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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 <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;
#include <boost/math/distributions/students_t.hpp>
using boost::math::students_t;
void two_samples_t_test_equal_sd(
double Sm1, // Sm1 = Sample Mean 1.
double Sd1, // Sd1 = Sample Standard Deviation 1.
unsigned Sn1, // Sn1 = Sample Size 1.
double Sm2, // Sm2 = Sample Mean 2.
double Sd2, // Sd2 = Sample Standard Deviation 2.
unsigned Sn2, // Sn2 = Sample Size 2.
double alpha) // alpha = Significance Level.
{
// A Students t test applied to two sets of data.
// We are testing the null hypothesis that the two
// samples have the same mean and that any difference
// if due to chance.
// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm
//
using namespace std;
// using namespace boost::math;
using boost::math::students_t;
// Print header:
cout <<
"_______________________________________________\n"
"Student t test for two samples (equal variances)\n"
"_______________________________________________\n\n";
cout << setprecision(5);
cout << setw(55) << left << "Number of Observations (Sample 1)" << "= " << Sn1 << "\n";
cout << setw(55) << left << "Sample 1 Mean" << "= " << Sm1 << "\n";
cout << setw(55) << left << "Sample 1 Standard Deviation" << "= " << Sd1 << "\n";
cout << setw(55) << left << "Number of Observations (Sample 2)" << "= " << Sn2 << "\n";
cout << setw(55) << left << "Sample 2 Mean" << "= " << Sm2 << "\n";
cout << setw(55) << left << "Sample 2 Standard Deviation" << "= " << Sd2 << "\n";
//
// Now we can calculate and output some stats:
//
// Degrees of freedom:
double v = Sn1 + Sn2 - 2;
cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n";
// Pooled variance and hence standard deviation:
double sp = sqrt(((Sn1-1) * Sd1 * Sd1 + (Sn2-1) * Sd2 * Sd2) / v);
cout << setw(55) << left << "Pooled Standard Deviation" << "= " << sp << "\n";
// t-statistic:
double t_stat = (Sm1 - Sm2) / (sp * sqrt(1.0 / Sn1 + 1.0 / Sn2));
cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n";
//
// 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 << "Sample 1 Mean != Sample 2 Mean " ;
if(q < alpha / 2)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << "Sample 1 Mean < Sample 2 Mean ";
if(cdf(dist, t_stat) < alpha)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << "Sample 1 Mean > Sample 2 Mean ";
if(cdf(complement(dist, t_stat)) < alpha)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << endl << endl;
}
void two_samples_t_test_unequal_sd(
double Sm1, // Sm1 = Sample Mean 1.
double Sd1, // Sd1 = Sample Standard Deviation 1.
unsigned Sn1, // Sn1 = Sample Size 1.
double Sm2, // Sm2 = Sample Mean 2.
double Sd2, // Sd2 = Sample Standard Deviation 2.
unsigned Sn2, // Sn2 = Sample Size 2.
double alpha) // alpha = Significance Level.
{
// A Students t test applied to two sets of data.
// We are testing the null hypothesis that the two
// samples have the same mean and
// that any difference is due to chance.
// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm
//
using namespace std;
using boost::math::students_t;
// Print header:
cout <<
"_________________________________________________\n"
"Student t test for two samples (unequal variances)\n"
"_________________________________________________\n\n";
cout << setprecision(5);
cout << setw(55) << left << "Number of Observations (Sample 1)" << "= " << Sn1 << "\n";
cout << setw(55) << left << "Sample 1 Mean" << "= " << Sm1 << "\n";
cout << setw(55) << left << "Sample 1 Standard Deviation" << "= " << Sd1 << "\n";
cout << setw(55) << left << "Number of Observations (Sample 2)" << "= " << Sn2 << "\n";
cout << setw(55) << left << "Sample 2 Mean" << "= " << Sm2 << "\n";
cout << setw(55) << left << "Sample 2 Standard Deviation" << "= " << Sd2 << "\n";
//
// Now we can calculate and output some stats:
//
// Degrees of freedom:
double v = Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2;
v *= v;
double t1 = Sd1 * Sd1 / Sn1;
t1 *= t1;
t1 /= (Sn1 - 1);
double t2 = Sd2 * Sd2 / Sn2;
t2 *= t2;
t2 /= (Sn2 - 1);
v /= (t1 + t2);
cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n";
// t-statistic:
double t_stat = (Sm1 - Sm2) / sqrt(Sd1 * Sd1 / Sn1 + Sd2 * Sd2 / Sn2);
cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n";
//
// 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 << "Sample 1 Mean != Sample 2 Mean " ;
if(q < alpha / 2)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << "Sample 1 Mean < Sample 2 Mean ";
if(cdf(dist, t_stat) < alpha)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << "Sample 1 Mean > Sample 2 Mean ";
if(cdf(complement(dist, t_stat)) < alpha)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << endl << endl;
}
int main()
{
//
// Run tests for Car Mileage sample data
// http://www.itl.nist.gov/div898/handbook/eda/section3/eda3531.htm
// from the NIST website http://www.itl.nist.gov. The data compares
// miles per gallon of US cars with miles per gallon of Japanese cars.
//
two_samples_t_test_equal_sd(20.14458, 6.414700, 249, 30.48101, 6.107710, 79, 0.05);
two_samples_t_test_unequal_sd(20.14458, 6.414700, 249, 30.48101, 6.107710, 79, 0.05);
return 0;
} // int main()
/*
Output is:
_______________________________________________
Student t test for two samples (equal variances)
_______________________________________________
Number of Observations (Sample 1) = 249
Sample 1 Mean = 20.145
Sample 1 Standard Deviation = 6.4147
Number of Observations (Sample 2) = 79
Sample 2 Mean = 30.481
Sample 2 Standard Deviation = 6.1077
Degrees of Freedom = 326
Pooled Standard Deviation = 6.3426
T Statistic = -12.621
Probability that difference is due to chance = 5.273e-030
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Sample 1 Mean != Sample 2 Mean NOT REJECTED
Sample 1 Mean < Sample 2 Mean NOT REJECTED
Sample 1 Mean > Sample 2 Mean REJECTED
_________________________________________________
Student t test for two samples (unequal variances)
_________________________________________________
Number of Observations (Sample 1) = 249
Sample 1 Mean = 20.14458
Sample 1 Standard Deviation = 6.41470
Number of Observations (Sample 2) = 79
Sample 2 Mean = 30.48101
Sample 2 Standard Deviation = 6.10771
Degrees of Freedom = 136.87499
T Statistic = -12.94627
Probability that difference is due to chance = 1.571e-025
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Sample 1 Mean != Sample 2 Mean NOT REJECTED
Sample 1 Mean < Sample 2 Mean NOT REJECTED
Sample 1 Mean > Sample 2 Mean REJECTED
*/
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