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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
- */
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