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AP Statistics Section 10.2 B
Comparative studies are more
convincing than single-sample
investigations. For that reason,
one-sample inference is less
common than comparative
inference. Paired t procedures are
called for in the following
A matched pairs design experiment.
1. Subjects are matched in pairs and
each treatment is given to one
subject in each pair.
2. Each subject receives both
treatments in some order.
Before-and-after observations on the same
To compare the responses to the two
treatments in a matched pairs design
or before-and-after measurements on
the same subjects, apply one-sample t
procedures to the observed difference.
Example: Our subjects are 11 people diagnosed
as being dependent on caffeine. Each subject
was barred from coffee, colas and other
substances containing caffeine for the duration
of the experiment. During one 2-day period
subjects took capsules containing their normal
caffeine intake. During a different 2-day period,
they took placebo capsules. The order in which
subjects took caffeine and the placebo was
At the end of each 2-day period, a test (Beck
Depression Inventory) was given to all 11 subjects.
The table at the right contains data on the subjects’
scores on the depression test.
Higher scores show more
symptoms of depression.
Construct a 90% confidence
interval for the mean change in
depression score.
Parameter The population of interest is all
people dependent on caffeine.
We want to estimate the mean difference
in depression score of individuals in the
population who take both the caffeine
capsule and the placebo.
Conditions Since the population standard
deviation of the differences in depression
scores is not known we will use
one sample t procedures
construct a confidence interval.
Subjects not an SRS - usually volunteers
Conclusions may not generalize to the population.
Normality: Unknown if population is Normal.
Sample too small for CLT.
if not normal:
Normal probability plot?
No serious deviations from Normal, so I will proceed.
If the distribution of x is not Normal, results may not
be accurate.
Based on the randomized design of the experiment,
it is reasonable to assume that the differences are
independent. Also safe to assume that the population
is at least 10 times as large as our sample size.
xdiff  7.364
sdiff  6.918
df  11  1  10
upper tail prob.  .05
t  1.812
n  11
x t
 7.364  1.812
 7.364  3.780
(3.584, 11.144)
We are 90% confident that the mean difference
in depression scores for the population of people
dependent on caffeine is between 3.584 and 11.144.