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The T-Test for Two Related
Samples (Dependent Samples)
Introduction to Statistics
Chapter 11
March 31, 2009
Class #20
Beyond the one-sample t test
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Are people happier before or after
therapy?
In married couples, is the husband or
wife happier?
Among people with the same IQ, is
performance higher on multiple choice or
essay exams?
How to do this?
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For each person or pair of people, create
difference score
Ignore the original scores
Calculate a t, comparing the mean of
difference scores to 0
df = number of pairs -1
The t Test for Dependent Samples
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Repeated-Measures Design
• When you have two sets of scores from the
same person in your sample, you have a
repeated-measures, or within-subjects design
The t Test for Dependent Samples
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Related-Measures Design
•
•
When each score in one sample is paired, on a oneto-one basis, with a single score in the other sample,
you have a related-measures or matched samples
design.
You use a related-measures design by matching pairs
of different subjects in terms of some uncontrolled
variable that appears to have a considerable impact
on the dependent variable.
The t Test for Dependent Samples

You do a t test for dependent samples
the same way you do a t test for a single
sample, except that:
• You use difference scores
• You assume the population mean is 0
• See formulas on page 292
Difference Scores
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The way to handle two scores per person, or a
matched pair, is to make difference scores.
•
•
For each person, or each pair, you subtract one score
from the other.
Once you have a difference score for each person, or
pair, in the study, you treat the study as if there were a
single sample of scores (scores that in this situation
happen to be difference scores).
A Population of Difference Scores with a
Mean of 0
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The null hypothesis in a repeated-measures
design is that on the average there is no
difference between the two groups of
scores.
This is the same as saying that the mean of
the population of the difference scores is 0.
HO : D  0
H A : D  0
The t Test for Dependent
Samples: Example 1
Step 1

State the statistical hypotheses:
HO : D  0
H A : D  0
Step 2
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Set  and locate the critical region.
  .05
df  number of difference scores  1  8  1  7
t crit  2.365
Step 3: Shade in critical region
Step 4: Calculate the t statistic
Individual
1
2
3
4
5
6
7
8
Before
After
D
D2
Step 4: Calculate the t statistic

Calculate the sample mean of the difference
scores
D

MD 

n
 -16
8
= - 2.00
Step 4: Calculate the t statistic

Calculate the sample variance (see page 292 for
formula)
• s2 = SS/df = SS/(n-1)
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First you need to calculate the SS (use computational
formula from page 93)
• SS = 42 – 162
8
= 42- 256 = 42-32=10
8
Step 4: Calculate the t statistic
 Then
you plug numbers into
variance formula:
 s2
= SS/df = SS/(n-1)
= 10/7 = 1.42
Step 4: Calculate the t statistic

Now compute the estimated standard error (see
formula on page 292)
estimated standard error =
s
MD
=
s2
n
Step 4: Calculate the t statistic
estimated standard error =
1.42 = .421
8
Step 4: Calculate the t statistic
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See new t formula (page 292)
t
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t = 2 - 0 = 4.750
.421
M D    
D
s MD
Step 5: Your decision
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Make a decision – compare t computed in Step 3
(tOBTAINED) with tCRITICAL found in the t table.
•
•
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If tOBT > tCRIT (ignoring signs)  Reject HO
If tOBT < tCRIT (ignoring signs)  Fail to reject HO
t obtained = 4.750
t critical = + 2.365
•
Answer: Reject HO
Interpret your results.
• After the pro-socialized medicine lecture, individuals’
attitudes toward socialized were significantly different
(more positive) than before the lecture.
The t Test for Dependent
Samples: Example 2
At the Olympic level of competition, even the smallest factors can make the
difference between winning and losing. For example, Pelton (1983) has
shown that Olympic marksmen shoot much better if they fire between
heartbeats, rather than squeezing the trigger during a heartbeat. The small
vibration caused by a heartbeat seems to be sufficient to affect the
marksman’s aim. The following hypothetical data demonstrate this
phenomenon. A sample of 6 Olympic marksmen fires a series of rounds while
a researcher records heartbeats. For each marksman, an accuracy score (out
of 100) is recorded for shots fired during heartbeats and for shots fired
between heartbeats. Do the data indicate a significant difference? Test with
an alpha of .05.
During Heartbeats
Between Heartbeats
93
90
95
92
95
91
98
94
96
91
97
97
Go through the same five steps
as in Example 1
Step 3: Shade in critical region
Issues with Repeated Measures
Designs
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Order effects.
•
•
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Use counterbalancing in order to eliminate any
potential bias in favor of one condition because most
subjects happen to experience it first (order effects).
Randomly assign half of the subjects to experience
the two conditions in a particular order.
Practice effects.
•
Do not repeat measurement if effects linger.
Credits
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http://myweb.liu.edu/~nfrye/psy53/ch11.ppt
http://psy.ucsd.edu/~sky/Psyc%2060%20t%20Tests.ppt#2