Page 108 - Modern Analytical Chemistry

1400-CH04 9/8/99 3:55 PM Page 91

Chapter 4 Evaluating Analytical Data 91

SOLUTION
We begin by summarizing the mean and standard deviation for the data
reported by each analyst. These values are
–
X A = 86.83%
s A = 0.32
–
X B = 82.71%
s B = 2.16
A two-tailed F-test of the following null and alternative hypotheses
2
2
H 0 : s A = s B 2 H A : s A ¹s B 2
is used to determine whether a pooled standard deviation can be calculated.
The test statistic is

s B 2 (. ) 2
6
21
F exp = = =45 .6
s 2 (. ) 2
03
2
A
Since F exp is larger than the critical value of 7.15 for F(0.05, 5, 5), the null
hypothesis is rejected and the alternative hypothesis that the variances are
significantly different is accepted. As a result, a pooled standard deviation
cannot be calculated.
The mean values obtained by the two analysts are compared using a two-
tailed t-test. The null and alternative hypotheses are
– – – –
H 0 : X A = X B H A: X A ≠ X B
Since a pooled standard deviation could not be calculated, the test statistic, t exp ,
is calculated using equation 4.19

X - X B 86 .83 -82 .71
A
t exp = = = . 462
2 2 2 2
6
21
+
2
6
03
n ) +
6
s (/ A ( s n ) [( . ) / ] [( . ) / ]
/ B
A B
and the degrees of freedom are calculated using equation 4.22
2
2
[( . 032 6 +( . 216 6 2
/ )]
/ )
n= -2 = . 5 3 » 5
2
2
1
/ ) /(6
{( . 032 6 2 + )} +{( . 216 6 2 +)}
/ ) /(6
1
The critical value for t(0.05, 5) is 2.57. Since the calculated value of t exp is
greater than t(0.05, 5) we reject the null hypothesis and accept the alternative
hypothesis that the mean values for %w/w Na 2 CO 3 reported by the two
analysts are significantly different at the chosen significance level.
Paired Data In some situations the variation within the data sets being compared
is more significant than the difference between the means of the two data sets. This
is commonly encountered in clinical and environmental studies, where the data
being compared usually consist of a set of samples drawn from several populations.
For example, a study designed to investigate two procedures for monitoring the
concentration of glucose in blood might involve blood samples drawn from ten pa-
tients. If the variation in the blood glucose levels among the patients is significantly
larger than the anticipated variation between the methods, then an analysis in which
the data are treated as unpaired will fail to find a significant difference between the