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1. Mary Anne Portuguez
MP-IP-1
The t Test for Two Related
Samples

2. PRESENTATION OF PROBLEM: THE WHY OF USING T TEST
FOR RELATED SAMPLES
Independent-measures design is prone to difficulty in
handling individual differences (e.g. IQ, age, gender, etc.)
which decreases its efficiency (Kantowitz et. al., 1988).
The mean difference between groups may be elucidated by
person characteristics rather than the effectiveness of a
particular treatment (Gravetter & Wallnau, 2012).
Through repeated-measures design, the problem is
addressed because the two sets of scores come from same
group of individuals. Therefore, the individual in one
treatment is perfectly matched with the individuals in the
other treatment (Gravetter & Wallnau, 2012).

3. THE RELATED-SAMPLES T TEST
The related-samples t-test is a parametric procedure used
with two related samples. Related samples happen when
researcher pair each scores in one sample with a
particular score in the other sample. Researchers generate
related samples to have more equivalent, hence, more
comparable samples. There are two types of research
designs that produce related samples namely: matched-
samples designs and repeated-measures designs
(Heiman, 2011).

4. THE TWO RESEARCH DESIGNS THAT PRODUCE RELATED
SAMPLES
 Matched-samples design refers to assigning
subjects to groups in which pairs of subjects are first
matched on some characteristic and then individually
assigned randomly to groups (Cozby, 2012; McGuigan,
1990; Gravetter & Wallnau, 2012).
 Repeated-measures design pertains to one in which
a single sample of individuals is measured or tested
under all conditions of independent variable
(Heiman, 2011; Gravetter & Wallnau, 2012;
McGuigan, 1990; Kantowitz et. al., 1988; Cozby,
2012).

5. THE T STATISTIC FOR REPEATED-MEASURES
DESIGN
The main distinction of related-samples t is that it is based
on difference scores rather than raw scores obtained from
a single group of participants. The test involves computing
t statistic and then consulting a statistical table to confirm
whether the t value obtained is adequate to indicate a
significant mean difference (Gravetter & Wallnau, 2012).

6. DIFFERENCE SCORES: THE DATA FOR REPEATED-MEASURES
STUDY
difference scores= D =
The sign of each D scores determines the direction of
change and the sample of difference scores (D values) will
serve as the sample data for the hypothesis test.

7. SAMPLE:

8. THE T STATISTIC FOR RELATED SAMPLES

9. HYPOTHESIS TESTS FOR REPEATED-MEASURES DESIGN
Step 1 State the hypotheses, and select the alpha level.
Step 2 Locate the critical region.
Step 3 Calculate the t statistic.
Step 4 Make a decision. If the t value falls or exceed on
the critical region, rejection of the null hypothesis
must be done.

10. EFFECT SIZE FOR REPEATED-MEASURES DESIGN
Cohen’s d formula
Evaluation of Effect size with Cohen’s d

11. PERCENTAGE OF VARIANCE ACCOUNTED FOR
Criteria for interpreting as proposed by Cohen

12. SITUATIONAL EXERCISE 1.0
One technique to help people deal with phobias is to have
them counteract the feared objects by using imagination to
move themselves to a place of safety. IN experimental test of
this technique, patients sit in front of a screen and are
instructed to relax. Then, they are shown an image of the
feared object. The patient signals the researcher as soon as
feelings of anxiety reach a point at which viewing the image
can no longer be tolerated. The researcher records the
amount of time that the patient was able to endure looking at
the image. The patient then spends 2 minutes imagining a
“safe scene” such as a tropical beach or a familiar room
before the image is presented again. As before, the patient
signals when the level of anxiety is intolerable. If patients can
tolerate the feared objects longer after the imagination
exercise, it is viewed as a reduction in phobia.

13. A sample of n =7 patients.
Question: Do the data (presented in the table) indicate
that the imagination technique effectively alters
phobia?
Step 1 State the hypotheses, and select the alpha level.
(There is no change in the phobia.)
(There is a change.)
For this test, we will use α= .01

14. EXERCISE 1.0
PATIENTS TIME BEFORE TIME AFTER D
A 15 24 9 81
B 10 23 13 169
C 7 11 4 16
D 18 25 7 49
E 5 14 9 81
F 9 14 5 25
G 12 21 9 81
Total = 56 = 502

15. Step 2 Locate the critical region.
For this sample, n= 7 has a df = n – 1.
7-1 = 6. Thus, df= 6.
From the t distribution, you should find the critical
value for α= .01 is ± 3.707.

16. Step 3 Calculate the t statistic.
In the sample data, and

17.  Step 4 Make a decision. If the t value falls or exceed on
the critical region, rejection of the null hypothesis must
be done.
In our example, the researcher rejects the null
hypothesis. The t value falls or exceed the critical region,
thus, concludes that the imagination technique does
affect the onset of anxiety when patients are exposed to
feared object.

18. MEASURING EFFECT SIZE
Cohen’s d
The treatment effect is statistically significant.
Percentage of Variance

19. REPORTING THE RESULTS
Imagining a safe scene increased the amount of
time that the patients could tolerate a feared stimulus
before feeling anxiety by M= 8. 00 with SD= 3. The
treatment effect is statistically significant, t (6) = 7.05,
p < .01, = 0.892.

20. REPEATED-MEASURES DESIGN VERSUS
INDEPENDENT-MEASURES DESIGN

21. Repeated-measures design
Advantages:
 Requires only few participants in a study or experiment (Goodwin,
2010; Gravetter & Wallnau, 2012; Cozby, 2012).
 It uses the subjects more efficiently for each individual is measured
in both treatment conditions (Gravetter & Wallnau, 2012;
McGuigan, 1990).
 It saves laboratory time or energy (McGuigan, 1990).
 It reduces problems caused by individual differences (e.g. IQ,
gender, age, personality, etc.) that differ from one person to another
(Gravetter & Wallnau, 2012). Therefore, greater ability to detect an
effect to independent variable (Cozby, 2012).
 It is a reasonable choice when it comes to study concerning
learning, development, physiological psychology, sensation and
perception, or other changes that happen over time. A researcher
can see difference in behaviors that develop or change over time
(McGuigan, 1990; Gravetter & Wallnau, 2012).
 It reduces error variance since in within-groups design, the
experimenter or researcher repeats measure on the same
participants. It removes individual differences from error variance
(McGuigan, 1990).

22. Disadvantages:
 There may be order effects in which changes in scores
of participants are caused by earlier treatments
(McGuigan, 1990; Gravetter & Wallnau, 2012).
 Since within-subject design involves measuring same
individuals at one time and then measuring them
again at a later time, outside factors that change over
time may be responsible for changes in the
participants’ scores (Gravetter & Wallnau, 2012).
 The researcher must be on guard on factors related to
time such as practice effects and fatigue effects. The
former refers to participants performing better in the
experimental task simply because of practice while
the latter refers to decrease in performance due to
tasks that are long, difficult, and boring (Kantowitz et.
al., 1988).

23. Independent-measures design:
Advantages:
 Each subject enters the study fresh and naive with
respect to the procedures to be tested (Goodwin,
2010).
 It is conservative. There is no chance that one
treatment will contaminate the other, since the same
person never receives both treatments (Kantowitz et.
al., 1988).
 The participants are randomly assigned in the
different groups differed in ability, thus, prevent
systematic biases, and the groups can be considered
equivalent in terms of their characteristics
(Kantowitz et. al., 1988; Cozby, 2012).
 Researcher or experimenter can minimize the
possibility of confounding (Kantowitz et. al., 1988).

24. Disadvantages:
 Requires more participants that need to be recruited,
tested, and debriefed (Goodwin, 2010).
 It must deal with differences among people and this
decreases its efficiency (Kantowitz et. al., 1988).

25. ASSUMPTIONS OF THE RELATED-SAMPLES T TEST
The related-samples t statistics requires two basic
assumptions (Gravetter & Wallnau, 2012):
 The observations within each treatment condition
must be independent. The assumption of
independence here pertains to scores within each
treatment. Within each treatment, the scores are
gathered from different persons and must be
independent of one another.
 The population distribution of difference scores (D
values) must be normal.

26. EXERCISES 2.0
a. A repeated-measures study with a sample of n= 9
participants produces a mean difference of = 3 with a
standard deviation of s= 6. Based on the mean and
standard deviation you should be able to visualize (or
sketch) the sample distribution. Use a two-tailed
hypothesis test with = .05 to determine whether it is likely
that this sample came from a population with = 0.
b. Now assume that the sample mean difference is = 12, and
once again visualize the sample distribution. Use a two-
tailed hypothesis test with = .05 to determine whether it is
likely that this sample came from a population with = 0.
c. Explain how the size of the sample mean difference
influences the likelihood of finding a significant mean
difference.

27. SOLUTION:
 The estimated standard error is 2 points and t(8) =
1.50. With a critical boundary of 2.306, fail to reject
the null hypothesis.
 With = 12, t (8) = 6. With a critical boundary of
2.306, reject the null hypothesis.
 The larger the mean difference, the greater the
likelihood of finding a significant difference.

28. REFERENCES
 McGuigan, F.J. (1990). Experimental psychology (5th ed.).
Englewood Cliffs, New Jersey: Prentice Hall.
 Kantowitz, B.H. et. al. (1988). Experimental psychology (3rd
ed.). New York: West Publishing Company.
 Gravetter, F.J. & Wallnau, L.B. (2012). Statistics for
behavioral sciences. Philippines: Cengage Learning Asia
Pte. Ltd.
 Cozby, P.C. & Bates, S.C. (2012). Methods in behavioral
research (11th ed.). New York: McGraw- Hill Companies,
Inc.
 Heiman, G.W. (2011). Basic statistics for the behavioral
sciences (6th ed.). Belmont, CA: Wadsworth, Cengage
Learning.
 Goodwin, C.J. (2010). Research in psychology methods and
design (6th ed.). USA: John Wiley & Sons, Inc.