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// Copyright John Maddock 2006
// Copyright Paul A. Bristow 2007, 2010
// 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)
#ifdef _MSC_VER
# pragma warning(disable: 4512) // assignment operator could not be generated.
# pragma warning(disable: 4510) // default constructor could not be generated.
# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required.
#endif
#include <boost/math/distributions/students_t.hpp>
// avoid "using namespace std;" and "using namespace boost::math;"
// to avoid potential ambiguity with names in std random.
#include <iostream>
using std::cout; using std::endl;
using std::left; using std::fixed; using std::right; using std::scientific;
#include <iomanip>
using std::setw;
using std::setprecision;
void confidence_limits_on_mean(double Sm, double Sd, unsigned Sn)
{
//
// Sm = Sample Mean.
// Sd = Sample Standard Deviation.
// Sn = Sample Size.
//
// Calculate confidence intervals for the mean.
// For example if we set the confidence limit to
// 0.95, we know that if we repeat the sampling
// 100 times, then we expect that the true mean
// will be between out limits on 95 occasions.
// Note: this is not the same as saying a 95%
// confidence interval means that there is a 95%
// probability that the interval contains the true mean.
// The interval computed from a given sample either
// contains the true mean or it does not.
// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm
using boost::math::students_t;
// Print out general info:
cout <<
"__________________________________\n"
"2-Sided Confidence Limits For Mean\n"
"__________________________________\n\n";
cout << setprecision(7);
cout << setw(40) << left << "Number of Observations" << "= " << Sn << "\n";
cout << setw(40) << left << "Mean" << "= " << Sm << "\n";
cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n";
//
// Define a table of significance/risk levels:
//
double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
//
// Start by declaring the distribution we'll need:
//
students_t dist(Sn - 1);
//
// Print table header:
//
cout << "\n\n"
"_______________________________________________________________\n"
"Confidence T Interval Lower Upper\n"
" Value (%) Value Width Limit Limit\n"
"_______________________________________________________________\n";
//
// Now print out the data for the table rows.
//
for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
{
// Confidence value:
cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
// calculate T:
double T = quantile(complement(dist, alpha[i] / 2));
// Print T:
cout << fixed << setprecision(3) << setw(10) << right << T;
// Calculate width of interval (one sided):
double w = T * Sd / sqrt(double(Sn));
// Print width:
if(w < 0.01)
cout << scientific << setprecision(3) << setw(17) << right << w;
else
cout << fixed << setprecision(3) << setw(17) << right << w;
// Print Limits:
cout << fixed << setprecision(5) << setw(15) << right << Sm - w;
cout << fixed << setprecision(5) << setw(15) << right << Sm + w << endl;
}
cout << endl;
} // void confidence_limits_on_mean
void single_sample_t_test(double M, double Sm, double Sd, unsigned Sn, double alpha)
{
//
// M = true mean.
// Sm = Sample Mean.
// Sd = Sample Standard Deviation.
// Sn = Sample Size.
// alpha = Significance Level.
//
// A Students t test applied to a single set of data.
// We are testing the null hypothesis that the true
// mean of the sample is M, and that any variation is down
// to chance. We can also test the alternative hypothesis
// that any difference is not down to chance.
// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm
using boost::math::students_t;
// Print header:
cout <<
"__________________________________\n"
"Student t test for a single sample\n"
"__________________________________\n\n";
cout << setprecision(5);
cout << setw(55) << left << "Number of Observations" << "= " << Sn << "\n";
cout << setw(55) << left << "Sample Mean" << "= " << Sm << "\n";
cout << setw(55) << left << "Sample Standard Deviation" << "= " << Sd << "\n";
cout << setw(55) << left << "Expected True Mean" << "= " << M << "\n\n";
//
// Now we can calculate and output some stats:
//
// Difference in means:
double diff = Sm - M;
cout << setw(55) << left << "Sample Mean - Expected Test Mean" << "= " << diff << "\n";
// Degrees of freedom:
unsigned v = Sn - 1;
cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n";
// t-statistic:
double t_stat = diff * sqrt(double(Sn)) / Sd;
cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n";
//
// Finally define our distribution, and get the probability:
//
students_t dist(v);
double q = cdf(complement(dist, fabs(t_stat)));
cout << setw(55) << left << "Probability that difference is due to chance" << "= "
<< setprecision(3) << scientific << 2 * q << "\n\n";
//
// Finally print out results of alternative hypothesis:
//
cout << setw(55) << left <<
"Results for Alternative Hypothesis and alpha" << "= "
<< setprecision(4) << fixed << alpha << "\n\n";
cout << "Alternative Hypothesis Conclusion\n";
cout << "Mean != " << setprecision(3) << fixed << M << " ";
if(q < alpha / 2)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << "Mean < " << setprecision(3) << fixed << M << " ";
if(cdf(complement(dist, t_stat)) > alpha)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << "Mean > " << setprecision(3) << fixed << M << " ";
if(cdf(dist, t_stat) > alpha)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << endl << endl;
} // void single_sample_t_test(
void single_sample_find_df(double M, double Sm, double Sd)
{
//
// M = true mean.
// Sm = Sample Mean.
// Sd = Sample Standard Deviation.
//
using boost::math::students_t;
// Print out general info:
cout <<
"_____________________________________________________________\n"
"Estimated sample sizes required for various confidence levels\n"
"_____________________________________________________________\n\n";
cout << setprecision(5);
cout << setw(40) << left << "True Mean" << "= " << M << "\n";
cout << setw(40) << left << "Sample Mean" << "= " << Sm << "\n";
cout << setw(40) << left << "Sample Standard Deviation" << "= " << Sd << "\n";
//
// Define a table of significance intervals:
//
double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
//
// Print table header:
//
cout << "\n\n"
"_______________________________________________________________\n"
"Confidence Estimated Estimated\n"
" Value (%) Sample Size Sample Size\n"
" (one sided test) (two sided test)\n"
"_______________________________________________________________\n";
//
// Now print out the data for the table rows.
//
for(unsigned i = 1; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
{
// Confidence value:
cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
// calculate df for single sided test:
double df = students_t::find_degrees_of_freedom(
fabs(M - Sm), alpha[i], alpha[i], Sd);
// convert to sample size, always one more than the degrees of freedom:
double size = ceil(df) + 1;
// Print size:
cout << fixed << setprecision(0) << setw(16) << right << size;
// calculate df for two sided test:
df = students_t::find_degrees_of_freedom(
fabs(M - Sm), alpha[i]/2, alpha[i], Sd);
// convert to sample size:
size = ceil(df) + 1;
// Print size:
cout << fixed << setprecision(0) << setw(16) << right << size << endl;
}
cout << endl;
} // void single_sample_find_df
int main()
{
//
// Run tests for Heat Flow Meter data
// see http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm
// The data was collected while calibrating a heat flow meter
// against a known value.
//
confidence_limits_on_mean(9.261460, 0.2278881e-01, 195);
single_sample_t_test(5, 9.261460, 0.2278881e-01, 195, 0.05);
single_sample_find_df(5, 9.261460, 0.2278881e-01);
//
// 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'.
//
confidence_limits_on_mean(37.8, 0.964365, 3);
// 95% test:
single_sample_t_test(38.9, 37.8, 0.964365, 3, 0.05);
// 90% test:
single_sample_t_test(38.9, 37.8, 0.964365, 3, 0.1);
// parameter estimate:
single_sample_find_df(38.9, 37.8, 0.964365);
return 0;
} // int main()
/*
Output:
------ Rebuild All started: Project: students_t_single_sample, Configuration: Release Win32 ------
students_t_single_sample.cpp
Generating code
Finished generating code
students_t_single_sample.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Release\students_t_single_sample.exe
__________________________________
2-Sided Confidence Limits For Mean
__________________________________
Number of Observations = 195
Mean = 9.26146
Standard Deviation = 0.02278881
_______________________________________________________________
Confidence T Interval Lower Upper
Value (%) Value Width Limit Limit
_______________________________________________________________
50.000 0.676 1.103e-003 9.26036 9.26256
75.000 1.154 1.883e-003 9.25958 9.26334
90.000 1.653 2.697e-003 9.25876 9.26416
95.000 1.972 3.219e-003 9.25824 9.26468
99.000 2.601 4.245e-003 9.25721 9.26571
99.900 3.341 5.453e-003 9.25601 9.26691
99.990 3.973 6.484e-003 9.25498 9.26794
99.999 4.537 7.404e-003 9.25406 9.26886
__________________________________
Student t test for a single sample
__________________________________
Number of Observations = 195
Sample Mean = 9.26146
Sample Standard Deviation = 0.02279
Expected True Mean = 5.00000
Sample Mean - Expected Test Mean = 4.26146
Degrees of Freedom = 194
T Statistic = 2611.28380
Probability that difference is due to chance = 0.000e+000
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Mean != 5.000 NOT REJECTED
Mean < 5.000 REJECTED
Mean > 5.000 NOT REJECTED
_____________________________________________________________
Estimated sample sizes required for various confidence levels
_____________________________________________________________
True Mean = 5.00000
Sample Mean = 9.26146
Sample Standard Deviation = 0.02279
_______________________________________________________________
Confidence Estimated Estimated
Value (%) Sample Size Sample Size
(one sided test) (two sided test)
_______________________________________________________________
75.000 2 2
90.000 2 2
95.000 2 2
99.000 2 2
99.900 3 3
99.990 3 3
99.999 4 4
__________________________________
2-Sided Confidence Limits For Mean
__________________________________
Number of Observations = 3
Mean = 37.8000000
Standard Deviation = 0.9643650
_______________________________________________________________
Confidence T Interval Lower Upper
Value (%) Value Width Limit Limit
_______________________________________________________________
50.000 0.816 0.455 37.34539 38.25461
75.000 1.604 0.893 36.90717 38.69283
90.000 2.920 1.626 36.17422 39.42578
95.000 4.303 2.396 35.40438 40.19562
99.000 9.925 5.526 32.27408 43.32592
99.900 31.599 17.594 20.20639 55.39361
99.990 99.992 55.673 -17.87346 93.47346
99.999 316.225 176.067 -138.26683 213.86683
__________________________________
Student t test for a single sample
__________________________________
Number of Observations = 3
Sample Mean = 37.80000
Sample Standard Deviation = 0.96437
Expected True Mean = 38.90000
Sample Mean - Expected Test Mean = -1.10000
Degrees of Freedom = 2
T Statistic = -1.97566
Probability that difference is due to chance = 1.869e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Mean != 38.900 REJECTED
Mean < 38.900 NOT REJECTED
Mean > 38.900 NOT REJECTED
__________________________________
Student t test for a single sample
__________________________________
Number of Observations = 3
Sample Mean = 37.80000
Sample Standard Deviation = 0.96437
Expected True Mean = 38.90000
Sample Mean - Expected Test Mean = -1.10000
Degrees of Freedom = 2
T Statistic = -1.97566
Probability that difference is due to chance = 1.869e-001
Results for Alternative Hypothesis and alpha = 0.1000
Alternative Hypothesis Conclusion
Mean != 38.900 REJECTED
Mean < 38.900 NOT REJECTED
Mean > 38.900 REJECTED
_____________________________________________________________
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
*/
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