XULAneXUS
,)1*#
7
//1# .0'!)#

Application of the Paired t-test
Stephanie Wilkerson
,)),30&'/+"""'0',+)3,.(/0 &9-/"'%'0)!,**,+/41)#"141)+#41/
8'/.0'!)#'/ .,1%&00,5,1$,.$.##+",-#+!!#// 5'%'0),**,+/0&/ ##+!!#-0#"$,.'+!)1/',+'++# 5+10&,.'6#"
#"'0,.,$'%'0),**,+/,.*,.#'+$,.*0',+-)#/#!,+0!0 (/'""#))41)#"1
#!,**#+"#"'00',+
')(#./,+0#-&+'#--)'!0',+,$0&#'.#"00#/0 XULAneXUS,)//.0'!)#
2') )#0 &9-/"'%'0)!,**,+/41)#"141)+#41/2,)'//
XULAneXUS: Xavier University of Louisiana’s Undergraduate Research Journal.
Scholarly Note. Vol. 5, No. 1, April 2008
Application of the Paired t-test
Stephanie D. Wilkerson, Mathematics
Faculty Mentors: Dr. Sindhu Unnithan, Mathematics; Dr. V.J. DuRapau, Jr., Mathematics
Abstract
This paper is aimed at introducing hypothesis testing, focusing on the paired t-test. It will
explain how the paired t-test is applied to statistical analyses using an example. Specific
formulas that are used to calculate values based on the data recorded in the example are
given. This paper was originally submitted as part of the required senior Colloquium
presentation for Mathematics majors at Xavier. It is required to research a topic in
mathematics or statistics, and present it to fellow students, faculty, and staff of the
mathematics department.
Key Terms:
Paired t-test; p-value; Student t-test; Hypothesis Testing
Introduction
Statistical analysis involves the calculation of the mean of a set of values in a
sample used for observational study. Statistical analysis can be applied in many fields.
There are now, many methods that are used to perform a statistical analysis. Hypothesis
testing is one method used in statistics. The objective of this paper to explain a form of
hypothesis testing, called the paired t-test.
Hypothesis Testing
Hypothesis testing is used to make an inference about a population that's under
study. The inference is based on the parameter(s) for the statistic, usually the sample
mean and standard deviation. Suppose it is believed that the mean of a population is zero,
the first step in hypothesis testing is to state the null hypothesis (
!
H
0
) and an alternative
hypothesis (
!
H
a
). The null hypothesis is the assumption that the mean will be equal to
zero. The alternative hypothesis is the assumption that the mean will be either greater
than zero, less than zero, or simply, not equal to zero. When the alternative hypothesis
states that the mean is less than zero, the test is called a left-tailed test. It is right tailed
when
!
H
a
states that the mean is greater than zero. The test is called two-tailed when
!
H
a
states that the mean is not equal to zero. The second step in hypothesis testing is to
calculate a test statistic. Based on the value of the test statistic, you will find the p-value
from the table of z-distributions, which is based upon the normal distribution known as
the bell curve. Normal distribution is a function expressed as
!
f (x) =
"1
#
2
$
e
"
1
2
#
2
(x"
µ
)
2
, where
1
Wilkerson: Application of the Paired t-test
Published by XULA Digital Commons, 2008
Paired t-test
2
!
µ
is the mean and
!
"
is the standard deviation of a population. It is displayed as a bell-
shaped curve about its mean. From the p-value you will make a decision to either reject
or not reject the null hypothesis.
The determined p-value is compared to the level of significance,
(alpha). The
level of significance is the probability of making a Type I error. A Type I error occurs
when
!
H
0
is rejected when it is true. If
!
H
0
is not rejected when it is actually false, the
error is called as a Type II error. The alpha level can be computed as one minus the
confidence level. The confidence level is equivalent to the area under the curve.
Common values for the confidence level are .90, .95, and .99. Of the three, the most
commonly used is the confidence level at .95. This suggests a 95% confidence that the
null hypothesis is true. Thus, the level of significance that is commonly used is .05( 1 -
.95). Thus,
!
"
set at .05 suggests that there is a 5% chance of making a Type I error. If
the p-value is less than or equal to
!
"
, the null hypothesis should be rejected. If the p-
value is greater than
!
"
,
!
H
0
should not be rejected. The last step is to explain the results
of the test and what it concludes about the analysis.
Paired t-test
The paired t-test is a type of hypothesis testing that is used when two sets of data
are being observed. The data in a paired t-test are dependent, because each value in the
first sample is paired with a value in the second sample. The parameter used to make the
inference is the difference of the means of both data sets. Similar to our previous
hypothesis test example, the null hypothesis states that the difference of the means is
equal to zero
!
(
µ
1
"
µ
2
= 0)
. This can also be understood as the means are equal. The
alternative hypothesis can be the mean of the first sample is greater than the mean of the
second
!
(
µ
1
"
µ
2
> 0)
, the mean of the first sample is less than the mean of the second
sample
!
(
µ
1
"
µ
2
< 0)
, or the means are not equal with no greater than or less than
comparison
!
(
µ
1
"
µ
2
# 0)
. Thus,
Example 1:
!
H
0
:
µ
1
=
µ
2
!
H
a
:
µ
1
"
µ
2
Example 2:
!
H
0
:
µ
1
=
µ
2
!
H
a
:
µ
1
>
µ
2
!
µ
1
<
µ
2
2
XULAneXUS, Vol. 5 [2008], Iss. 1, Art. 7
https://digitalcommons.xula.edu/xulanexus/vol5/iss1/7
Paired t-test
3
Student's t-test
The Student's t-test is the test statistic used in a paired t-test. The Student's t is a
distribution discovered by a statistician, W.S. Gosset. Because he worked for a company
that did not approve of its employees publishing their research, when publishing his work
he used the name Student. When a population is too large for testing, samples are taken
from the population and used in a test. Sampling is expensive and the t-distribution is
ideal when the sample is small (i.e. < 30). The t-distribution was created for use when
!
"
is unknown. The student t-distribution is similar to normal distribution in that it is bell
shaped and symmetrical. Its shape depends on the degrees of freedom, which is one less
than the sample size (
!
n "1
). As degrees of freedom increase, the t-distribution
approaches normal distribution. Gosset developed the table of t-distributions to find the
p-value that corresponds to the test statistic and the degrees of freedom. The values in the
Student t-table are calculated by formulas developed by Gosset. This falls under another
subject called mathematical statistics, and is beyond the scope of this scholarly note.
Formula to calculate the t test statistic:
!
d = x
1
" x
2
difference
!
d
=
"
d
n
mean difference
!
s
d
=
"
d # d
( )
2
n #1
sample standard deviation
!
t =
d
s
d
" n t test statistic
Demonstrating the Paired t-test
The following sample problem will be used to show how to apply the paired t-
test. The purpose of the test is to determine whether a person's physical condition
improves after jogging. An investigator obtains maximal VO
2
before subjects start
jogging and again six months later. The first sample contains the values of VO
2
that was
recorded for each participant before they started jogging. The second sample contains the
values of VO
2
recorded after jogging.
The data used is given in the following display:
3
Wilkerson: Application of the Paired t-test
Published by XULA Digital Commons, 2008
Paired t-test
4
(In this table, Diffsqrd represents
!
(d " d)
2
.)
!
d
=
"310.1
25
= "12.404
!
"
(d # d)
2
$ 60.86
!
s
d
=
60.86
24
" 1.59
!
"
d = #310.1
4
XULAneXUS, Vol. 5 [2008], Iss. 1, Art. 7
https://digitalcommons.xula.edu/xulanexus/vol5/iss1/7
Paired t-test
5
Step 1:
!
H
0
: The physical condition is the same after jogging
!
(
µ
1
=
µ
2
)
!
"
= 0.05
!
H
a
: The physical condition improves after jogging
!
(
µ
1
"
µ
2
)
This is a two tailed test.
Step 2:
!
t =
"12.404
1.59
25 # "39
Step 3: Since -39 is not on the t-distribution table, the p-value is approximately
zero.
Step 4: Since
!
0 < 0.05
, reject the
!
H
0
in favor of the alternative hypothesis.
Step 5: Because
!
H
0
was rejected, we can conclude that jogging does improve a
person's physical condition.
Conclusion
This scholarly note has presented a brief overview of hypothesis testing and focused
on understanding the paired t-test. An example using the paired t-test was given
developing the null and alternative hypothesis, calculating the test statistics, and drawing
an inference based upon the level of confidence desired. For additional information
regarding any of the topics covered herein, the reader should refer to the books in the
bibliography.
References
Brase, Charles Henry, Brase, Corrinne Pellillo. Understanding Basic Statistics. Boston:
Houghton Mifflin Company. 2007.
Freund, John E. Mathematical Statistics. New Jersey: Prentice-Hall, Inc., 1971.
Koopmans, Lambert H. Introduction to Contemporary Statistical Methods. Boston:
PWS-KENT, 1987.
Acknowledgements
I would like to thank Dr.V.J.DuRapau of Xavier University’s Department of Mathematics
for assisting me in developing this paper.
5
Wilkerson: Application of the Paired t-test
Published by XULA Digital Commons, 2008