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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.t_test"></a><a class="link" href="t_test.html" title="t-tests"><span class="emphasis"><em>t</em></span>-tests</a>
  28 </h2></div></div></div>
  29 <h4>
  30 <a name="math_toolkit.t_test.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.t_test.synopsis"></a></span><a class="link" href="t_test.html#math_toolkit.t_test.synopsis">Synopsis</a>
  32     </h4>
  33 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">t_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34
  35 <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span> <span class="special">{</span>
  36
  37 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  38 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</span> <span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">sample_mean</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">sample_variance</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">num_samples</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
  39
  40 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
  41 <span class="keyword">auto</span> <span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">begin</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">end</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">iterator_traits</span><span class="special">&lt;</span><span class="identifier">ForwardIterator</span><span class="special">&gt;::</span><span class="identifier">value_type</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
  42
  43 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</span>
  44 <span class="keyword">auto</span> <span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
  45
  46 <span class="special">}}}</span>
  47 </pre>
  48 <h4>
  49 <a name="math_toolkit.t_test.h1"></a>
  50       <span class="phrase"><a name="math_toolkit.t_test.background"></a></span><a class="link" href="t_test.html#math_toolkit.t_test.background">Background</a>
  51     </h4>
  52 <p>
  53       A one-sample <span class="emphasis"><em>t</em></span>-test attempts to answer the question "given
  54       a sample mean, is it likely that the population mean of my data is a certain
  55       value?" The test statistic is
  56     </p>
  57 <p>
  58       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/one_sample_t_test_statistic.svg" width="97" height="59"></object></span>
  59     </p>
  60 <p>
  61       where &#181;<sub>0</sub> is the assumed mean, <span class="emphasis"><em>s</em></span><sup>2</sup> is the sample variance,
  62       and <span class="emphasis"><em>n</em></span> is the number of samples. If the absolute value
  63       of the test statistic is large, then we have low confidence that the population
  64       mean is equal to &#181;<sub>0</sub>, and if the absolute value of the test statistic is small,
  65       we have high confidence. We now ask the question "what constitutes large
  66       and small in this context?"
  67     </p>
  68 <p>
  69       Under reasonable assumptions, the test statistic <span class="emphasis"><em>t</em></span> can
  70       be assumed to come from a Student's <span class="emphasis"><em>t</em></span>-distribution. Since
  71       we wish to know if the sample mean deviates from the true mean in either direction,
  72       the test is two-tailed. Hence the <span class="emphasis"><em>p</em></span>-value is straightforward
  73       to calculate from the Student's <span class="emphasis"><em>t</em></span>-distribution on <span class="emphasis"><em>n</em></span>
  74       - 1 degrees of freedom, but nonetheless it is convenient to have it computed
  75       here.
  76     </p>
  77 <p>
  78       An example usage is as follows:
  79     </p>
  80 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
  81 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
  82 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">statistics</span><span class="special">/</span><span class="identifier">t_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  83
  84 <span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
  85 <span class="identifier">std</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">gen</span><span class="special">{</span><span class="identifier">rd</span><span class="special">()};</span>
  86 <span class="identifier">std</span><span class="special">::</span><span class="identifier">normal_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">dis</span><span class="special">{</span><span class="number">0</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
  87 <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">(</span><span class="number">1024</span><span class="special">);</span>
  88 <span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&amp;</span> <span class="identifier">x</span> <span class="special">:</span> <span class="identifier">v</span><span class="special">)</span> <span class="special">{</span>
  89   <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">dis</span><span class="special">(</span><span class="identifier">gen</span><span class="special">);</span>
  90 <span class="special">}</span>
  91
  92 <span class="keyword">auto</span> <span class="special">[</span><span class="identifier">t</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">statistics</span><span class="special">::</span><span class="identifier">one_sample_t_test</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="number">0.0</span><span class="special">);</span>
  93 </pre>
  94 <p>
  95       The test statistic is the first element of the pair, and the <span class="emphasis"><em>p</em></span>-value
  96       is the second element.
  97     </p>
  98 <h4>
  99 <a name="math_toolkit.t_test.h2"></a>
 100       <span class="phrase"><a name="math_toolkit.t_test.performance"></a></span><a class="link" href="t_test.html#math_toolkit.t_test.performance">Performance</a>
 101     </h4>
 102 <p>
 103       There are two cases: Where the mean and sample variance have already been computed,
 104       and the case where the mean and sample variance must be computed on the fly.
 105     </p>
 106 <pre class="programlisting"><span class="special">----------------------------------------------</span>
 107 <span class="identifier">Benchmark</span>                                <span class="identifier">Time</span>
 108 <span class="special">----------------------------------------------</span>
 109 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>               <span class="number">291</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">210.058</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 110 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>             <span class="number">1064</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">114.697</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 111 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>              <span class="number">407</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">599.213</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 112 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>              <span class="number">595</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">821.086</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 113 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>            <span class="number">1475</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">662.071</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 114 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>            <span class="number">1746</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1118.85</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 115 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>            <span class="number">3303</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.15492</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 116 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>           <span class="number">6404</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.19139</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 117 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>          <span class="number">12461</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.2245</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 118 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>          <span class="number">24805</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23029</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 119 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>          <span class="number">49639</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.22956</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 120 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>         <span class="number">98685</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23698</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 121 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>        <span class="number">197434</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23656</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 122 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>        <span class="number">393929</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23952</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 123 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>       <span class="number">790967</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23466</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 124 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>      <span class="number">1582366</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.23434</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 125 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>      <span class="number">3141112</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24358</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 126 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>     <span class="number">6260407</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24792</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 127 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>    <span class="number">12521811</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24784</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 128 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span>    <span class="number">25076257</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24619</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 129 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span>    <span class="number">50226183</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.2444</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 130 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>  <span class="number">100522789</span> <span class="identifier">ns</span> <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">1.24353</span><span class="identifier">G</span><span class="special">/</span><span class="identifier">s</span>
 131 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>           <span class="number">5.99</span> <span class="identifier">N</span>
 132 <span class="identifier">OneSampleTTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span>               <span class="number">0</span> <span class="special">%</span>
 133 <span class="identifier">OneSampleTTestKnownMeanAndVariance</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span>        <span class="number">207</span> <span class="identifier">ns</span>
 134 </pre>
 135 </div>
 136 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 137 <td align="left"></td>
 138 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 139       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 140       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 141       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 142       Daryle Walker and Xiaogang Zhang<p>
 143         Distributed under the Boost Software License, Version 1.0. (See accompanying
 144         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>)
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