 1.123
|
|
|
|
CDF of test statistic: = 8.148e-001
|
|
Upper Critical Value at alpha: = 1.238e+000
|
|
Upper Critical Value at alpha/2: = 1.289e+000
|
|
Lower Critical Value at alpha: = 8.080e-001
|
|
Lower Critical Value at alpha/2: = 7.756e-001
|
|
|
|
Results for Alternative Hypothesis and alpha = 0.0500
|
|
|
|
Alternative Hypothesis Conclusion
|
|
Standard deviations are unequal (two sided test) REJECTED
|
|
Standard deviation 1 is less than standard deviation 2 REJECTED
|
|
Standard deviation 1 is greater than standard deviation 2 REJECTED
|
|
</pre>
|
|
<p>
|
|
In this case we are unable to reject the null-hypothesis, and must instead
|
|
reject the alternative hypothesis.
|
|
</p>
|
|
<p>
|
|
By contrast let's see what happens when we use some different <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc32.htm" target="_top">sample
|
|
data</a>:, once again from the NIST Engineering Statistics Handbook:
|
|
A new procedure to assemble a device is introduced and tested for possible
|
|
improvement in time of assembly. The question being addressed is whether
|
|
the standard deviation of the new assembly process (sample 2) is better
|
|
(i.e., smaller) than the standard deviation for the old assembly process
|
|
(sample 1).
|
|
</p>
|
|
<pre class="programlisting">____________________________________
|
|
F test for equal standard deviations
|
|
____________________________________
|
|
|
|
Sample 1:
|
|
Number of Observations = 11.00000
|
|
Sample Standard Deviation = 4.90820
|
|
|
|
Sample 2:
|
|
Number of Observations = 9.00000
|
|
Sample Standard Deviation = 2.58740
|
|
|
|
Test Statistic = 3.59847
|
|
|
|
CDF of test statistic: = 9.589e-001
|
|
Upper Critical Value at alpha: = 3.347e+000
|
|
Upper Critical Value at alpha/2: = 4.295e+000
|
|
Lower Critical Value at alpha: = 3.256e-001
|
|
Lower Critical Value at alpha/2: = 2.594e-001
|
|
|
|
Results for Alternative Hypothesis and alpha = 0.0500
|
|
|
|
Alternative Hypothesis Conclusion
|
|
Standard deviations are unequal (two sided test) REJECTED
|
|
Standard deviation 1 is less than standard deviation 2 REJECTED
|
|
Standard deviation 1 is greater than standard deviation 2 ACCEPTED
|
|
</pre>
|
|
<p>
|
|
In this case we take our null hypothesis as "standard deviation 1
|
|
is less than or equal to standard deviation 2", since this represents
|
|
the "no change" situation. So we want to compare the upper critical
|
|
value at <span class="emphasis"><em>alpha</em></span> (a one sided test) with the test statistic,
|
|
and since 3.35 < 3.6 this hypothesis must be rejected. We therefore
|
|
conclude that there is a change for the better in our standard deviation.
|
|
</p>
|
|
</div>
|
|
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
|
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
|
Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
|
</p>
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