# Statistics for Environmental Engineers

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n

2<J2(Zg/2 -I- Zp)

Д2

d = observed difference of two treatments

FIGURE 23.1 Definition of type I and type II errors for a one-sided test of the difference between two means.

where A = |n — q2 and a and в are probabilities of type I and type II errors. If the variance a2 is not known, it is replaced with the sample variance s .

The sample size for a one-sided test on whether a mean is above a fixed standard level (i.e., a regulatory threshold) is:

2a2( Za + Zp) + 1 2

n = -A- + 2 Za

This is an approximate, but very accurate, sample size estimate (U.S.EPA, 1994).

Example 23.4

A decision rule is being developed for directing loads of contaminated material to a sanitary landfill or a hazardous waste landfill. Each load is fairly homogeneous but there is considerable variation between loads. The standard deviation of contaminant in a given load is 0.06 mg/kg. The decision rule conditions are (1) a probability of 95% of declaring a load hazardous when the true mean concentration is 1.0 mg/kg and (2) a probability of 10% of declaring a load hazardous when the true mean concentration is 0.75 mg/kg. What size sample should be analyzed from each load, assuming samples can be collected at random?

For the stated conditions, a = 0.05 and в = 0.10, giving z o.os/2 = 1.96 and Zo.rn = 1.28. With a = 0.06 and A = 1.00 — 0.75:

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