[ceph.git] / ceph / src / boost / libs / math / example / students_t_single_sample.cpp

// 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 occations.
   // 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

*/
Commit	Line	Data
7c673cae FG	1	// Copyright John Maddock 2006
	2	// Copyright Paul A. Bristow 2007, 2010
	3
	4	// Use, modification and distribution are subject to the
	5	// Boost Software License, Version 1.0.
	6	// (See accompanying file LICENSE_1_0.txt
	7	// or copy at http://www.boost.org/LICENSE_1_0.txt)
	8
	9	#ifdef _MSC_VER
	10	# pragma warning(disable: 4512) // assignment operator could not be generated.
	11	# pragma warning(disable: 4510) // default constructor could not be generated.
	12	# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required.
	13	#endif
	14
	15	#include <boost/math/distributions/students_t.hpp>
	16
	17	// avoid "using namespace std;" and "using namespace boost::math;"
	18	// to avoid potential ambiguity with names in std random.
	19	#include <iostream>
	20	using std::cout; using std::endl;
	21	using std::left; using std::fixed; using std::right; using std::scientific;
	22	#include <iomanip>
	23	using std::setw;
	24	using std::setprecision;
	25
	26	void confidence_limits_on_mean(double Sm, double Sd, unsigned Sn)
	27	{
	28	//
	29	// Sm = Sample Mean.
	30	// Sd = Sample Standard Deviation.
	31	// Sn = Sample Size.
	32	//
	33	// Calculate confidence intervals for the mean.
	34	// For example if we set the confidence limit to
	35	// 0.95, we know that if we repeat the sampling
	36	// 100 times, then we expect that the true mean
	37	// will be between out limits on 95 occations.
	38	// Note: this is not the same as saying a 95%
	39	// confidence interval means that there is a 95%
	40	// probability that the interval contains the true mean.
	41	// The interval computed from a given sample either
	42	// contains the true mean or it does not.
	43	// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm
	44
	45	using boost::math::students_t;
	46
	47	// Print out general info:
	48	cout <<
	49	"__________________________________\n"
	50	"2-Sided Confidence Limits For Mean\n"
	51	"__________________________________\n\n";
	52	cout << setprecision(7);
	53	cout << setw(40) << left << "Number of Observations" << "= " << Sn << "\n";
	54	cout << setw(40) << left << "Mean" << "= " << Sm << "\n";
	55	cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n";
	56	//
	57	// Define a table of significance/risk levels:
	58	//
	59	double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
	60	//
	61	// Start by declaring the distribution we'll need:
	62	//
	63	students_t dist(Sn - 1);
	64	//
65	// Print table header:
66	//
67	cout << "\n\n"
68	"_______________________________________________________________\n"
69	"Confidence T Interval Lower Upper\n"
70	" Value (%) Value Width Limit Limit\n"
71	"_______________________________________________________________\n";
72	//
73	// Now print out the data for the table rows.
74	//
75	for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
76	{
77	// Confidence value:
78	cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
79	// calculate T:
80	double T = quantile(complement(dist, alpha[i] / 2));
81	// Print T:
82	cout << fixed << setprecision(3) << setw(10) << right << T;
83	// Calculate width of interval (one sided):
84	double w = T * Sd / sqrt(double(Sn));
85	// Print width:
86	if(w < 0.01)
87	cout << scientific << setprecision(3) << setw(17) << right << w;
88	else
89	cout << fixed << setprecision(3) << setw(17) << right << w;
90	// Print Limits:
91	cout << fixed << setprecision(5) << setw(15) << right << Sm - w;
92	cout << fixed << setprecision(5) << setw(15) << right << Sm + w << endl;
93	}
94	cout << endl;
95	} // void confidence_limits_on_mean
96
97	void single_sample_t_test(double M, double Sm, double Sd, unsigned Sn, double alpha)
98	{
99	//
100	// M = true mean.
101	// Sm = Sample Mean.
102	// Sd = Sample Standard Deviation.
103	// Sn = Sample Size.
104	// alpha = Significance Level.
105	//
106	// A Students t test applied to a single set of data.
107	// We are testing the null hypothesis that the true
108	// mean of the sample is M, and that any variation is down
109	// to chance. We can also test the alternative hypothesis
110	// that any difference is not down to chance.
111	// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm
112
113	using boost::math::students_t;
114
115	// Print header:
116	cout <<
117	"__________________________________\n"
118	"Student t test for a single sample\n"
119	"__________________________________\n\n";
120	cout << setprecision(5);
121	cout << setw(55) << left << "Number of Observations" << "= " << Sn << "\n";
122	cout << setw(55) << left << "Sample Mean" << "= " << Sm << "\n";
123	cout << setw(55) << left << "Sample Standard Deviation" << "= " << Sd << "\n";
124	cout << setw(55) << left << "Expected True Mean" << "= " << M << "\n\n";
125	//
126	// Now we can calculate and output some stats:
127	//
128	// Difference in means:
129	double diff = Sm - M;
130	cout << setw(55) << left << "Sample Mean - Expected Test Mean" << "= " << diff << "\n";
131	// Degrees of freedom:
132	unsigned v = Sn - 1;
133	cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n";
134	// t-statistic:
135	double t_stat = diff * sqrt(double(Sn)) / Sd;
136	cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n";
137	//
138	// Finally define our distribution, and get the probability:
139	//
140	students_t dist(v);
141	double q = cdf(complement(dist, fabs(t_stat)));
142	cout << setw(55) << left << "Probability that difference is due to chance" << "= "
143	<< setprecision(3) << scientific << 2 * q << "\n\n";
144	//
145	// Finally print out results of alternative hypothesis:
146	//
147	cout << setw(55) << left <<
148	"Results for Alternative Hypothesis and alpha" << "= "
149	<< setprecision(4) << fixed << alpha << "\n\n";
150	cout << "Alternative Hypothesis Conclusion\n";
151	cout << "Mean != " << setprecision(3) << fixed << M << " ";
152	if(q < alpha / 2)
153	cout << "NOT REJECTED\n";
154	else
155	cout << "REJECTED\n";
156	cout << "Mean < " << setprecision(3) << fixed << M << " ";
157	if(cdf(complement(dist, t_stat)) > alpha)
158	cout << "NOT REJECTED\n";
159	else
160	cout << "REJECTED\n";
161	cout << "Mean > " << setprecision(3) << fixed << M << " ";
162	if(cdf(dist, t_stat) > alpha)
163	cout << "NOT REJECTED\n";
164	else
165	cout << "REJECTED\n";
166	cout << endl << endl;
167	} // void single_sample_t_test(
168
169	void single_sample_find_df(double M, double Sm, double Sd)
170	{
171	//
172	// M = true mean.
173	// Sm = Sample Mean.
174	// Sd = Sample Standard Deviation.
175	//
176
177	using boost::math::students_t;
178
179	// Print out general info:
180	cout <<
181	"_____________________________________________________________\n"
182	"Estimated sample sizes required for various confidence levels\n"
183	"_____________________________________________________________\n\n";
184	cout << setprecision(5);
185	cout << setw(40) << left << "True Mean" << "= " << M << "\n";
186	cout << setw(40) << left << "Sample Mean" << "= " << Sm << "\n";
187	cout << setw(40) << left << "Sample Standard Deviation" << "= " << Sd << "\n";
188	//
189	// Define a table of significance intervals:
190	//
191	double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
192	//
193	// Print table header:
194	//
195	cout << "\n\n"
196	"_______________________________________________________________\n"
197	"Confidence Estimated Estimated\n"
198	" Value (%) Sample Size Sample Size\n"
199	" (one sided test) (two sided test)\n"
200	"_______________________________________________________________\n";
201	//
202	// Now print out the data for the table rows.
203	//
204	for(unsigned i = 1; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
205	{
206	// Confidence value:
207	cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
208	// calculate df for single sided test:
209	double df = students_t::find_degrees_of_freedom(
210	fabs(M - Sm), alpha[i], alpha[i], Sd);
211	// convert to sample size, always one more than the degrees of freedom:
212	double size = ceil(df) + 1;
213	// Print size:
214	cout << fixed << setprecision(0) << setw(16) << right << size;
215	// calculate df for two sided test:
216	df = students_t::find_degrees_of_freedom(
217	fabs(M - Sm), alpha[i]/2, alpha[i], Sd);
218	// convert to sample size:
219	size = ceil(df) + 1;
220	// Print size:
221	cout << fixed << setprecision(0) << setw(16) << right << size << endl;
222	}
223	cout << endl;
224	} // void single_sample_find_df
225
226	int main()
227	{
228	//
229	// Run tests for Heat Flow Meter data
230	// see http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm
231	// The data was collected while calibrating a heat flow meter
232	// against a known value.
233	//
234	confidence_limits_on_mean(9.261460, 0.2278881e-01, 195);
235	single_sample_t_test(5, 9.261460, 0.2278881e-01, 195, 0.05);
236	single_sample_find_df(5, 9.261460, 0.2278881e-01);
237
238	//
239	// Data for this example from:
240	// P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64.
241	// from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55
242	// J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907
243	//
244	// Determination of mercury by cold-vapour atomic absorption,
245	// the following values were obtained fusing a trusted
246	// Standard Reference Material containing 38.9% mercury,
247	// which we assume is correct or 'true'.
248	//
249	confidence_limits_on_mean(37.8, 0.964365, 3);
250	// 95% test:
251	single_sample_t_test(38.9, 37.8, 0.964365, 3, 0.05);
252	// 90% test:
253	single_sample_t_test(38.9, 37.8, 0.964365, 3, 0.1);
254	// parameter estimate:
255	single_sample_find_df(38.9, 37.8, 0.964365);
256
257	return 0;
258	} // int main()
259
260	/*
261
262	Output:
263
264	------ Rebuild All started: Project: students_t_single_sample, Configuration: Release Win32 ------
265	students_t_single_sample.cpp
266	Generating code
267	Finished generating code
268	students_t_single_sample.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Release\students_t_single_sample.exe
269	__________________________________
270	2-Sided Confidence Limits For Mean
271	__________________________________
272
273	Number of Observations = 195
274	Mean = 9.26146
275	Standard Deviation = 0.02278881
276
277
278	_______________________________________________________________
279	Confidence T Interval Lower Upper
280	Value (%) Value Width Limit Limit
281	_______________________________________________________________
282	50.000 0.676 1.103e-003 9.26036 9.26256
283	75.000 1.154 1.883e-003 9.25958 9.26334
284	90.000 1.653 2.697e-003 9.25876 9.26416
285	95.000 1.972 3.219e-003 9.25824 9.26468
286	99.000 2.601 4.245e-003 9.25721 9.26571
287	99.900 3.341 5.453e-003 9.25601 9.26691
288	99.990 3.973 6.484e-003 9.25498 9.26794
289	99.999 4.537 7.404e-003 9.25406 9.26886
290
291	__________________________________
292	Student t test for a single sample
293	__________________________________
294
295	Number of Observations = 195
296	Sample Mean = 9.26146
297	Sample Standard Deviation = 0.02279
298	Expected True Mean = 5.00000
299
300	Sample Mean - Expected Test Mean = 4.26146
301	Degrees of Freedom = 194
302	T Statistic = 2611.28380
303	Probability that difference is due to chance = 0.000e+000
304
305	Results for Alternative Hypothesis and alpha = 0.0500
306
307	Alternative Hypothesis Conclusion
308	Mean != 5.000 NOT REJECTED
309	Mean < 5.000 REJECTED
310	Mean > 5.000 NOT REJECTED
311
312
313	_____________________________________________________________
314	Estimated sample sizes required for various confidence levels
315	_____________________________________________________________
316
317	True Mean = 5.00000
318	Sample Mean = 9.26146
319	Sample Standard Deviation = 0.02279
320
321
322	_______________________________________________________________
323	Confidence Estimated Estimated
324	Value (%) Sample Size Sample Size
325	(one sided test) (two sided test)
326	_______________________________________________________________
327	75.000 2 2
328	90.000 2 2
329	95.000 2 2
330	99.000 2 2
331	99.900 3 3
332	99.990 3 3
333	99.999 4 4
334
335	__________________________________
336	2-Sided Confidence Limits For Mean
337	__________________________________
338
339	Number of Observations = 3
340	Mean = 37.8000000
341	Standard Deviation = 0.9643650
342
343
344	_______________________________________________________________
345	Confidence T Interval Lower Upper
346	Value (%) Value Width Limit Limit
347	_______________________________________________________________
348	50.000 0.816 0.455 37.34539 38.25461
349	75.000 1.604 0.893 36.90717 38.69283
350	90.000 2.920 1.626 36.17422 39.42578
351	95.000 4.303 2.396 35.40438 40.19562
352	99.000 9.925 5.526 32.27408 43.32592
353	99.900 31.599 17.594 20.20639 55.39361
354	99.990 99.992 55.673 -17.87346 93.47346
355	99.999 316.225 176.067 -138.26683 213.86683
356
357	__________________________________
358	Student t test for a single sample
359	__________________________________
360
361	Number of Observations = 3
362	Sample Mean = 37.80000
363	Sample Standard Deviation = 0.96437
364	Expected True Mean = 38.90000
365
366	Sample Mean - Expected Test Mean = -1.10000
367	Degrees of Freedom = 2
368	T Statistic = -1.97566
369	Probability that difference is due to chance = 1.869e-001
370
371	Results for Alternative Hypothesis and alpha = 0.0500
372
373	Alternative Hypothesis Conclusion
374	Mean != 38.900 REJECTED
375	Mean < 38.900 NOT REJECTED
376	Mean > 38.900 NOT REJECTED
377
378
379	__________________________________
380	Student t test for a single sample
381	__________________________________
382
383	Number of Observations = 3
384	Sample Mean = 37.80000
385	Sample Standard Deviation = 0.96437
386	Expected True Mean = 38.90000
387
388	Sample Mean - Expected Test Mean = -1.10000
389	Degrees of Freedom = 2
390	T Statistic = -1.97566
391	Probability that difference is due to chance = 1.869e-001
392
393	Results for Alternative Hypothesis and alpha = 0.1000
394
395	Alternative Hypothesis Conclusion
396	Mean != 38.900 REJECTED
397	Mean < 38.900 NOT REJECTED
398	Mean > 38.900 REJECTED
399
400
401	_____________________________________________________________
402	Estimated sample sizes required for various confidence levels
403	_____________________________________________________________
404
405	True Mean = 38.90000
406	Sample Mean = 37.80000
407	Sample Standard Deviation = 0.96437
408
409
410	_______________________________________________________________
411	Confidence Estimated Estimated
412	Value (%) Sample Size Sample Size
413	(one sided test) (two sided test)
414	_______________________________________________________________
415	75.000 3 4
416	90.000 7 9
417	95.000 11 13
418	99.000 20 22
419	99.900 35 37
420	99.990 50 53
421	99.999 66 68
422
423	*/
424