### Math 225 Course Notes

### Section 7.4: Paired Comparisons

Designing an experiment where two measurements may be taken
on the same individual is an effective way of limiting the variablity
caused by variables which are not of primary interest.
In the analysis of such data,
the proper methodology is to take differences between the measurements
on each individual,
and to treat these differences as a single sample.

"Breathe-Right" nasal strips were developed to assist people who have severe
snoring problems and other breathing disorders.
Recently, many athletes have begun to use them as a device for enhancing
athletic performance.
A team of researchers desired to test whether "Breathe-Right" nasal strips
increased oxygenation to the body.
One variable of interest is maximal oxygen consumption (VO_{2} Max).
The researchers tested a sample of 16 male athletes on an exercise bicycle,
with the strip and without.
(There was a time period of 2 days between tests for each athlete.)
The data from the experiment is summarized as (Mean +/- SD):
Without Breathe-Right 3326 +/- 520
With Breathe-Right 3367 +/- 615
Difference 41.4 +/- 438.3

There are many variables that may affect VO_{2} Max besides
whether the "Breathe-Right" nasal strips were worn or not.
However,
most of these variables (such as age, cardiovascular fitness, etc.)
are controlled for by having the same athlete tested twice.
Each measurement without the strip
is matched to a corresponding measurement on the same individual
with the strip.
The sixteen individual differences in VO_{2} Max consist
of a single sample of differences.
To test the hypothesis that the nasal strips improve VO_{2} Max,
the test statistic is

t = (41.4 - 0) / (438.3 / sqrt{16}) = 0.38

In other words,
the observed difference in sample means is 0.38 estimated standard errors
above the mean,
a value that is easily attributable to chance variation.
A one-sided p-value is 0.35.
Since we expect to see a result this extreme 35% of the time
when the null hypothesis is true
(more than once every three experiments),
we are not surprised that it occured.
Hence, the data is consistent with the hypothesis
that "Breathe-Right" nasal strips have no effect on VO_{2} Max.

Last modified: April 15, 1996

Bret Larget,
larget@mathcs.duq.edu