preface to fifteenth edition

2.126 SECTION 2
Should there be more than one known material, a weighted average of the individual differences
(x) should be taken. The value of s should be based on the combined estimate from the two or more
materials (perhaps different primary standards for bases). Should the materials differ markedly in
composition, a plot of the individual constant errors against composition should be made. If the
constant error appear to depend upon the composition, they should not be pooled in a weighted
average.
The t test is also used to judge whether a given lot of material conforms to a particular specification.
If both plus and minus departures from the known value are to be guarded against, a twotailed
test is involved. If departures in only one direction are undesirable, then the 10% level values
for t are appropriate for the 5% level in one direction. Similarly, the 2% level should be used to
obtain the 1% level to test the departure from the known value in one direction only; these constitute
a one-tailed test. More on this subject will be in the next section.
Sometimes just one determination is available on each of several known materials similar in
composition. A single determination by each of two procedures (or two analysts) on a series of
material may be used to test for a relative bias between the two methods, as in Example 2.4. Of
course, the average difference does not throw any light on which procedure has the larger constant
error. It only supplies a test as to whether the two procedures are in disagreement.
2.3.7 Hypotheses About Means
Statistical methods are frequently used to give a “yes” or “no” answer to a particular question
concerning the significance of data. When performing hypothesis tests on real data, we cannot set
an absolute cutoff as to where we can expect to find no values from the population against which
we are testing data, but we can set a limit beyond which we consider it very unlikely to find a
member of the population. If a measurement is made that does in fact fall outside the specified range,
the probability of its happening by chance alone can be rejected; something beyond the randomness
of the reference population must be operating. In other words, hypothesis testing is an attempt
to determine whether a given measured statistic could have come from some hypothesized population.
In attempting to reach decisions, it is useful to make assumptions or guesses about the populations
involved. Such assumptions, which may or may not be true, are called statistical hypotheses and in
general are statements about the probability distributions of the populations. A common procedure
is toset up a null hypothesis, denoted by H 0 , which states that there is nosignificant difference
between two sets of data or that a variable exerts no significant effect. Any hypothesis which differs
from a null hypothesis is called an alternative hypothesis, denoted by H 1 .
Our answer is qualified by a confidence level (or level of significance) indicating the degree of
certainty of the answer. Generally confidence levels of 95% and 99% are chosen to express the
probability that the answer is correct. These are also denoted as the 0.05 and 0.01 level of significance,
respectively. When the hypothesis can be rejected at the 0.05 level of significance, but not at
the 0.01 level, we can say that the sample results are probably significant. If, however, the hypothesis
is alsorejected at the 0.01 level, the results become highly significant.
The abbreviated table on the next page, which gives critical values of z for both one-tailed and
two-tailed tests at various levels of significance, will be found useful for purposes of reference.
Critical values of z for other levels of significance are found by the use of Table 2.26b. For a small
number of samples we replace z, obtained from above or from Table 2.26b, byt from Table 2.27,
and we replace by:
[ N/(N 1)] s
p