Appendix a. to Appendix IX -Statistics  

A.1 Determination of Enrichment Factor

After at least 10 initial emissions tests are performed, an enrichment factor for each metal must be determined. At the 95% confidence level, the enrichment factor, EF_95% s, is based on the test results and is statistically determined so there is only a 5% chance that the enrichment factor at any given time will be larger than EF_95%. Similarly, at the 99% confidence level, the enrichment factor, EF_99%, is statistically determined so there is only a 1% chance that the enrichment factor at any given time will be larger than EF_99%.

For a large number of samples (n > 30), EF_95% is based on a normal distribution, and is equal to:

EF_95% = EF + z_cs(1)

where:

For a 95% confidence level, z_c is equal to 1.645.

For a small number of samples (n < 30), EF_95% is based on the t - distribution and is equal to:

EF_95% = EF + t_c S (4)

where the standard deviation, S, is defined as:

t_c is a function of the number of samples and the confidence level that is desired. It increases in value as the sample size decreases and the confidence level increases. The 95% confidence level is used in this method to calculate the “violation” kiln dust metal concentration limit; and the 99% confidence level is sometimes used to calculate the “conservative” kiln dust metal concentration limit. Values of t_c are shown in table A-1 for various degrees of freedom (degrees of freedom = sample size-1) at the 95% and 99% confidence levels. As the sample size approaches infinity, the normal distribution is approached.

A.2 Comparison of Enrichment Factor Groups

To determine if the enrichment factors measured in the quarterly tests are significantly different from the enrichment factors determined in the initial Step 2 tests, the t-test is used. In this test, the value t_meas:

is compared to t_crit at the desired confidence level. The 95% confidence level is used in this method. Values of t_crit are shown in table A-1 for various degrees of freedom (degrees of freedom n₁ + n₂ - 2) at the 95% and 99% confidence levels. If t_meas is greater then t_crit, it can be concluded with 95% confidence that the two groups are not from the same population.

A.3 Rejection of Data

If the concentration of any hazardous metal in the “required” kiln dust sample exceeds the kiln dust metal concentration limit, the “spare” samples are analyzed. If the average of the combined “required” and “spare” values is still above the limit, a statistical test is used to decide if the upper value can be rejected.

The “Q-test” is used to determine if a data point can be rejected. The difference between the questionable result and its neighbor is divided by the spread of the entire data set. The resulting ratio, Q _meas, is then compared with rejection values that are critical for a particular degree of confidence, where Q _meas is:

The 90% confidence level for data rejection is used in this method. Table A-2 provides the values of Q_crit at the 90% confidence level. If Q at the 90% confidence level. If Q_meas is larger than Q_crit, the data point can be discarded. Only one data point from a sample group can be rejected using this method.