Example 19-1

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Introduction to
Analytical Chemistry
CHAPTER 19
STATISTICAL AIDS TO
HYPOTHESIS TESTING
AND GROSS ERRORS
19A Statistical Aids To Hypothesis
Testing
 Scientists and engineers frequently must judge whether
a numerical difference is a manifestation of the random
errors inevitable in all measurements.
 Tests of this kind make use of a null hypothesis, which
assumes that the numerical quantities being compared
are, in fact, the same.
19-2
Copyright © 2011 Cengage Learning
19A-1 Comparing an Experimental
Mean with the True Value
 A common way of testing for bias in an analytical
method is to use the method to analyze a sample
whose composition is accurately known in Figure 19-1.
 Method A has no bias, so the population mean μA is
the true value xt . Method B has a systematic error, or
bias, that is given by
(19-1)
19-3
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19A-1 Comparing an Experimental
Mean with the True Value
 It is likely that the experimental mean
will differ
from the accepted value xt as shown in the figure; the
judgment must then be made whether this difference is
the consequence of random error or, alternatively, a
systematic error.
19-4
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19A-1 Comparing an Experimental
Mean with the True Value
 In treating this type of problem statistically, the
difference
is compared with the difference that
could be caused by random error. If the observed
difference is less than that computed for a chosen
probability level, the null hypothesis that
are
the same cannot be rejected; that is, no significant
systematic error has been demonstrated.
19-5
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19A-1 Comparing an Experimental
Mean with the True Value
 The critical value for rejecting the null hypothesis is
(19-2)
 where N is the number of replicate measurements used
in the test. If a good estimate of σis available, Equation
19-2 can be modified by replacing t with z and s with σ.
19-6
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Example 19-1
 A new procedure for the rapid determination of sulfur
in kerosenes was tested on a sample known from its
method of preparation to contain 0.123% S (xt). The
results were % S = 0.112, 0.118, 0.115, and 0.119. Do
the data indicate that there is bias in the method?
19-7
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Example 19-1
19-8
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Example 19-1
 From Table 3-6 (Chapter 3), we find that at the 95%
confidence level, t has a value of 3.18 for three degrees
of freedom.
 The values of t from the tables are often called critical
values and symbolized tcrit . The test value is calculated
from
19-9
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Example 19-1
 If
we reject the null hypothesis at the
confidence level chosen. The absolute value of t is
used.
 This type of test is often called a two-tailed test. In our
case
 Since 4.375 > 3.18, the critical value of t at the 95%
confidence level, we conclude that a difference this
large is significant and reject the null hypothesis.
19-10
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19A-2 Comparing Two
Experimental Means
 Since we know from Equation 6-5 that the standard
deviation of the mean
is
 and likewise for
19-11
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19A-2 Comparing Two
Experimental Means
 Thus, the variance sd2 of the difference (d = x1 – x2)
between the means is given by
19-12
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19A-2 Comparing Two
Experimental Means
 If we then assume that the spooled standard deviation
spooled is a good estimate of both sm1 and sm2 , then
19-13
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19A-2 Comparing Two
Experimental Means
 Substituting this equation into Equation 19-2 (and also
for xt), we find that
(19-3)
 test value of t is given by
(19-4)
19-14
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Example 19-2
 Two barrels of wine were analyzed for their alcohol
content to determine whether they were from different
sources. On the basis of six analyses, the average
content of the first barrel was established to be 12.61%
ethanol. Four analyses of the second barrel gave a
mean of 12.53% alcohol. The ten analyses yielded a
pooled value of s = 0.070%. Do the data indicate a
difference between the wines? Here we employ
Equation 19-4 to calculate the test statistic t.
19-15
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Example 19-2
 The critical value of t at the 95% confidence level for 10
– 2 = 8 degrees of freedom is 2.31. Since 1.771 < 2.31,
we accept the null hypothesis at the 95% confidence
level and conclude that there is no difference in the
alcohol content of the wines.
19-16
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19A-2 Comparing Two
Experimental Means
 In Example 19-2, no significant difference between the
alcohol content of the two wines was indicated at the
95% confidence level. Note that this statement is
equivalent to saying that is equal to with a certain
probability, but the tests do not prove that the wines
come from the same source.
19-17
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19A-2 Comparing Two
Experimental Means
 In contrast, the finding of one significant difference in
any test would clearly show that the two wines are
different. Thus, the establishment of a significant
difference by a single test is much more revealing than
the establishment of an absence of difference.
19-18
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19B Detecting Gross Errors
 A data point that differs excessively from the mean in a
data set is termed an outlier.
19-19
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19B-1 Using the Q Test
 the absolute value of the difference between the
questionable result xq and its nearest neighbor xn is
divided by the spread w of the entire set to give the
quantity Qexp :
(19-5)
 This ratio is then compared with rejection values Qcrit
found in Table 19-1. If Qexp is greater than Qcrit , the
questionable result can be rejected with the indicated
degree of confidence.
19-20
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Table 19-1
19-21
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Example 19-3
 The analysis of a calcite sample yielded CaO
percentages of 55.95, 56.00, 56.04, 56.08, and 56.23.
The last value appears anomalous; should it be retained
or rejected?
 The difference between 56.23 and 56.08 is 0.15%. The
spread (56.23-55.95) is 0.28%. Thus,
 For five measurements, Qcrit at the 90% confidence
level is 0.64. Because 0.54<0.64, we must retain the
outlier at the 90% confidence level.
19-22
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19B-2 A Word of Caution about
Rejecting Outliers
 Statistical tests like the Q test, assume that the
distribution of the population data is normal, or
Gaussian. Unfortunately, this condition cannot be
proved or disproved for samples that have many fewer
than 50 results. Consequently, statistical rules, should
be used with extreme caution when applied to samples
containing only a few data.
19-23
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THE END
19-24
Copyright © 2011 Cengage Learning

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