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88 Modern Analytical Chemistry
unpaired data
Two sets of data consisting of results mean is derived from the analysis of several samples drawn from the same source.
obtained using several samples drawn Paired data are encountered when analyzing a series of samples drawn from differ-
from a single source.
paired data
Two sets of data consisting of results Unpaired Data Consider two samples, A and B, for which mean values, X A and X B ,
obtained using several samples drawn and standard deviations, s A and s B , have been measured. Confidence intervals for m A
from different sources.
4 3 Comparing Two Sample Variances
F.
The F-test can be extended to the comparison of variances for two samples, A and
B, by rewriting equation 4.16 as
s 2 A
F exp = 2
s
B
2
2
where A and B are defined such that s A is greater than or equal to s B . An example of
this application of the F-test is shown in the following example.
4
EXAMPLE .18
Tables 4.1 and 4.8 show results for two separate experiments to determine the
mass of a circulating U.S. penny. Determine whether there is a difference in the
precisions of these analyses at a= 0.05.
SOLUTION
Letting A represent the results in Table 4.1 and B represent the results in Table
2
2
4.8, we find that the variances are s A = 0.00259 and s B = 0.00138. A two-tailed
significance test is used since there is no reason to suspect that the results for
one analysis will be more precise than that of the other. The null and
alternative hypotheses are
2
2
H 0 : s A = s B 2 H A : s A ¹s B 2
and the test statistic is
s 2 A . 0 00259
.
F exp = 2 = =188
s . 0 00138
B
The critical value for F(0.05, 6, 4) is 9.197. Since F exp is less than F(0.05, 6, 4),
the null hypothesis is retained. There is no evidence at the chosen significance
level to suggest that the difference in precisions is significant.
4 4 Comparing Two Sample Means
F.
The result of an analysis is influenced by three factors: the method, the sample, and
the analyst. The influence of these factors can be studied by conducting a pair of ex-
periments in which only one factor is changed. For example, two methods can be
compared by having the same analyst apply both methods to the same sample and
examining the resulting means. In a similar fashion, it is possible to compare two
analysts or two samples.
Significance testing for comparing two mean values is divided into two cate-
gories depending on the source of the data. Data are said to be unpaired when each
ent sources.
– –
and m B can be written for both samples