...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

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 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 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 silicon wafer data from the NIST/SEMATECH e-Handbook of Statistical Methods.. 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:

_____________________________________________________________ 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.