| // 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 |
| |
| */ |
| |
| |