Environmental Sampling Design - Fundamentals of Environmental Sampling and Analysis - page 62

A particular equation used to estimate sample size for solid wastes is given by
the U.S. EPA, which relate required samp le number (n) to the sample variance (s 2 )
and the closeness of the estimate mean (x) to the regulatory threshold (RT):

t 2 s 2
e 2

n ¼ ð3 : 11Þ

where e is the acceptable level of error, e ¼ RTx for hazardous materials. Eq. 3.11
indicates that the sample number is directly proportional to the contaminant
variability and more samples should be collected if contaminant concentration is
closer to its regulatory standard. Since the tabulated t-value in Eq. 3.11 depends on
the number of sample (df ¼ n1), a trial-and-error procedure is typically needed to
estimate the optimal sample size (n).
It is common that the sample sizes calculated using simple random sampling
formulae exceed the study budget. Then adjustment needs to be done, including
redefining the study goal or the DQOs (a smaller study), and modifying the required
precision. Most often, choosing other sampling designs may reduce a significant
number of samples. For instance, stratified random sampling may result in a smaller
sample size and composite sampling may reduce a significant cost if the goal is to
determine the mean.

EXAMPLE 3.4. Historical data from a contaminated site suggested Hg concentrations
in the range of 2-20 mg/kg, and a standard deviation of 3.25 mg/kg. A thorough survey is
needed for a planned remediation of this site. Estimate the number of samples required so
that the sample mean would be within 1.5 mg/kg of the population mean at a 95%
confidence level.

SOLUTION: From Eq. 3.11, it is apparent that an initial guess of n is required. Let us assume
n ¼ 10 (i.e., df ¼ 9), this gives a t-value of 2.262 (Appendix C2). Plug these known values
(s ¼ 3:25, e ¼ 1:5, t ¼ 2:262) into Eq. 3.11, we have:

t 2 s 2
e 2 ¼ð2:2623:25=1:5Þ

2

n ¼ ¼ 24:01:

Since the calculated n ¼ 24 is much larger than the assumed n ¼ 10, a further iteration is
needed. The new t-value for n ¼ 24 (95%) is 2.069 (Appendix C2),

then calculate

2

n ¼ð2:0693:25=1:5Þ ¼ 20:10 20, which is closer to the assumed value of 24. A third

iteration at n ¼ 24 gives a new value of n ¼ð2:0933:25=1:5Þ

2

¼ 20:56 21. Therefore a

total of 21 samples should be tested.

EXAMPLE 3.5. Calculation of required numbers of samples. An abandoned waste
pile needs to be sampled and analyzed due to recent complaints from local residents.
Historical data show that the wastes were mainly from a solvent recovery facility in
1970s with a PCB concentration of 0.70 mg/kg (average of five samples) and standard
deviation of 0.12 mg/kg. The regulatory soil screening level for PCB is 0.74 mg/kg. Use a
80% confidence level. (a) Estimate the number of samples required; (b) Estimate the
number of sample required if the mean was 0.58 mg/kg.