Epa 402-R-98-008 - Statistical Procedures For Certifying Phosphogypsum For Entry Into Commerce - U.s. Environmental Protection Agency - 1998 Page 11
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17The null hypothesis is rejected only if the evidence is persuasive that the true
concentration is less than 10 pCi/g. This means that the phosphogypsum is determined to be
acceptable for agricultural purposes only if the radium-226 concentration can be shown to be less
than 10 pCi/g by means of the statistical test described in this document.
To illustrate hypothesis testing, assume an area of a hypothetical phosphogypsum stack
has a true concentration equal 10 pCi/g. This concentration is used because it is the highest
concentration permissible for use in agriculture, and therefore, the highest concentration not
meeting the conditions of the null hypothesis. The distribution of the mean of a sample drawn
from this area would be expected to be normally distribute with a mean of 10 pCi/g. The
standard deviation of this distribution could be estimated from a sample drawn from this area
(when dealing with the distribution of the sample mean, the standard deviation is usually termed
the standard error). Assume, for this illustration, that the standard error is 1.5 pCi/g. (A sample
of 30 with a standard deviation equal 8.2 pCi/g would yield a standard error of the mean equal
1.5 pCi/g.) The standard deviations of radium-226 measurements on five phosphogypsum
stacks, as reported in the document: A Long-Term Study of Radon and Airborne Particulates at
Phosphogypsum Stacks in Centra Florida, (EPA 520/5-88-021, Oct. 1988), range from about
1.65 to 18.3 pCi/g, so a standard deviation equal 8.2 pCi/g is reasonable. The sampling
distribution is show in Figure 2. This distribution is entirely hypothetical, and is used only to
illustrate this discussion. The shading in the left hand tail is explained below.
When testing an area of the stack with an unknown radium concentration, the mean of the
sample can be compared to this distribution to determine how likely it is that the sample was
drawn from an area of a phosphogypsum stack with a mean concentration greater than 10 pCi/g.
The null hypothesis would be rejected only if the sample mean were small enough that it is
unlikely that it could have been drawn from an area with a rue concentration greater than 10
pCi/g.
The probability of erroneously rejecting the null hypothesis expresses the likelihood of
mistakenly concluding that phosphogypsum can be entered into commerce when, in fact its
radium-226 concentration is greater than 10 pCi/g. This probability must be established before
the null hypothesis can be tested, and, when decided upon, determines the critical value.
The critical value divides the sampling distribution into a region for accepting the null
hypothesis and a region for rejecting the null.
There are no rules for determining an acceptable probability or erroneously rejecting the
null hypothesis. The choice should be made in the context of the consequences of an incorrect
decision. If there are expected to be substantial consequences from incorrectly rejecting the null
(i.e., concluding the concentration is less than or equal 10 pCi/g when it is, in fact, greater than
10 pCi/g), then it is reasonable to establish a small chance of error. If the consequences are not
expected to be substantial, it is reasonable to establish a larger chance of error.
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