Statistics for Environmental Engineers

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The one-sided tolerance interval is of the form y + ks, where s, the standard deviation for each constituent, has been estimated from an available sample of nb background measurements. Values of are tabulated in Gibbons (1994, Table 4.2).


Suppose that we want a tolerance interval that has 95% confidence (a = 0.05) of including 99% of all values in the interval (coverage p = 0.99). This is nb = 20, 1 — a = 0.95, and p = 0.99, k = 3.295 and:


y + 3.295s


For the failure rate of (1 — 0.99) = 1%, we expect that k = 0.01(500) = 5 comparisons might exceed the published standards. If there are more than the expected five exceedances, the offending wells should be resampled, but only for those constituents that failed.


The resampling data should be compared to a 95% prediction interval for the expected number of exceedances and not the number that happens to be observed. The one-sided prediction interval is:


y + ks 1 +


V nb


If nb is reasonably large (i.e., nb > 40), the quantity under the square root is approximately 1.0 and can be ignored. Assuming this to be true, this case study uses k = 2.43, which is from Gibbons (1994, Table 1.2) for nb = 40, 1 — a = 0.95, and p = 0.99. Thus, the prediction interval is y + 2.43s.

Case Study: Water Quality Compliance


A company is required to meet a water quality limit of 300 ppm in a river. This has been monitored by collecting specimens of river water during the first week of each of the past 27 quarters. The data are from Hahn and Meeker (1991).

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