Generic TStatistic Formula
Note that for all situations in which the TStatistic
is used, the formula involves the same generic structure:
TTests for 2 Related Samples
Chapter 11 presents TTests for the situation where
there are two related samples of scores. This situation
commonly occurs in the repeatedmeasures experimental
study. It can also occur in the matchedsubject
experimental study. These designs are sometimes called
dependent sample studies or withinsubject
designs.
 RepeatedMeasures Study
 A repeatedmeasures study is one in which a single
sample of subjects is used to compare two (or more)
different treatment conditions. Each individual
is measured in one treatment, and then the same
individual is measured again in the second treatment.
Thus, a repeatedmeasures study produces two (or
more) samples of scores, but each sample of scores
is obtained from the same sample of subjects. (Note
we have an ambiguous usage of the word "sample"
here).
 MatchedSubject Study
 A matchedsubject study is one in which each individual
in one sample is matched with a a subject in another
sample. The matching is done so that the two individuals
are as equivalent as possible with respect to a
specific variable (or variables) that the researcher
would like to control.
 Dependent Samples
 Related samples are sometimes called "dependent
samples" because the values observed in one sample
of scores "depend" on those observed in the other
sample of scores.
 Within Subject Design
 Related samples are sometimes called "within subject
designs" because the values observed "within subjects"
(for the same subject) across conditions.
TStatistic for Related Samples
The TStatistic for two related samples uses the generic
TStatistic formula:
The specific TStatistic for two related samples is
identical to the TStatistic for a single sample, except
that the statistic is defined on the differences
between the scores for the two related samples. Thus:
 Sample Statistic:
 The difference score for the subjects is simply
the second score minus the first:
 Population Parameter:
 The hypothesized difference between the two population
means:
 Estimated Standard Error:
 The estimated standard error of the difference
scores is defined exactly as for the single sample
TStatistic, except that everything (mean, standard
deviation, n, etc.) is defined on the differences:
 TStatistic:
 T is defined just as with a single sample, except
that is defined on difference scores. It becomes:
 Hypothesis Testing:
 Hypotheses are constructed just as before, except
that they are about the differences (usually involving
hypotheses about zero differences). The possible
pairs of onetail and twotail alternative and null
hypotheses are:
One Tail, Less Than 0

Two Tail

One Tail, Greater Than 0

 DegreesofFreedom:
 The Tvalue is then evaluated just as before,
using degreesoffreedom of
Independent Samples TTest Example:
Asthma Attacks
A researcher in behavioral medicine believes that stress
often makes asthma symptoms worse for people who suffer
from this respiratory disorder. Therefore, the researcher
decides to study the effect of relaxation training on
the severity of their symptoms.
A sample of 5 patients is selected. During the week
before treatment, the investigator records the severity
of their symptoms by measuring how many doses of medication
are needed for asthma attacks. Then the patients receive
relaxation training. For the week following the training
the research once again records the number of doses
used by each patient.

Hand Calculations:
The classical hypothesis testing steps are:
 State the hypothesis: We test the hypothesis
that there will be fewer medication doses used
after relaxation than before (before minus after
will be positive). Note that this differs from
the book, where they use a nondirectional test.
Thus we have:
 Set the decision criteria: We use alpha=.05,
one tail, for df=4. We find the critical Tvalue
is 2.132.
 Gather the data: Here is ViSta's datasheet
for the Asthma data:
 Evaluate the hypothesis: The mean difference
is 3.2 (3.2 fewer doses after relaxation). The
variance of the differences is 3.7. The estimate
of the standard error of the difference scores
is .86. We now calculate T=3.72. Since T is
in the critical region we reject the null hypothesis
that there is no reduction in the number of
medication doses after relaxation.

Computer Calculations:
ViSta can be used to analyze these data, as specified
in the ViSta Applet.
We specified a directional TTest: There will be
fewer medication doses used after relaxation than
before. Note that this differs from the book, where
they use a nondirectional test. We selected the
"before" variable as the first variable and the
"after" variable as the second one (the second variable
is subtracted from the first).
We obtained the following workmap:
ReportModel: The analysis produces the
following report, which corresponds with the hand
calculations. In particular, p=.0102, which suggest
we can safely reject the null hypothesis that
there is no reduction in the number of medication
doses after relaxation. The homogeneity of variance
test indicates that the two conditions have about
the same variance, an assumption required for
the TTest to be valid.
VisualizeModel: The visualization shown
below suggests that the data are not symmetrically
distributed, since the jagged lines don't follow
the straight lines in the quantile plots and in
the quantilequantile plot, and since the boxes
in the box and diamond plot are not symmetric.
The quantilequantile plot indicates that the
two conditions have about the same variance. The
normalprobability plots suggest that the data
may not be normal.
This implies that the assumption of normality
underlying the TTest may be violated. Consequently,
the p value (p=.0102) may be to optimistic. This
value may be misleading, particularly with a small
sample size.
