Range of Application of Program

The program has been verified to converge as long as

• The sample standard deviation s is in the range [0.1, 10.0] (Note that this range is for the log-transformed values, if one is dealing with the lognormal distribution)
• The degrees of freedom is in the range [2,1000] (in most cases, this is equivalent to the number of data points being in the range [3,1001])

How Can You Be Sure that the Calculated Confidence Limits are Correct?

One simple method of verification involves the following:

• Assume values for the mean m, s², and number of samples n
• For any possible sample mean y and sample variance s², it is known that
  • y = y_p = m + (s n^-1/2) F(p) for some p in (0,1), where F(p) is the pth percentile of the standard normal distribution
  • s² = s²_q = s² c²(q) / (n-1) for some q (0,1), where c²(q) is the qth percentile of the chi-squared distribution with n-1 degrees of freedom
• Perform Monte Carlo sampling for p and q from (0,1), thus obtaining samples of the sample mean and variance
• For each Monte Carlo sample, calculate the confidence limits for (m + s²/2) of level a
• Verify that the fraction of samples for which the calculated confidence limits fail is approximately equal to 1-a (we can tell whether they fail or not because we know what the actual mean and variance are)

A problem with this method is that it is vulnerable to sampling variation in the Monte Carlo sampling itself. It is possible, however, to perform a more precise test. Basically, with this method one calculates the area of the region in (p,q) space for which the calculated confidence limits fail (see report for details). Due to nature of the particular sampling distributions involved, the area of this region can be calculated without having to deal with impacts of sampling variation. A program that calculates this area was implemented and used to verify the accuracy of the confidence limits. This program is available on the download page. It was applied over the range of values shown in the table below. You can use this program to convince yourself that the program is actually working over the range of s, a, and/or n that you expect to encounter in your application (note that the program is not sensitive to the value of the sample mean y or l, since a normalization is possible that allows the most general case to be determined from the special case y=0 and l=1/2).