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Chapter 13: Confidence Intervals – Small Samples
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In this section, the
t
test is used when we have
dependent
samples. Samples
are considered to be dependent when they are paired or matched in some
way. For example, a research physician may want to investigate the effect
of a certain drug in the treatment of migraine headaches. The physician can
select a group of patients he or she sees in a clinic with the same level of
pain. The physician can then administer the same dosage of the same
medication to the patients and record the pain level after a certain fixed
amount of time. The pain level at the beginning and at the end of the test
period for the test subjects will be recorded. Although we have two different
sets of data, it was obtained from the same set of test subjects (assuming all
patients remain in the experiment). Thus, we say that the data are dependent
since the same experimental units (patients) were used. Another situation in
which we may have dependent samples is for example, when patients are
matched or paired according to some variable of interest. Patients may then
be assigned to two different groups. For instance, patients may be paired
according to their age (blood pressure, cholesterol level, etc.). That is, two
patients with the same age will be paired then one will be assigned to one
sample group and the other to another sample group. Caution should be
taken when matching experimental units. In this example we matched by
age, but this does not eliminate the influence of other variables.
In constructing confidence intervals for dependent data, we use the
difference of the values of the before and after or the difference of the values
of the matched pairs. In dealing with the differences we will be
accommodating for the dependency in the data. Also by doing this, we will
have a single sample of differences with which to construct a confidence
interval.
If we will let
d
represent the differences then we will be dealing with a
single variable of the differences. We will assume that these differences are
normally distributed with unknown variance. Thus, the confidence interval
for the difference between two means for two dependent samples will be
reduced to a simple one-sample
t
interval.
The general equation used in constructing a (1 -
)
100 percent confidence
interval for the differences
d
is given by