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   4 <title>Estimating Sample Sizes for a Binomial Distribution.</title>
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  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.stat_tut.weg.binom_eg.binom_size_eg"></a><a class="link" href="binom_size_eg.html" title="Estimating Sample Sizes for a Binomial Distribution.">Estimating
  28           Sample Sizes for a Binomial Distribution.</a>
  29 </h5></div></div></div>
  30 <p>
  31             Imagine you have a critical component that you know will fail in 1 in
  32             N "uses" (for some suitable definition of "use").
  33             You may want to schedule routine replacement of the component so that
  34             its chance of failure between routine replacements is less than P%. If
  35             the failures follow a binomial distribution (each time the component
  36             is "used" it either fails or does not) then the static member
  37             function <code class="computeroutput"><span class="identifier">binomial_distibution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_maximum_number_of_trials</span></code>
  38             can be used to estimate the maximum number of "uses" of that
  39             component for some acceptable risk level <span class="emphasis"><em>alpha</em></span>.
  40           </p>
  41 <p>
  42             The example program <a href="../../../../../../example/binomial_sample_sizes.cpp" target="_top">binomial_sample_sizes.cpp</a>
  43             demonstrates its usage. It centres on a routine that prints out a table
  44             of maximum sample sizes for various probability thresholds:
  45           </p>
  46 <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">find_max_sample_size</span><span class="special">(</span>
  47    <span class="keyword">double</span> <span class="identifier">p</span><span class="special">,</span>              <span class="comment">// success ratio.</span>
  48    <span class="keyword">unsigned</span> <span class="identifier">successes</span><span class="special">)</span>    <span class="comment">// Total number of observed successes permitted.</span>
  49 <span class="special">{</span>
  50 </pre>
  51 <p>
  52             The routine then declares a table of probability thresholds: these are
  53             the maximum acceptable probability that <span class="emphasis"><em>successes</em></span>
  54             or fewer events will be observed. In our example, <span class="emphasis"><em>successes</em></span>
  55             will be always zero, since we want no component failures, but in other
  56             situations non-zero values may well make sense.
  57           </p>
  58 <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>
  59 </pre>
  60 <p>
  61             Much of the rest of the program is pretty-printing, the important part
  62             is in the calculation of maximum number of permitted trials for each
  63             value of alpha:
  64           </p>
  65 <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>
  66 <span class="special">{</span>
  67    <span class="comment">// Confidence value:</span>
  68    <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>
  69    <span class="comment">// calculate trials:</span>
  70    <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">binomial</span><span class="special">::</span><span class="identifier">find_maximum_number_of_trials</span><span class="special">(</span>
  71                   <span class="identifier">successes</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
  72    <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">t</span><span class="special">);</span>
  73    <span class="comment">// Print Trials:</span>
  74    <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">5</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</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">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  75 <span class="special">}</span>
  76 </pre>
  77 <p>
  78             Note that since we're calculating the maximum number of trials permitted,
  79             we'll err on the safe side and take the floor of the result. Had we been
  80             calculating the <span class="emphasis"><em>minimum</em></span> number of trials required
  81             to observe a certain number of <span class="emphasis"><em>successes</em></span> using
  82             <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code>
  83             we would have taken the ceiling instead.
  84           </p>
  85 <p>
  86             We'll finish off by looking at some sample output, firstly for a 1 in
  87             1000 chance of component failure with each use:
  88           </p>
  89 <pre class="programlisting">________________________
  90 Maximum Number of Trials
  91 ________________________
  92
  93 Success ratio                           =  0.001
  94 Maximum Number of "successes" permitted =  0
  95
  96
  97 ____________________________
  98 Confidence        Max Number
  99  Value (%)        Of Trials
 100 ____________________________
 101     50.000            692
 102     75.000            287
 103     90.000            105
 104     95.000             51
 105     99.000             10
 106     99.900              0
 107     99.990              0
 108     99.999              0
 109 </pre>
 110 <p>
 111             So 51 "uses" of the component would yield a 95% chance that
 112             no component failures would be observed.
 113           </p>
 114 <p>
 115             Compare that with a 1 in 1 million chance of component failure:
 116           </p>
 117 <pre class="programlisting">________________________
 118 Maximum Number of Trials
 119 ________________________
 120
 121 Success ratio                           =  0.0000010
 122 Maximum Number of "successes" permitted =  0
 123
 124
 125 ____________________________
 126 Confidence        Max Number
 127  Value (%)        Of Trials
 128 ____________________________
 129     50.000         693146
 130     75.000         287681
 131     90.000         105360
 132     95.000          51293
 133     99.000          10050
 134     99.900           1000
 135     99.990            100
 136     99.999             10
 137 </pre>
 138 <p>
 139             In this case, even 1000 uses of the component would still yield a less
 140             than 1 in 1000 chance of observing a component failure (i.e. a 99.9%
 141             chance of no failure).
 142           </p>
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 146 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 147       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 148       Holin, Bruno Lalande, John Maddock, Johan R&#229;de, Gautam Sewani, Benjamin Sobotta,
 149       Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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