The two-sample test for proportions can be used to compare two proportions (or percentiles) and is based on an independent random sample from a study-area population and an independent random sample from a reference-area population. For example, the proportion of samples in a study-area population, which exceeds a limit \(C\) = 10 ppm, can be compared to the proportion of samples that exceeds \(C\) = 10 ppm in a reference-area (background) population. The Action Level is a specified difference in true proportions for the study area and the reference area. Please consult EPA's guidance document, Guidance for the Data Quality Objectives Process (EPA 2006a), to put this test in the context of environmental decision-making.
Before deciding to develop a sampling plan based on using the two-sample proportion test, consider the assumptions and limitations involved. For a discussion of these assumptions, limitations, and for the details of the test, please consult EPA's Data Quality Assessment: Statistical Methods for Practitioners (EPA. 2006b.) This document, as well as the DQO guidance document, is currently available at: http://www.epa.gov/quality/qa_docs.html.
The number of samples is calculated using a modified version of Eq. (1) (EPA 2000b, p. 3-29). Note that the denominator is different in the VSP equation. See Visual Sample Plan (VSP) Models and Code Verification (Gilbert et al. 2001, pp. 3.14 and 3.15) for a brief discussion of this correction. No MQO option is currently provided with this option.
\begin{equation} m = n = \frac{2( Z_{1- \alpha} + Z_{1- \beta} )^2 \bar P (1 - \bar P )}{( \delta_0 - \delta_1 )^2} \end{equation}
where:
\( n \) |
is the minimum number of samples to be collected in the study area |
\( m \) |
is the minimum number of samples to be collected in the reference area |
\( Z_{1 - \alpha } \) |
is the value of the standard normal distribution for which the proportion of the distribution to the left of \( Z_{1 - \alpha } \) is \( 1 - \alpha \) |
\( Z_{1 - \beta } \) |
is the value of the standard normal distribution for which the proportion of the distribution to the left of \( Z_{1 - \beta } \) is \( 1 - \beta \) |
\( \bar P \) |
is the average proportion, \( (P1 + P2)/2 \) |
\( P1 \) |
is the true proportion in the study area |
\( P2 \) |
is the true proportion in the reference area |
\( \delta_0 \) |
is the difference in true proportions that defines the Action Level |
\( \delta_1 \) |
is the difference in true proportions that defines the outer bound of the gray region |
Note:
The Action Level is the specified difference between the true proportion in the study area (P1) and the true proportion in the reference area (P2) that will be used to decide if the site is clean or dirty.
EPA. 2006a. Guidance on Systematic Planning Using the Data Quality Objectives Process. EPA QA/G-4, EPA/240/B-06/001, U.S. Environmental Protection Agency, Office of Environmental Information, Washington DC.
EPA, February 2006. Data Quality Assessment: Statistical Methods for Practitioners, EPA QA/G-9S, Office of Environmental Information, U.S. Environmental Protection Agency, Washington, DC.
Gilbert, RO, JR Davidson, JE Wilson, BA Pulsipher. 2001. Visual Sample Plan (VSP) models and code verification. PNNL-13450, Pacific Northwest National Laboratory, Richland, Washington.
For Null Hypothesis = Site is Dirty:
Width of Gray Area (Delta) / LBGR / UBGR
For Null Hypothesis = Site is Clean:
Width of Gray Area (Delta) / LBGR / UBGR
Estimated Proportion in Reference Area
Estimated Proportion in Survey Unit