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 <div class="titlepage"><div><div><h6 class="title">
 <a name="math_toolkit.dist.stat_tut.weg.st_eg.tut_mean_size"></a><a class="link" href="tut_mean_size.html" title="Estimating how large a sample size would have to become in order to give a significant Students-t test result with a single sample test">
             Estimating how large a sample size would have to become in order to give
             a significant Students-t test result with a single sample test</a>
 </h6></div></div></div>
 <p>
               Imagine you have conducted a Students-t test on a single sample in
               order to check for systematic errors in your measurements. Imagine
               that the result is borderline. At this point one might go off and collect
               more data, but it might be prudent to first ask the question "How
               much more?". The parameter estimators of the students_t_distribution
               class can provide this information.
             </p>
 <p>
               This section is based on the example code in <a href="../../../../../../../../example/students_t_single_sample.cpp" target="_top">students_t_single_sample.cpp</a>
               and we begin by defining a procedure that will print out a table of
               estimated sample sizes for various confidence levels:
             </p>
 <pre class="programlisting"><span class="comment">// Needed includes:
 </span><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">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iomanip</span><span class="special">&gt;</span>
 <span class="comment">// Bring everything into global namespace for ease of use:
 </span><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>
 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>

 <span class="keyword">void</span> <span class="identifier">single_sample_find_df</span><span class="special">(</span>
    <span class="keyword">double</span> <span class="identifier">M</span><span class="special">,</span>          <span class="comment">// M = true mean.
 </span>   <span class="keyword">double</span> <span class="identifier">Sm</span><span class="special">,</span>         <span class="comment">// Sm = Sample Mean.
 </span>   <span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">)</span>         <span class="comment">// Sd = Sample Standard Deviation.
 </span><span class="special">{</span>
 </pre>
 <p>
               Next we define a table of significance levels:
             </p>
 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
 </pre>
 <p>
               Printing out the table of sample sizes required for various confidence
               levels begins with the table header:
             </p>
 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span>
         <span class="string">"_______________________________________________________________\n"</span>
         <span class="string">"Confidence       Estimated          Estimated\n"</span>
         <span class="string">" Value (%)      Sample Size        Sample Size\n"</span>
         <span class="string">"              (one sided test)    (two sided test)\n"</span>
         <span class="string">"_______________________________________________________________\n"</span><span class="special">;</span>
 </pre>
 <p>
               And now the important part: the sample sizes required. Class <code class="computeroutput"><span class="identifier">students_t_distribution</span></code> has a static
               member function <code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>
               that will calculate how large a sample size needs to be in order to
               give a definitive result.
             </p>
 <p>
               The first argument is the difference between the means that you wish
               to be able to detect, here it's the absolute value of the difference
               between the sample mean, and the true mean.
             </p>
 <p>
               Then come two probability values: alpha and beta. Alpha is the maximum
               acceptable risk of rejecting the null-hypothesis when it is in fact
               true. Beta is the maximum acceptable risk of failing to reject the
               null-hypothesis when in fact it is false. Also note that for a two-sided
               test, alpha must be divided by 2.
             </p>
 <p>
               The final parameter of the function is the standard deviation of the
               sample.
             </p>
 <p>
               In this example, we assume that alpha and beta are the same, and call
               <code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>
               twice: once with alpha for a one-sided test, and once with alpha/2
               for a two-sided test.
             </p>
 <pre class="programlisting">   <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</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">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
    <span class="special">{</span>
       <span class="comment">// Confidence value:
 </span>      <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
       <span class="comment">// calculate df for single sided test:
 </span>      <span class="keyword">double</span> <span class="identifier">df</span> <span class="special">=</span> <span class="identifier">students_t</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
          <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">M</span> <span class="special">-</span> <span class="identifier">Sm</span><span class="special">),</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">Sd</span><span class="special">);</span>
       <span class="comment">// convert to sample size:
 </span>      <span class="keyword">double</span> <span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
       <span class="comment">// Print size:
 </span>      <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">size</span><span class="special">;</span>
       <span class="comment">// calculate df for two sided test:
 </span>      <span class="identifier">df</span> <span class="special">=</span> <span class="identifier">students_t</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
          <span class="identifier">fabs</span><span class="special">(</span><span class="identifier">M</span> <span class="special">-</span> <span class="identifier">Sm</span><span class="special">),</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">Sd</span><span class="special">);</span>
       <span class="comment">// convert to sample size:
 </span>      <span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
       <span class="comment">// Print size:
 </span>      <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">size</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
    <span class="special">}</span>
    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 <span class="special">}</span>
 </pre>
 <p>
               Let's now look at some sample output using data taken from <span class="emphasis"><em>P.K.Hou,
               O. W. Lau &amp; M.C. Wong, Analyst (1983) vol. 108, p 64. and from
               Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C.
               Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907.</em></span>
               The values result from the determination of mercury by cold-vapour
               atomic absorption.
             </p>
 <p>
               Only three measurements were made, and the Students-t test above gave
               a borderline result, so this example will show us how many samples
               would need to be collected:
             </p>
 <pre class="programlisting">_____________________________________________________________
 Estimated sample sizes required for various confidence levels
 _____________________________________________________________

 True Mean                               =  38.90000
 Sample Mean                             =  37.80000
 Sample Standard Deviation               =  0.96437


 _______________________________________________________________
 Confidence       Estimated          Estimated
  Value (%)      Sample Size        Sample Size
               (one sided test)    (two sided test)
 _______________________________________________________________
     75.000               3               4
     90.000               7               9
     95.000              11              13
     99.000              20              22
     99.900              35              37
     99.990              50              53
     99.999              66              68
 </pre>
 <p>
               So in this case, many more measurements would have had to be made,
               for example at the 95% level, 14 measurements in total for a two-sided
               test.
             </p>
 </div>
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       Thijs van den Berg<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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	<div class="section" lang="en">
	<div class="titlepage"><div><div><h6 class="title">
	<a name="math_toolkit.dist.stat_tut.weg.st_eg.tut_mean_size"></a><a class="link" href="tut_mean_size.html" title="Estimating how large a sample size would have to become in order to give a significant Students-t test result with a single sample test">
	Estimating how large a sample size would have to become in order to give
	a significant Students-t test result with a single sample test</a>
	</h6></div></div></div>
	<p>
	Imagine you have conducted a Students-t test on a single sample in
	order to check for systematic errors in your measurements. Imagine
	that the result is borderline. At this point one might go off and collect
	more data, but it might be prudent to first ask the question "How
	much more?". The parameter estimators of the students_t_distribution
	class can provide this information.
	</p>
	<p>
	This section is based on the example code in <a href="../../../../../../../../example/students_t_single_sample.cpp" target="_top">students_t_single_sample.cpp</a>
	and we begin by defining a procedure that will print out a table of
	estimated sample sizes for various confidence levels:
	</p>
	<pre class="programlisting"><span class="comment">// Needed includes:
	</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">distributions</span><span class="special">/</span><span class="identifier">students_t</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
	<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span>
	<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span>
	<span class="comment">// Bring everything into global namespace for ease of use:
	</span><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>
	<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>

	<span class="keyword">void</span> <span class="identifier">single_sample_find_df</span><span class="special">(</span>
	<span class="keyword">double</span> <span class="identifier">M</span><span class="special">,</span> <span class="comment">// M = true mean.
	</span> <span class="keyword">double</span> <span class="identifier">Sm</span><span class="special">,</span> <span class="comment">// Sm = Sample Mean.
	</span> <span class="keyword">double</span> <span class="identifier">Sd</span><span class="special">)</span> <span class="comment">// Sd = Sample Standard Deviation.
	</span><span class="special">{</span>
	</pre>
	<p>
	Next we define a table of significance levels:
	</p>
	<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
	</pre>
	<p>
	Printing out the table of sample sizes required for various confidence
	levels begins with the table header:
	</p>
	<pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n\n"</span>
	<span class="string">"_______________________________________________________________\n"</span>
	<span class="string">"Confidence Estimated Estimated\n"</span>
	<span class="string">" Value (%) Sample Size Sample Size\n"</span>
	<span class="string">" (one sided test) (two sided test)\n"</span>
	<span class="string">"_______________________________________________________________\n"</span><span class="special">;</span>
	</pre>
	<p>
	And now the important part: the sample sizes required. Class <code class="computeroutput"><span class="identifier">students_t_distribution</span></code> has a static
	member function <code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>
	that will calculate how large a sample size needs to be in order to
	give a definitive result.
	</p>
	<p>
	The first argument is the difference between the means that you wish
	to be able to detect, here it's the absolute value of the difference
	between the sample mean, and the true mean.
	</p>
	<p>
	Then come two probability values: alpha and beta. Alpha is the maximum
	acceptable risk of rejecting the null-hypothesis when it is in fact
	true. Beta is the maximum acceptable risk of failing to reject the
	null-hypothesis when in fact it is false. Also note that for a two-sided
	test, alpha must be divided by 2.
	</p>
	<p>
	The final parameter of the function is the standard deviation of the
	sample.
	</p>
	<p>
	In this example, we assume that alpha and beta are the same, and call
	<code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>
	twice: once with alpha for a one-sided test, and once with alpha/2
	for a two-sided test.
	</p>
	<pre class="programlisting"> <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</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="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
	<span class="special">{</span>
	<span class="comment">// Confidence value:
	</span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <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">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
	<span class="comment">// calculate df for single sided test:
	</span> <span class="keyword">double</span> <span class="identifier">df</span> <span class="special">=</span> <span class="identifier">students_t</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
	<span class="identifier">fabs</span><span class="special">(</span><span class="identifier">M</span> <span class="special">-</span> <span class="identifier">Sm</span><span class="special">),</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">Sd</span><span class="special">);</span>
	<span class="comment">// convert to sample size:
	</span> <span class="keyword">double</span> <span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
	<span class="comment">// Print size:
	</span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">size</span><span class="special">;</span>
	<span class="comment">// calculate df for two sided test:
	</span> <span class="identifier">df</span> <span class="special">=</span> <span class="identifier">students_t</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
	<span class="identifier">fabs</span><span class="special">(</span><span class="identifier">M</span> <span class="special">-</span> <span class="identifier">Sm</span><span class="special">),</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]/</span><span class="number">2</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">Sd</span><span class="special">);</span>
	<span class="comment">// convert to sample size:
	</span> <span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
	<span class="comment">// Print size:
	</span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">size</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
	<span class="special">}</span>
	<span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
	<span class="special">}</span>
	</pre>
	<p>
	Let's now look at some sample output using data taken from <span class="emphasis"><em>P.K.Hou,
	O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64. and from
	Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55 J. C.
	Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907.</em></span>
	The values result from the determination of mercury by cold-vapour
	atomic absorption.
	</p>
	<p>
	Only three measurements were made, and the Students-t test above gave
	a borderline result, so this example will show us how many samples
	would need to be collected:
	</p>
	<pre class="programlisting">_____________________________________________________________
	Estimated sample sizes required for various confidence levels
	_____________________________________________________________

	True Mean = 38.90000
	Sample Mean = 37.80000
	Sample Standard Deviation = 0.96437


	_______________________________________________________________
	Confidence Estimated Estimated
	Value (%) Sample Size Sample Size
	(one sided test) (two sided test)
	_______________________________________________________________
	75.000 3 4
	90.000 7 9
	95.000 11 13
	99.000 20 22
	99.900 35 37
	99.990 50 53
	99.999 66 68
	</pre>
	<p>
	So in this case, many more measurements would have had to be made,
	for example at the 95% level, 14 measurements in total for a two-sided
	test.
	</p>
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	<td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow,
	Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and
	Thijs van den Berg<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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