Page 104 - Modern Analytical Chemistry

1400-CH04 9/8/99 3:54 PM Page 87

Chapter 4 Evaluating Analytical Data 87
4 F.2 Comparing s to s 2
2
When a particular type of sample is analyzed on a regular basis, it may be possible
2
to determine the expected, or true variance, s , for the analysis. This often is the
case in clinical labs where hundreds of blood samples are analyzed each day. Repli-
2
cate analyses of any single sample, however, results in a sample variance, s . A statis-
2
2
tical comparison of s to s provides useful information about whether the analysis
2
2
is in a state of “statistical control.” The null hypothesis is that s and s are identical,
and the alternative hypothesis is that they are not identical.
The test statistic for evaluating the null hypothesis is called an F-test, and is F-test
given as either Statistical test for comparing two
s 2 s 2 variances to see if their difference is too
F exp = 2 or F exp = 2 large to be explained by indeterminate
s s 4.16 error.
2
2
2
2
(s > s ) (s > s )
2
2
depending on whether s is larger or smaller than s . Note that F exp is defined such
that its value is always greater than or equal to 1.
If the null hypothesis is true, then F exp should equal 1. Due to indeterminate er-
rors, however, the value for F exp usually is greater than 1. A critical value, F(a, n num ,
n den ), gives the largest value of F that can be explained by indeterminate error. It is
chosen for a specified significance level, a, and the degrees of freedom for the vari-
ances in the numerator, n num , and denominator, n den . The degrees of freedom for s 2
is n – 1, where n is the number of replicates used in determining the sample’s vari-
ance. Critical values of F for a= 0.05 are listed in Appendix 1C for both one-tailed
and two-tailed significance tests.
4 7
EXAMPLE .1
A manufacturer’s process for analyzing aspirin tablets has a known variance of
25. A sample of ten aspirin tablets is selected and analyzed for the amount of
aspirin, yielding the following results
254 249 252 252 249 249 250 247 251 252
Determine whether there is any evidence that the measurement process is not
under statistical control at a= 0.05.
SOLUTION
The variance for the sample of ten tablets is 4.3. A two-tailed significance test is
used since the measurement process is considered out of statistical control if
the sample’s variance is either too good or too poor. The null hypothesis and
alternative hypotheses are
2
2
H 0 : s = s 2 H A : s ≠ s 2
The test statistic is
s 2 25
.
F exp = = =58
s 2 . 43
The critical value for F(0.05, ¥, 9) from Appendix 1C is 3.33. Since F is greater
than F(0.05,¥, 9), we reject the null hypothesis and accept the alternative
hypothesis that the analysis is not under statistical control. One explanation for
the unreasonably small variance could be that the aspirin tablets were not
selected randomly.