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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.z_test"></a><a class="link" href="z_test.html" title="z-tests"><span class="emphasis"><em>z</em></span>-tests</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.z_test.h0"></a>
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<span class="phrase"><a name="math_toolkit.z_test.synopsis"></a></span><a class="link" href="z_test.html#math_toolkit.z_test.synopsis">Synopsis</a>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</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">z_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">statistics</span> <span class="special">{</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">one_sample_z_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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ForwardIterator</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</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"><</span><span class="identifier">ForwardIterator</span><span class="special">>::</span><span class="identifier">value_type</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">one_sample_z_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="identifier">Real</span> <span class="identifier">assumed_mean</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">one_sample_z_test</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ForwardIterator</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</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"><</span><span class="identifier">ForwardIterator</span><span class="special">>::</span><span class="identifier">value_type</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">two_sample_z_test</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">begin_1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">end_1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">begin_2</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">begin_2</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</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="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">two_sample_z_test</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ForwardIterator</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</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"><</span><span class="identifier">ForwardIterator</span><span class="special">>::</span><span class="identifier">value_type</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">paired_samples_z_test</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">begin_1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">end_1</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">begin_2</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">begin_2</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Real</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="special">></span>
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||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">paired_samples_z_test</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v</span><span class="special">);</span>
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<span class="special">}}}</span>
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</pre>
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<h4>
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<a name="math_toolkit.z_test.h1"></a>
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<span class="phrase"><a name="math_toolkit.z_test.background"></a></span><a class="link" href="z_test.html#math_toolkit.z_test.background">Background</a>
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</h4>
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<p>
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A set of C++11 compatible functions for various one sample and independent
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sample <span class="emphasis"><em>z</em></span>-tests. The input can be any real number or set
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of real numbers. In the event that the input is an integer or a set of integers
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typename Real will be deduced as a double precision type.
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</p>
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<h4>
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<a name="math_toolkit.z_test.h2"></a>
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||
<span class="phrase"><a name="math_toolkit.z_test.one_sample_z_test"></a></span><a class="link" href="z_test.html#math_toolkit.z_test.one_sample_z_test">One-sample
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<span class="emphasis"><em>z</em></span>-test</a>
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</h4>
|
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<p>
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A one-sample <span class="emphasis"><em>z</em></span>-test is used to determine whether two population
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means are different when the variances are known and the sample sizes are large.
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The <span class="emphasis"><em>z</em></span>-test is closely related to the <span class="emphasis"><em>t</em></span>-test
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but can be performed using a large sample size.
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/one_sample_z_test_statistic.svg"></object></span>
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</p>
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<p>
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where
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/one_sample_z_s_value.svg" width="51" height="34"></object></span>
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</p>
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<p>
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with <span class="emphasis"><em>X</em></span> being the test-statistic and µ<sub>0</sub> being the assumed
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mean
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</p>
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<p>
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||
the test statistic <span class="emphasis"><em>X</em></span> can be assumed to come from a uniform
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real distribution. Since we wish to know if the sample mean deviates from the
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true mean in either direction, the test is two-tailed. Hence the <span class="emphasis"><em>p</em></span>-value
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is straightforward to calculate from the uniform real distribution on <span class="emphasis"><em>n</em></span>
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- 1 degrees of freedom, but nonetheless it is convenient to have it computed
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here.
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||
</p>
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<p>
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An example usage is as follows:
|
||
</p>
|
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">vector</span><span class="special">></span>
|
||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">random</span><span class="special">></span>
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||
<span class="preprocessor">#include</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">z_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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||
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
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||
<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>
|
||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">normal_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">(</span><span class="number">1024</span><span class="special">);</span>
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<span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&</span> <span class="identifier">x</span> <span class="special">:</span> <span class="identifier">v</span><span class="special">)</span> <span class="special">{</span>
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<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>
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<span class="special">}</span>
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<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_z_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>
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||
</pre>
|
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<p>
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The test statistic is the first element of the pair, and the <span class="emphasis"><em>p</em></span>-value
|
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is the second element.
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</p>
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<h4>
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<a name="math_toolkit.z_test.h3"></a>
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<span class="phrase"><a name="math_toolkit.z_test.independent_two_sample_z_test"></a></span><a class="link" href="z_test.html#math_toolkit.z_test.independent_two_sample_z_test">Independent
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two-sample <span class="emphasis"><em>z</em></span>-test</a>
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</h4>
|
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<p>
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A two-sample <span class="emphasis"><em>z</em></span>-test determines if the means of two sets
|
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of data have a statistically significant difference from each other.
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</p>
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<p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/two_sample_z_statistic.svg"></object></span> An example of usage
|
||
is as follows:
|
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</p>
|
||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">vector</span><span class="special">></span>
|
||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">random</span><span class="special">></span>
|
||
<span class="preprocessor">#include</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">z_test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||
|
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
|
||
<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>
|
||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">normal_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></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>
|
||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">u</span><span class="special">(</span><span class="number">1024</span><span class="special">);</span>
|
||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">v</span><span class="special">(</span><span class="number">1024</span><span class="special">);</span>
|
||
<span class="keyword">for</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="identifier">u</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
|
||
<span class="special">{</span>
|
||
<span class="identifier">u</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">dis</span><span class="special">(</span><span class="identifier">gen</span><span class="special">);</span>
|
||
<span class="identifier">v</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">dis</span><span class="special">(</span><span class="identifier">gen</span><span class="special">);</span>
|
||
<span class="special">}</span>
|
||
|
||
<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">two_sample_z_test</span><span class="special">(</span><span class="identifier">u</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span>
|
||
</pre>
|
||
<p>
|
||
<span class="emphasis"><em>Nota bene:</em></span> The sample sizes for the two sets of data do
|
||
not need to be equal.
|
||
</p>
|
||
</div>
|
||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
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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>)
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</tr></table>
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