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Chapter 4 Evaluating Analytical Data 83

The process by which we determine the probability that there is a significant
difference between two samples is called significance testing or hypothesis testing.
Before turning to a discussion of specific examples, however, we will first establish a
general approach to conducting and interpreting significance tests.

4 E.2 Constructing a Significance Test
A significance test is designed to determine whether the difference between two significance test
or more values is too large to be explained by indeterminate error. The first step A statistical test to determine if the
in constructing a significance test is to state the experimental problem as a yes- difference between two values is
significant.
or-no question, two examples of which were given at the beginning of this sec-
tion. A null hypothesis and an alternative hypothesis provide answers to the ques-
null hypothesis
tion. The null hypothesis, H 0 , is that indeterminate error is sufficient to explain
A statement that the difference between
any difference in the values being compared. The alternative hypothesis, H A , is two values can be explained by
that the difference between the values is too great to be explained by random indeterminate error; retained if the
error and, therefore, must be real. A significance test is conducted on the null hy- significance test does not fail (H 0).
pothesis, which is either retained or rejected. If the null hypothesis is rejected,
then the alternative hypothesis must be accepted. When a null hypothesis is not alternative hypothesis
rejected, it is said to be retained rather than accepted. A null hypothesis is re- A statement that the difference between
two values is too great to be explained by
tained whenever the evidence is insufficient to prove it is incorrect. Because of the
indeterminate error; accepted if the
way in which significance tests are conducted, it is impossible to prove that a null significance test shows that null
hypothesis is true. hypothesis should be rejected (H A ).
The difference between retaining a null hypothesis and proving the null hy-
pothesis is important. To appreciate this point, let us return to our example on de-
termining the mass of a penny. After looking at the data in Table 4.12, you might
pose the following null and alternative hypotheses
H 0 : Any U.S. penny in circulation has a mass that falls in the range of
2.900–3.200 g
H A : Some U.S. pennies in circulation have masses that are less than
2.900 g or more than 3.200 g.
To test the null hypothesis, you reach into your pocket, retrieve a penny, and deter-
mine its mass. If the mass of this penny is 2.512 g, then you have proved that the
null hypothesis is incorrect. Finding that the mass of your penny is 3.162 g, how-
ever, does not prove that the null hypothesis is correct because the mass of the next
penny you sample might fall outside the limits set by the null hypothesis.
After stating the null and alternative hypotheses, a significance level for the
analysis is chosen. The significance level is the confidence level for retaining the null
hypothesis or, in other words, the probability that the null hypothesis will be incor-
rectly rejected. In the former case the significance level is given as a percentage (e.g.,
95%), whereas in the latter case, it is given as a, where ais defined as

confidence level

a=1 -
100
Thus, for a 95% confidence level, ais 0.05.
Next, an equation for a test statistic is written, and the test statistic’s critical
value is found from an appropriate table. This critical value defines the breakpoint
between values of the test statistic for which the null hypothesis will be retained or
rejected. The test statistic is calculated from the data, compared with the critical
value, and the null hypothesis is either rejected or retained. Finally, the result of the
significance test is used to answer the original question.