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298 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

will be unknown. Furthermore, we may not be certain that the population is well modeled by a
normal distribution. In these situations if n is large (say n 40) the sample standard deviation s
can be substituted for in the test procedures with little effect. Thus, while we have given a test
2
for the mean of a normal distribution with known , it can be easily converted into a large-
2
sample test procedure for unknown that is valid regardless of the form of the distribution
of the population. This large-sample test relies on the central limit theorem just as the large-
sample confidence interval on that was presented in the previous chapter did. Exact treatment
2
of the case where the population is normal, is unknown, and n is small involves use of the
t distribution and will be deferred until Section 9-3.

9-2.6 Some Practical Comments on Hypothesis Tests

The Eight-Step Procedure
In Section 9-1.4 we described an eight-step procedure for statistical hypothesis testing. This
procedure was illustrated in Example 9-2 and will be encountered many times in both this
chapter and Chapter 10. In practice, such a formal and (seemingly) rigid procedure is not
always necessary. Generally, once the experimenter (or decision maker) has decided on
the question of interest and has determined the design of the experiment (that is, how the data
are to be collected, how the measurements are to be made, and how many observations are
required), only three steps are really required:
1. Specify the test statistic to be used (such as Z 0 ).
2. Specify the location of the critical region (two-tailed, upper-tailed, or lower-tailed).
3. Specify the criteria for rejection (typically, the value of , or the P-value at which
rejection should occur).

These steps are often completed almost simultaneously in solving real-world problems,
although we emphasize that it is important to think carefully about each step. That is why we
present and use the eight-step process: it seems to reinforce the essentials of the correct
approach. While you may not use it every time in solving real problems, it is a helpful frame-
work when you are first learning about hypothesis testing.
Statistical versus Practical Significance
We noted previously that reporting the results of a hypothesis test in terms of a P-value is very
useful because it conveys more information than just the simple statement “reject H ” or “fail
0
to reject H ”. That is, rejection of H at the 0.05 level of significance is much more meaning-
0
0
ful if the value of the test statistic is well into the critical region, greatly exceeding the 5% crit-
ical value, than if it barely exceeds that value.
Even a very small P-value can be difficult to interpret from a practical viewpoint when
we are making decisions because, while a small P-value indicates statistical significance in
the sense that H 0 should be rejected in favor of H 1 , the actual departure from H 0 that has been
detected may have little (if any) practical significance (engineers like to say “engineering
significance”). This is particularly true when the sample size n is large.
For example, consider the propellant burning rate problem of Example 9-3 where we are
testing H 0 : 50 centimeters per second versus H 1 : 50 centimeters per second with
2. If we suppose that the mean rate is really 50.5 centimeters per second, this is not a seri-
ous departure from H 0 : 50 centimeters per second in the sense that if the mean really is
50.5 centimeters per second there is no practical observable effect on the performance of the
aircrew escape system. In other words, concluding that 50 centimeters per second when
it is really 50.5 centimeters per second is an inexpensive error and has no practical signifi-
x
cance. For a reasonably large sample size, a true value of 50.5 will lead to a sample that