chi_squared_examples.qbk

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   cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "=  "
      << setprecision(3) << scientific << lcv2 << "\n\n";

Now that we have the critical values, we can compare these to our test
statistic, and print out the result of each hypothesis and test:

   cout << setw(55) << left <<
      "Results for Alternative Hypothesis and alpha" << "=  "
      << setprecision(4) << fixed << alpha << "\n\n";
   cout << "Alternative Hypothesis              Conclusion\n";

   cout << "Standard Deviation != " << setprecision(3) << fixed << D << "            ";
   if((ucv2 < t_stat) || (lcv2 > t_stat))
      cout << "ACCEPTED\n";
   else
      cout << "REJECTED\n";

   cout << "Standard Deviation  < " << setprecision(3) << fixed << D << "            ";
   if(lcv > t_stat)
      cout << "ACCEPTED\n";
   else
      cout << "REJECTED\n";

   cout << "Standard Deviation  > " << setprecision(3) << fixed << D << "            ";
   if(ucv < t_stat)
      cout << "ACCEPTED\n";
   else
      cout << "REJECTED\n";
   cout << endl << endl;

To see some example output we'll use the
[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm 
gear data] from the __handbook.
The data represents measurements of gear diameter from a manufacturing
process.  The program output is deliberately designed to mirror 
the DATAPLOT output shown in the 
[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm 
NIST Handbook Example].

[pre'''
______________________________________________
Chi Squared test for sample standard deviation
______________________________________________

Number of Observations                                 =  100
Sample Standard Deviation                              =  0.00628
Expected True Standard Deviation                       =  0.10000

Test Statistic                                         =  0.39030
CDF of test statistic:                                 =  1.438e-099
Upper Critical Value at alpha:                         =  1.232e+002
Upper Critical Value at alpha/2:                       =  1.284e+002
Lower Critical Value at alpha:                         =  7.705e+001
Lower Critical Value at alpha/2:                       =  7.336e+001

Results for Alternative Hypothesis and alpha           =  0.0500

Alternative Hypothesis              Conclusion'''
Standard Deviation != 0.100            ACCEPTED
Standard Deviation  < 0.100            ACCEPTED
Standard Deviation  > 0.100            REJECTED
]

In this case we are testing whether the sample standard deviation is 0.1,
and the null-hypothesis is rejected, so we conclude that the standard
deviation ['is not] 0.1.

For an alternative example, consider the 
[@http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm 
silicon wafer data] again from the __handbook.
In this scenario a supplier of 100 ohm.cm silicon wafers claims 
that his fabrication  process can produce wafers with sufficient 
consistency so that the standard deviation of resistivity for 
the lot does not exceed 10 ohm.cm. A sample of N = 10 wafers taken 
from the lot has a standard deviation of 13.97 ohm.cm, and the question
we ask ourselves is "Is the suppliers claim correct?".

The program output now looks like this:

[pre'''
______________________________________________
Chi Squared test for sample standard deviation
______________________________________________

Number of Observations                                 =  10
Sample Standard Deviation                              =  13.97000
Expected True Standard Deviation                       =  10.00000

Test Statistic                                         =  17.56448
CDF of test statistic:                                 =  9.594e-001
Upper Critical Value at alpha:                         =  1.692e+001
Upper Critical Value at alpha/2:                       =  1.902e+001
Lower Critical Value at alpha:                         =  3.325e+000
Lower Critical Value at alpha/2:                       =  2.700e+000

Results for Alternative Hypothesis and alpha           =  0.0500

Alternative Hypothesis              Conclusion'''
Standard Deviation != 10.000            REJECTED
Standard Deviation  < 10.000            REJECTED
Standard Deviation  > 10.000            ACCEPTED
]

In this case, our null-hypothesis is that the standard deviation of 
the sample is less than 10: this hypothesis is rejected in the analysis
above, and so we reject the manufacturers claim.

[endsect][/section:chi_sq_test Chi-Square Test for the Standard Deviation]

[section:chi_sq_size Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation]

Suppose we conduct a Chi Squared test for standard deviation and the result
is borderline, a legitimate question to ask is "How large would the sample size
have to be in order to produce a definitive result?"

The class template [link math_toolkit.dist.dist_ref.dists.chi_squared_dist 
chi_squared_distribution] has a static method 
`find_degrees_of_freedom` that will calculate this value for
some acceptable risk of type I failure /alpha/, type II failure 
/beta/, and difference from the standard deviation /diff/.  Please
note that the method used works on variance, and not standard deviation
as is usual for the Chi Squared Test.

The code for this example is located in [@../../../example/chi_square_std_dev_test.cpp 
chi_square_std_dev_test.cpp].

We begin by defining a procedure to print out the sample sizes required
for various risk levels:

   void chi_squared_sample_sized(
        double diff,      // difference from variance to detect
        double variance)  // true variance
   {

The procedure begins by printing out the input data:

   using namespace std;
   using namespace boost::math;

   // 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 Variance" << "=  " << variance << "\n";
   cout << setw(40) << left << "Difference to detect" << "=  " << diff << "\n";

And defines a table of significance levels for which we'll calculate sample sizes:

   double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };

For each value of alpha we can calculate two sample sizes: one where the
sample variance is less than the true value by /diff/ and one
where it is greater than the true value by /diff/.  Thanks to the
asymmetric nature of the Chi Squared distribution these two values will
not be the same, the difference in their calculation differs only in the
sign of /diff/ that's passed to `find_degrees_of_freedom`.  Finally
in this example we'll simply things, and let risk level /beta/ be the 
same as /alpha/:

   cout << "\n\n"
           "_______________________________________________________________\n"
           "Confidence       Estimated          Estimated\n"
           " Value (%)      Sample Size        Sample Size\n"
           "                (lower one         (upper one\n"
           "                 sided test)        sided test)\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 df for a lower single sided test:
      double df = chi_squared::find_degrees_of_freedom(
         -diff, alpha[i], alpha[i], variance);
      // convert to sample size:
      double size = ceil(df) + 1;
      // Print size:
      cout << fixed << setprecision(0) << setw(16) << right << size;
      // calculate df for an upper single sided test:
      df = chi_squared::find_degrees_of_freedom(
         diff, alpha[i], alpha[i], variance);
      // convert to sample size:
      size = ceil(df) + 1;
      // Print size:
      cout << fixed << setprecision(0) << setw(16) << right << size << endl;
   }
   cout << endl;

For some example output, consider the 
[@http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm 
silicon wafer data] from the __handbook.
In this scenario a supplier of 100 ohm.cm silicon wafers claims 
that his fabrication  process can produce wafers with sufficient 
consistency so that the standard deviation of resistivity for 
the lot does not exceed 10 ohm.cm. A sample of N = 10 wafers taken 
from the lot has a standard deviation of 13.97 ohm.cm, and the question
we ask ourselves is "How large would our sample have to be to reliably
detect this difference?".

To use our procedure above, we have to convert the
standard deviations to variance (square them), 
after which the program output looks like this:

[pre'''
_____________________________________________________________
Estimated sample sizes required for various confidence levels
_____________________________________________________________

True Variance                           =  100.00000
Difference to detect                    =  95.16090


_______________________________________________________________
Confidence       Estimated          Estimated
 Value (%)      Sample Size        Sample Size
                (lower one         (upper one
                 sided test)        sided test)
_______________________________________________________________
    50.000               2               2
    75.000               2              10
    90.000               4              32
    95.000               5              51
    99.000               7              99
    99.900              11             174
    99.990              15             251
    99.999              20             330'''
]

In this case we are interested in a upper single sided test.  
So for example, if the maximum acceptable risk of falsely rejecting
the null-hypothesis is 0.05 (Type I error), and the maximum acceptable 
risk of failing to reject the null-hypothesis is also 0.05 
(Type II error), we estimate that we would need a sample size of 51.

[endsect][/section:chi_sq_size Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation]

[endsect][/section:cs_eg Chi Squared Distribution]

[/ 
  Copyright 2006 John Maddock and Paul A. Bristow.
  Distributed under 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).
]