APPENDIX B
GENERALIZED EQUATIONS FOR ESTIMATING ALIQUOT
WEIGHT AND NUMBER OF ALIQUOTS PER SAMPLE
Results for Am
Suppose that the relationship between the standard deviation (s)
counts and aliquot weight (w) in grams is
s = aw? .
for Am Ge(Li)
(Bl)
In addition, assume the total cost C for Ge(Li) counting n aliquots of any
size w is
C = nk,
(B2)
where K is the cost per aliquot.
Since the variance of the estimated mean
Am concentration is Va = s‘*/n, we see from Equation Al and A2 that
VA = at /nweP = a-K/CweP .
(B3)
This can be solved for w to give
1
1
w= (a°/V,n)?? = (a’K/cv,)P ,
(B4)
Hence, for fixed counting costs C and K, and a desired precision V, of xX we
can estimate both the number (n) and weight (w) of aliquots that are required
by solving Equations A2 and A4&.
This assumes, of course, that we have previously obtained data that give reliable estimates of a and b in Equation Bl.
For a specified precision Va the cost C will be minimized by taking n = 1.
In that case, Equation A4 gives
i
we (a°/v,)7? :
(B5)
Conversely, if C and K are fixed (equivalent to fixing n) then Vag will be
minimized by using the largest possible aliquot size w up to the limit
where the flat rate of K dollars per aliquot no longer applies.
This can
be seen from Equation B3.
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