Page 208 - Modern Analytical Chemistry

1400-CH07 9/8/99 4:03 PM Page 191

Chapter 7 Obtaining and Preparing Samples for Analysis 191
7B.4 How Many Samples to Collect

In the previous section we considered the amount of sample needed to minimize
the sampling variance. Another important consideration is the number of samples
required to achieve a desired maximum sampling error. If samples drawn from the
target population are normally distributed, then the following equation describes
the confidence interval for the sampling error
ts s
m=X ±
n s
where n s is the number of samples and s s is the sampling standard deviation. Rear-
–
ranging and substituting e for the quantity (m–X), gives the number of samples as
22
ts s
n s =
e 2 7.7
2
2
where s s and e are both expressed as absolute uncertainties or as relative uncertain-
ties. Finding a solution to equation 7.7 is complicated by the fact that the value of t
depends on n s . As shown in Example 7.8, equation 7.7 is solved iteratively.
7
EXAMPLE .8
In Example 7.6 we found that an analysis for the inorganic ash content of a
breakfast cereal required a sample of 1.5 g to establish a relative standard
deviation for sampling of ±2.0%. How many samples are needed to obtain a
relative sampling error of no more than 0.80% at the 95% confidence level?

SOLUTION
Because the value of t depends on n s , and the value of n s is not yet known, we
begin by letting n s = ¥ and use the associated value of t. From Appendix 1B,
the value for t is 1.96. Substituting known values into equation 7.7
2
0
(. ) ( . ) 2
6
19
2
n s = =24
(. ) 2
08
0
Letting n s = 24, the value of t from Appendix 1B is 2.075. Recalculating n s gives
2
2
(. 2 075 ) ( . ) 0 2
26
n s = = .9 » 27
(. ) 2
080
When n s = 27, the value of t is 2.066. Recalculating n s , we find
2
2
(. 2 066 ) ( . ) 0 2
26
n s = 2 = .7 » 27
080
(. )
Since two successive calculations give the same value for n s , an iterative
solution has been found. Thus, 27 samples are needed to achieve the desired
sampling error.
Equation 7.7 only provides an estimate for the smallest number of samples ex-
pected to produce the desired sampling error. The actual sampling error may be
substantially higher if the standard deviation for the samples that are collected is
significantly greater than the standard deviation due to sampling used to calculate n s .