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10-2 INFERENCE FOR A DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES KNOWN 333

where is the true difference in means of interest. Then by following a procedure similar
to that used to obtain Equation 9-17, the expression for can be obtained for the case
where n n n .
1
2
EXAMPLE 10-3 To illustrate the use of these sample size equations, consider the situation described in
Example 10-1, and suppose that if the true difference in drying times is as much as 10 min-
utes, we want to detect this with probability at least 0.90. Under the null hypothesis, 0.
0
We have a one-sided alternative hypothesis with 10, 0.05 (so z z 0.05 1.645),

and since the power is 0.9, 0.10 (so z z 0.10 1.28). Therefore we may find the re-

quired sample size from Equation 10-6 as follows:
2 2 2 2 2 2
1z z 2 1 1 2 2 11.645 1.282 3182 182 4

n 2 2 11
1 2 110 02
0
This is exactly the same as the result obtained from using the O.C. curves.

10-2.3 Identifying Cause and Effect

Engineers and scientists are often interested in comparing two different conditions to deter-
mine whether either condition produces a significant effect on the response that is observed.
These conditions are sometimes called treatments. Example 10-1 illustrates such a situation;
the two different treatments are the two paint formulations, and the response is the drying
time. The purpose of the study is to determine whether the new formulation results in a
significant effect—reducing drying time. In this situation, the product developer (the experi-
menter) randomly assigned 10 test specimens to one formulation and 10 test specimens to the
other formulation. Then the paints were applied to the test specimens in random order until all
20 specimens were painted. This is an example of a completely randomized experiment.
When statistical significance is observed in a randomized experiment, the experimenter can
be confident in the conclusion that it was the difference in treatments that resulted in the differ-
ence in response. That is, we can be confident that a cause-and-effect relationship has been found.
Sometimes the objects to be used in the comparison are not assigned at random to the
treatments. For example, the September 1992 issue of Circulation (a medical journal pub-
lished by the American Heart Association) reports a study linking high iron levels in the body
with increased risk of heart attack. The study, done in Finland, tracked 1931 men for five years
and showed a statistically significant effect of increasing iron levels on the incidence of heart
attacks. In this study, the comparison was not performed by randomly selecting a sample of
men and then assigning some to a “low iron level” treatment and the others to a “high iron
level” treatment. The researchers just tracked the subjects over time. Recall from Chapter 1
that this type of study is called an observational study.
It is difficult to identify causality in observational studies, because the observed statisti-
cally significant difference in response between the two groups may be due to some other
underlying factor (or group of factors) that was not equalized by randomization and not due to
the treatments. For example, the difference in heart attack risk could be attributable to the dif-
ference in iron levels, or to other underlying factors that form a reasonable explanation for the
observed results—such as cholesterol levels or hypertension.
The difficulty of establishing causality from observational studies is also seen in the
smoking and health controversy. Numerous studies show that the incidence of lung cancer and
other respiratory disorders is higher among smokers than nonsmokers. However, establishing