students_t_example1.cpp

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// students_t_example1.cpp

// Copyright Paul A. Bristow 2006, 2007.

// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt
// or copy at http://www.boost.org/LICENSE_1_0.txt)

// Example 1 of using Student's t

// http://en.wikipedia.org/wiki/Student's_t-test  says:
// The t statistic was invented by William Sealy Gosset
// for cheaply monitoring the quality of beer brews.
// "Student" was his pen name.
// WS Gosset was statistician for Guinness brewery in Dublin, Ireland,
// hired due to Claude Guinness's innovative policy of recruiting the
// best graduates from Oxford and Cambridge for applying biochemistry
// and statistics to Guinness's industrial processes.
// Gosset published the t test in Biometrika in 1908,
// but was forced to use a pen name by his employer who regarded the fact
// that they were using statistics as a trade secret.
// In fact, Gosset's identity was unknown not only to fellow statisticians
// but to his employer - the company insisted on the pseudonym
// so that it could turn a blind eye to the breach of its rules.

// Data for this example from:
// P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64.
// from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55
// J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907

// Determination of mercury by cold-vapour atomic absorption,
// the following values were obtained fusing a trusted
// Standard Reference Material containing 38.9% mercury,
// which we assume is correct or 'true'.
double standard = 38.9;

const int values = 3;
double value[values] = {38.9, 37.4, 37.1};

// Is there any evidence for systematic error?

// The Students't distribution function is described at
// http://en.wikipedia.org/wiki/Student%27s_t_distribution
#include <boost/math/distributions/students_t.hpp>
   using boost::math::students_t;  // Probability of students_t(df, t).

#include <iostream>
   using std::cout;    using std::endl;
#include <iomanip>
   using std::setprecision;
#include <cmath>
   using std::sqrt;

int main()
{
  cout << "Example 1 using Student's t function. " << endl;

  // Example/test using tabulated value
  // (deliberately coded as naively as possible).

  // Null hypothesis is that there is no difference (greater or less)
  // between measured and standard.

  double degrees_of_freedom = values-1; // 3-1 = 2
  cout << "Measurement 1 = " << value[0] << ", measurement 2 = " << value[1] << ", measurement 3 = " << value[2] << endl;
  double mean = (value[0] + value[1] + value[2]) / static_cast<double>(values);
  cout << "Standard = " << standard << ", mean = " << mean << ", (mean - standard) = " << mean - standard  << endl;
  double sd = sqrt(((value[0] - mean) * (value[0] - mean) + (value[1] - mean) * (value[1] - mean) + (value[2] - mean) * (value[2] - mean))/ static_cast<double>(values-1));
  cout << "Standard deviation = " << sd << endl;
  if (sd == 0.)
  {
      cout << "Measured mean is identical to SRM value," << endl;
      cout << "so probability of no difference between measured and standard (the 'null hypothesis') is unity." << endl;
      return 0;
  }

  double t = (mean - standard) * std::sqrt(static_cast<double>(values)) / sd;
  cout << "Student's t = " << t << endl;
  cout.precision(2); // Useful accuracy is only a few decimal digits.
  cout << "Probability of Student's t is " << cdf(students_t(degrees_of_freedom), std::abs(t)) << endl;
  //  0.91, is 1 tailed.
  // So there is insufficient evidence of a difference to meet a 95% (1 in 20) criterion.

  return 0;
}  // int main()

/*

Output is:

Example 1 using Student's t function. 
Measurement 1 = 38.9, measurement 2 = 37.4, measurement 3 = 37.1
Standard = 38.9, mean = 37.8, (mean - standard) = -1.1
Standard deviation = 0.964365
Student's t = -1.97566
Probability of Student's t is 0.91

*/