The t Test for Related Samples Program Transcript DR. JENNIFER ANN MORROW: Welcome to the t test for related samples. My name is Dr. Jennifer Ann Morrow. In today's demonstration, I will go over with you the definition of a t test for related samples. I will give you some of the alternative names for this statistic. I will give you a couple of sample research questions. I will talk about the advantages and disadvantages of this statistic. I will give you the formulas. I'll discuss the assumptions of this analysis. And I will show you how to calculate the effect size. And I will give you examples using both the formula and SPSS. OK, let's get started. A t test for related samples is a statistic that is used when you have only one group of participants. And you want to measure them twice on the same dependent variable. Your sample will then contain two scores for each participant. There are many different names for a t test for related samples. Dependent t test, repeated samples t test, paired samples t test, correlated samples t test, within subjects t test, and within groups t test, these are all the same thing. Some examples of research questions that can be addressed by using a t test for related samples are, question one, does self-esteem change from the beginning to the end of treatment? My independent variable would be time. Before treatment and end of treatment are the two groups, or two levels of that independent variable. And my dependent variable would be self-esteem. Question two, Does alcohol impact memory? My independent variable here would be alcohol. And my two levels or groups would be none and alcohol. And my dependent variable would be memory. And again, don't forget the same participants are in both groups, or levels, of your independent variable. There are many advantages of a related samples t test. First, you need fewer participants than you do for a t test for independent samples. You are able to study changes over time. It also reduces the impact of individual differences. And lastly, it is more powerful, or more sensitive, than a t test for independent samples. There are also some disadvantages. The first disadvantage is called carryover effects. And these can occur when the participant's response in the second treatment is altered by the lingering aftereffects of the first treatment. For example, if you were looking at the impact of caffeine on memory and you first gave your participants 100 milligrams of caffeine. And then you tested their memory. And then gave those same participants 200 milligrams of caffeine. And then tested their ©2012 Laureate Education, Inc. 1 memory. If you don't leave enough time in between your treatments, participants could still have some remnants of caffeine in their body when they get that second treatment. And that is considered carryover effects. The second disadvantage of a related samples t test is called progressive error. An ...

1. The t Test for Related Samples
Program Transcript
DR. JENNIFER ANN MORROW: Welcome to the t test for
related samples. My name is
Dr. Jennifer Ann Morrow.
In today's demonstration, I will go over with you the definition
of a t test for related
samples. I will give you some of the alternative names for this
statistic. I will give you a
couple of sample research questions. I will talk about the
advantages and disadvantages
of this statistic. I will give you the formulas. I'll discuss the

2. assumptions of this analysis.
And I will show you how to calculate the effect size. And I will
give you examples using
both the formula and SPSS. OK, let's get started.
A t test for related samples is a statistic that is used when you
have only one group of
participants. And you want to measure them twice on the same
dependent variable. Your
sample will then contain two scores for each participant.
There are many different names for a t test for related samples.
Dependent t test, repeated
samples t test, paired samples t test, correlated samples t test,
within subjects t test, and
within groups t test, these are all the same thing.
Some examples of research questions that can be addressed by
using a t test for related
samples are, question one, does self-esteem change from the
beginning to the end of
treatment? My independent variable would be time. Before
treatment and end of
treatment are the two groups, or two levels of that independent
variable. And my
dependent variable would be self-esteem.
Question two, Does alcohol impact memory? My independent
variable here would be
alcohol. And my two levels or groups would be none and
alcohol. And my dependent
variable would be memory. And again, don't forget the same
participants are in both
groups, or levels, of your independent variable.
There are many advantages of a related samples t test. First, you

3. need fewer participants
than you do for a t test for independent samples. You are able to
study changes over time.
It also reduces the impact of individual differences. And lastly,
it is more powerful, or
more sensitive, than a t test for independent samples.
There are also some disadvantages. The first disadvantage is
called carryover effects.
And these can occur when the participant's response in the
second treatment is altered by
the lingering aftereffects of the first treatment.
For example, if you were looking at the impact of caffeine on
memory and you first gave
your participants 100 milligrams of caffeine. And then you
tested their memory. And
then gave those same participants 200 milligrams of caffeine.
And then tested their
©2012 Laureate Education, Inc. 1

4. memory. If you don't leave enough time in between your
treatments, participants could
still have some remnants of caffeine in their body when they get
that second treatment.
And that is considered carryover effects.
The second disadvantage of a related samples t test is called
progressive error. And this
can occur when the participant's performance changes
consistently over time due to
fatigue or practice.
For example, you're testing the impact of exercise on stress.
You first have your
participants run a mile. And then you measure their stress level.
And then you have them
do 30 minutes of cardio. And then you measure their stress
level. And again, if you don't
leave enough time in between the two treatments, your
participants can be very fatigued
during your second treatment. And again, this is called
progressive error.
The basic formula for a t test for related samples is as follows,
x sub 2 minus x sub 1,
where x sub 2 is the mean of the second treatment, or second
group, minus the mean of
the first group, or first treatment. And then you divide that by
the standard error of the
difference. Your mean of the second treatment minus the mean
of the first treatment is
also known as the difference score.

5. And your standard error of the difference is your sample
variance divided by n. And you
take the square root of that. The degrees of freedom for a t test
related samples is n minus
1.
There's also another formula that you can use. The
computational formula is as follows,
first, you calculate the standard error of the difference. You
sum up the squared
difference scores and subtract from it the sum of all the
difference scores, square that,
divide by n, divide that by n times n minus 1. And that becomes
the denominator in your t
statistic.
You then take your difference score, or the mean of the second
group, minus the mean of
the first group, and divide that by your standard error of the
difference. And that is your t
test for related samples.
All right, let's recap. So far, I've gone over with you the
definition for a t test for related
samples. I'm given you some of the alternative names for this
statistic. I've given you two

6. examples of sample research questions that can be addressed
using this analysis. I've
talked about the advantages and disadvantages of this analysis.
And I've given you the
formulas to calculate a t test for related samples.
Now let's review t test for related samples in more detail. There
are two basic
assumptions that must be addressed. The first one is that
observations within each
treatment condition must be independent. So no two scores
within one group or level
should be related to each other.
©2012 Laureate Education, Inc. 2

7. The second assumption is that the population distribution of
difference scores, or D
values, must be normally distributed. If you violate either of
these assumptions, you
should not be using a t test for related samples.
For a t test for related samples you can report two measures of
effect size, Cohen's d and
percentage of variance explained. Cohen's d is the mean of the
second group minus the
mean of the first group divided by your standard deviation. And
the percentage of
variance explained is equal to t squared divided by t squared
plus your degrees of
freedom.
Now let's go over a couple of examples. For my first example,
we'll use the formula to
calculate the t test for related samples. My research question is,
does expressive writing
reduce students' drinking? My null hypothesis is that the mean
difference is equal to zero.
Expressive writing has no impact on drinking. My alternative
hypothesis, or my research
hypothesis, is that the mean difference doesn't equal zero.
Expressive writing has an
impact on drinking.
Now let's go through the example. I have five participants. So

8. my degrees of freedom is n
minus 1, which is four. Here my independent variable is equal
to time before and after
expressive writing. And my dependent variable is number of
drinks per week.
So for time one my five participants have these scores for
drinking, nine, four, five, four,
and five. Then those same participants, at time two, after they
have gone through the
expressive writing intervention report drinking four drinks, one,
five, zero, and one.
The mean number of drinks for the first time is 5.4. And the
mean number drinks for the
second time is 2.2. I have a standard deviation of 3.7. I have an
alpha level that I'm
choosing a priori of 0.05. And I'm going to choose a two tailed
test.
So I look in my t distribution table, and then I find that my
critical value that I must
surpass is equal to t plus or minus 2.776. So that is the critical
value that I must surpass in
order for me to say that I have a significant result.
My formula for my t test related samples is equal to t is equal to
the mean of the second
group minus the mean of the first group divided by the square
root of the standard
deviation over n equal to then negative 3.2 over the square root
of 0.74. And it comes out
to be t equals negative 3.72.
So how do I interpret this? It would be t and four degrees of
freedom equals negative 3.72

9. comma p less than 0.05, because it is significant, two tailed. So
what would that be
saying? It would be saying that the number of drinks
significantly decreased from time
one, for a mean of 5.4, to time two, a mean 2.2. Expressive
writing had an impact.
I can also calculate the percent of variance explained. Again,
that's t squared divided by t
squared plus my degrees of freedom. And in this case, my
percent of variance explained
is equal to 0.78.
©2012 Laureate Education, Inc. 3
Now, let's move on to an example using SPSS. First, open up

10. your SPSS program. Now
find the data set that you want to use to conduct your analysis.
Click on File, Open, Data.
Now find the data set that you want to use.
Once you have found the data set, click on it. And then click on
Open. And make sure
your data view window appears on your screen.
For this example, I want to do a t test for related samples to test
if there is a change in
self-esteem from first semester to second semester. My
independent variable would be
time. And my two levels would be first semester and second
semester. And my dependent
variable would be self-esteem.
I'm going to choose an alpha level of 0.05 and a two tailed test.
So to calculate a t test for
related samples click on Analyze, Compare Means, paired
samples t test. Once you do
that, you're t test for related samples, or your paired samples t
test, dialog box will appear
on your screen.
And now, what you have to do is click the two levels of your
independent variable in the
box on the left. So scroll down, we need to find our first level,
self-esteem first semester,
click on that. Once you click on that, you'll see here at the
bottom left, under current
selections, it's now put the variable s esteem, which is self-
esteem first semester.
Go back up to the box and click on your second level, self-
esteem second semester. Once

11. you've clicked on that, you'll see here it also says variable two,
s esteem two, which is the
variable self-esteem second semester.
Once both of the variables appear in the current selections box,
go up to the right arrow to
click it. And now your two variables appear in your paired
variables dialog box on the
right. Now click OK.
As you'll see in your output window, SPSS will give you the
results of your t test for
related samples. Here, self-esteem first semester your mean is
1.72. You have 149
participants. Your standard deviation is 0.74. And your standard
error of the mean 0.06.
Your second level, self-esteem second semester your mean is
1.36. Sample size is the
same 149 because the same participants are in both levels of
your independent variable.
Your standard deviation is 0.60. And your standard error of the
mean is 0.05.
Now let's scroll down in your output box. We'll ignore the
second table. It's not
important. And the third table is your results for your t test for
related samples. The first
thing it will give you is the mean difference. Here, in this case,
it's 0.36. If you scroll to
the right, you'll see your t value of 7.84. Your degrees of
freedom, n minus 1, 148. And
you're significance is 0.000.
So how do we interpret this? You would have t your degrees of
freedom, 148, equals 7.84

12. comma-- you could do one of two things. You could put the
exact p value, p equals
©2012 Laureate Education, Inc. 4
0.000, or you could put p less than 0.001. Again, both are APA
format. Both are
acceptable. And then comma two tailed.
So what is this saying? It is saying that self-esteem significantly
decreased from first
semester, mean equals 1.72, and my standard deviation is 0.74,
to second semester. And
my mean is 1.36. And my standard deviation is 0.60.
I can also calculate the percent of variance explained. And
again, that's equal to t squared
divided by t squared plus your degrees of freedom. And so here,
in this case, my
percentage of variance explained would be 0.29.
OK, let's recap. I went over with you the assumptions for a t
test for related samples. I
showed you how to calculate the different measures of effect
size. And we did an
example using a formula and using SPSS.

13. We have now come to the end of our demonstration. Again,
don't forget to practice on
your own calculating a t test for related samples, both using the
formula and SPSS. Thank
you, and have a great day.
©2012 Laureate Education, Inc. 5