Frequent Questions

Use of 2-tailed vs. 1-tailed t-values in SW-846 Chapter 9 formula for calculating confidence limits.

The Chapter 9 formula for calculating confidence limit appears to use the 2‐tailed t‐value to evaluate sample data against a regulatory limit at 80% confidence. However, this type of evaluation involves a one‐sided hypothesis (sample mean > regulatory limit) and requires a 1‐tailed t‐value which is a much smaller value than indicated in the Chapter 9 t‐value table.

Is this a mistake in the text or is there some reason for using the 2‐tailed t‐value when a 1‐tailed t‐value is typically used?

 

There is no mistake in the Chapter 9 text regarding use of the 2-tailed t-value. Using a 0.2 probability (80% confidence interval) with the 2-tailed t-value and only looking at the upper end to determine if it exceeds the hazardous waste level, is in effect, using a 1-tailed t-value at 0.1 probability (90% confidence interval).  The equivalency is mentioned in the parentheses in footnote b under Table 9.2:  "Tabulated "t" values are for a two-tailed confidence interval and a probability of 0.20 (the same values are applicable to a one-tailed confidence interval and a probability of 0.10)."

Calculation wise, use Equation 6 where the confidence interval is expressed as the mean plus or minus the 0.2 t-statistic times the standard deviation. For hazardous waste determination, we are only using the mean plus the 0.2 t-statistic times the standard deviation and ignoring the mean minus the 0.2 t-statistic times the standard deviation.

Finally, Section 9.1.1.1 states, "For the purposes of evaluating solid wastes, the probability level (confidence interval) of 80% has been selected.  That is, for each chemical contaminant of concern, a confidence interval (CI) is described within which μ occurs if the sample is representative, which is expected of about 80 out of 100 samples. The upper limit of the 80% CI is then compared with the appropriate regulatory threshold.  If the upper limit is less than the threshold, the chemical contaminant is not considered to be present in the waste at a hazardous level; otherwise, the opposite conclusion is drawn.  One last point merits explanation. Even if the upper limit of an estimated 80% CI is only slightly less than the regulatory threshold (the worst case of chemical contamination that would be judged acceptable), there is only a 10% (not 20%) chance that the threshold is equaled or exceeded. That is because values of a normally distributed contaminant that are outside the limits of an 80% CI are equally distributed between the left (lower) and right (upper) tails of the normal curve.  Consequently, the CI employed to evaluate solid wastes is, for all practical purposes, a 90% interval." In other words, the 80% two-tailed value effectively becomes a 90% one-tailed value when we only consider the upper end.

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