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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"><</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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Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani
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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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