Student's t-Test for Matched Pairs
© 1998 by Dr. Thomas W. MacFarland -- All Rights Reserved
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stu_pair.doc
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Background: Student's t-test is a very common (and possibly
overused) test for determining differences
between two groups. The t-test (developed in
1915 by Gosset for the Guinness Breweries of
Dublin) is the appropriate test for small samples,
as opposed to samples with greater than 30 or
more observations. And recall:
-- Student's t-test is still the appropriate test
with greater than 30 observations.
-- With n => 30 observations, t approximates z.
Student's t-test is typically used to determine
if the difference between two independent groups
is indeed a true difference, or if the difference
between the two groups is instead due only to chance.
It is not suggested, however, that the t-test is
only used to test the difference between separate
groups. On the contrary, Student's t-test can
also be used to determine differences among the
same (or similar, i.e., matched) subjects for a
specific phenomenon.
Scenario: This study examines if separating students who
previously worked on programming assignments
as a pair has any impact on project quality
when:
-- One member of the pair is required to work
on their own.
-- The other member of the pair is assigned a
new programming partner.
Dr. Kieta teaches Cobol I and Cobol II at a
local community college. There are 60 students
in Cobol I, with instruction offered in a
laboratory-type classroom setting with only 30
computers. Logically, Dr. Kieta used random
assignment to segregate the 60 students into
30 sets of paired students.
During the 10-day break between Cobol I and Cobol
II, more computers are brought into the classroom.
Further, many of the students who enrolled in
Cobol I immediately enrolled for Dr. Kieta's
class in Cobol II.
Seeing an opportunity for a research project on
group vs. individual work with the introduction
of these new computers, Dr. Kieta makes new
assignments:
-- 20 students who were enrolled in Cobol I,
working with an assigned partner, now have the
chance to work on their own in Cobol II.
-- Their counterparts from Cobol I are now assigned
a new partner for Cobol II.
This test will determine if there are differences
in the Cobol II final examination among these 40
Cobol I and Cobol II students.
Dr. Kieta previously examined this data set with
the non-parametric Wilcoxon test. However, he
has since reviewed the final examination and now
is convenienced that the data represent interval
data (i.e., the data are parametric, with the
difference between "89" and "90" equal to the
difference between "75" and "76").
Accordingly, Dr. Kieta will now use Student's
t-Test for Matched Pairs to determine if there
are differences, instead of the non-parametric
Wilcoxon test.
A summary of the study is presented in Table 1.
Table 1
Final Exam Scores in Cobol II of Students Who Were
Previously Paired Together in Cobol I
===================================================
Exam Score for Previously
Paired Students
----------------------------------------
Pair Member: Pair Member:
Worked on Own Worked With a New
Pair in Cobol II Partner in Cobol II
---------------------------------------------------
01 090 083
02 084 089
03 078 058
04 079 093
05 065 049
06 049 082
07 092 093
08 071 100
09 085 083
10 076 055
11 068 065
12 071 083
13 088 081
14 089 088
15 076 083
16 061 078
17 088 088
18 058 080
19 073 070
20 082 083
_______________________________________________________
Ho: Null Hypothesis: There is no difference in final
examination test scores in Cobol II between students
who were previously paired together in Cobol I, with
the pairs now redesigned so that one student in Cobol
II now works on their own and their matched counterpart
now works with another student in Cobol II (p <= .05).
Files: 1. stu_pair.doc
2. stu_pair.dat
3. stu_pair.r01
4. stu_pair.o01
5. stu_pair.con
6. stu_pair.lis
Command: At the Unix prompt (%), key:
%spss -m < stu_pair.r01 > stu_pair.o01
************
stu_pair.dat
************
01 090 083
02 084 089
03 078 058
04 079 093
05 065 049
06 049 082
07 092 093
08 071 100
09 085 083
10 076 055
11 068 065
12 071 083
13 088 081
14 089 088
15 076 083
16 061 078
17 088 088
18 058 080
19 073 070
20 082 083
************
stu_pair.r01
************
SET WIDTH = 80
SET LENGTH = NONE
SET CASE = UPLOW
SET HEADER = NO
TITLE = Student's t-Test for Matched Pairs
COMMENT = This file examines if separating students who
previously worked on programming assignments
as a pair has any impact on project quality
when: one member of the pair is required to
work on their own and the other member of the
pair is assigned a new programming partner.
Dr. Kieta teaches Cobol I and Cobol II at a
local community college. There are 60 students
in Cobol I, with instruction offered in a
laboratory-type classroom setting with only 30
computers. Logically, Dr. Kieta used random
assignment to segregate the 60 students into
30 sets of paired students.
During the 10-day break between Cobol I and Cobol
II more computers are brought into the classroom.
Further, many of the students who enrolled in
Cobol I immediately enrolled for Dr. Kieta's
class in Cobol II.
Seeing an opportunity for a research project on
group vs. individual work with the introduction
of these new computers, Dr. Kieta makes new
assignments:
-- 20 students who were enrolled in Cobol I,
working with an assigned partner, now have the
chance to work on their own in Cobol II
-- their counterparts from Cobol I are now assigned
a new partner for Cobol II
This test will determine if there are differences
in the Cobol II final examination among these 40
Cobol I and Cobol II students.
Dr. Kieta previously examined this data set with
the non-parametric Wilcoxon test. However, he
has since reviewed the final examination and now
is convenienced that the data are represent
interval data (i.e., the data are parametric,
with the difference between "89" and "90" equal
to the difference between "75" and "76").
Accordingly, Dr. Kieta will now use Student's
t-Test for Matched Pairs to determine if there
are differences, instead of the non-parametric
Wilcoxon test.
DATA LIST FILE = 'stu_pair.dat' FIXED
/ Pair 20-21
Wk_Own 34-36
Wk_New 49-51
Variable Labels
Pair "Pair Number"
/ Wk_Own "Exam Score: Student Who Worked on Their Own"
/ Wk_New "Exam Score: Student Assigned a New Partner"
T-TEST PAIRS = Wk_Own, Wk_New
***********
reading.o01
***********
1 SET WIDTH = 80
2 SET LENGTH = NONE
3 SET CASE = UPLOW
4 SET HEADER = NO
5 TITLE = Student's t-Test for Matched Pairs
6 COMMENT = This file examines if separating students who
7 previously worked on programming assignments
8 as a pair has any impact on project quality
9 when: one member of the pair is required to
10 work on their own and the other member of the
11 pair is assigned a new programming partner.
12
13 Dr. Kieta teaches Cobol I and Cobol II at a
14 local community college. There are 60 students
15 in Cobol I, with instruction offered in a
16 laboratory-type classroom setting with only 30
17 computers. Logically, Dr. Kieta used random
18 assignment to segregate the 60 students into
19 30 sets of paired students.
20
21 During the 10-day break between Cobol I and Cobol
22 II more computers are brought into the classroom.
23 Further, many of the students who enrolled in
24 Cobol I immediately enrolled for Dr. Kieta's
25 class in Cobol II.
26
27 Seeing an opportunity for a research project on
28 group vs. individual work with the introduction
29 of these new computers, Dr. Kieta makes new
30 assignments:
31
32 -- 20 students who were enrolled in Cobol I,
33 working with an assigned partner, now have the
34 chance to work on their own in Cobol II
35
36 -- their counterparts from Cobol I are now assigned
37 a new partner for Cobol II
38
39 This test will determine if there are differences
40 in the Cobol II final examination among these 40
41 Cobol I and Cobol II students.
42
43 Dr. Kieta previously examined this data set with
44 the non-parametric Wilcoxon test. However, he
45 has since reviewed the final examination and now
46 is convenienced that the data are represent
47 interval data (i.e., the data are parametric,
48 with the difference between "89" and "90" equal
49 to the difference between "75" and "76").
50
51 Accordingly, Dr. Kieta will now use Student's
52 t-Test for Matched Pairs to determine if there
53 are differences, instead of the non-parametric
54 Wilcoxon test.
55 DATA LIST FILE = 'stu_pair.dat' FIXED
56 / Pair 20-21
57 Wk_Own 34-36
58 Wk_New 49-51
59
This command will read 1 records from stu_pair.dat
Variable Rec Start End Format
PAIR 1 20 21 F2.0
WK_OWN 1 34 36 F3.0
WK_NEW 1 49 51 F3.0
60 Variable Labels
61 Pair "Pair Number"
62 / Wk_Own "Exam Score: Student Who Worked on Their Own"
63 / Wk_New "Exam Score: Student Assigned a New Partner"
64
65 T-TEST PAIRS = Wk_Own, Wk_New
T-TEST requires 104 bytes of workspace for execution.
t-tests for Paired Samples
Number of 2-tail
Variable pairs Corr Sig Mean SD SE of
Mean
-------------------------------------------------------------------------------
WK_OWN Exam Score: Student Who Worke 76.1500 11.753
2.628
20 .312 .181
WK_NEW Exam Score: Student Assigned 79.2000 13.344
2.984
-------------------------------------------------------------------------------
Paired Differences |
Mean SD SE of Mean | t-value df 2-tail
Sig
----------------------------------|--------------------------------------------
-3.0500 14.781 3.305 | -.92 19
.368
95% CI (-9.968, 3.868) |
***********
stu_pair.con
***********
Outcome: Computed t = |-0.92|
Criterion t = |-2.09| (alpha = .05, df = 19)
Computed t |-0.92| < Criterion t |-2.09|
Note. The | and | characters are used to indicate
absolute value.
Therefore, the null hypothesis is accepted and it can
be claimed that there is no difference (p <= .05) in
final examination test scores in Cobol II between
students who were previously paired together in
Cobol I, with the pairs now redesigned so that one
student in Cobol II now works on their own and their
matched counterpart now works with another student in
Cobol II.
The p value is another way to view differences in
the three graded activities:
-- The calculated p value is .368.
-- The delcared p value is .05.
The calculated p value exceeds the declared p value
and there is, accordingly, no difference in final
examination scores between the two groups of Cobol
II students.
To be more exact:
1. Students from Cobol I who later worked on
their own in Cobol II had a mean score of
76.15 (SD = 11.75).
2. Students from Cobol I who later worked with
an assigned partner in Cobol II had a mean
score of 79.20 (SD = 13.34).
3. There is no difference in these test scores
(p <= .05) and it can be claimed that pairing
and/or not pairing had no influence on test
scores.
If final examination test scores are indeed a
measure of summative learning, then this study
serves as a source of evidence that the practice
of pairing students, when there are insufficient
computers to allow one-on-one programming
assignments, can be defended. Pairing does not
influence final examination scores.
Of course, this example is a "one-shot"
experiment. Replication, including diverse
locations and variety in subjects, would be
needed before purporting the value of this
teaching methodology.
************
stu_pair.lis
************
% minitab
MTB > outfile 'stu_pair.lis'
Collecting Minitab session in file: stu_pair.lis
MTB > # MINITAB addendum to 'stu_pair.dat'
MTB > #
MTB > read 'stu_pair.dat' c1 c2 c3
Entering data from file: stu_pair.dat
20 rows read.
MTB > print c1 c2 c3
ROW C1 C2 C3
1 1 90 83
2 2 84 89
3 3 78 58
4 4 79 93
5 5 65 49
6 6 49 82
7 7 92 93
8 8 71 100
9 9 85 83
10 10 76 55
11 11 68 65
12 12 71 83
13 13 88 81
14 14 89 88
15 15 76 83
16 16 61 78
17 17 88 88
18 18 58 80
19 19 73 70
20 20 82 83
MTB > describe c2 c3
N MEAN MEDIAN TRMEAN STDEV SEMEAN
C2 20 76.15 77.00 76.78 11.75 2.63
C3 20 79.20 83.00 79.72 13.34 2.98
MIN MAX Q1 Q3
C2 49.00 92.00 68.75 87.25
C3 49.00 100.00 72.00 88.00
MTB > # In MINITAB, the task here is to demonstrate that
MTB > # the average population difference between the
MTB > # pairs is equal to zero.
MTB > #
MTB > name c6 = 'Differen'
MTB > let 'Differen' = c2 - c3
MTB > print c1-c6
ROW C1 C2 C3 Differen
1 1 90 83 7
2 2 84 89 -5
3 3 78 58 20
4 4 79 93 -14
5 5 65 49 16
6 6 49 82 -33
7 7 92 93 -1
8 8 71 100 -29
9 9 85 83 2
10 10 76 55 21
11 11 68 65 3
12 12 71 83 -12
13 13 88 81 7
14 14 89 88 1
15 15 76 83 -7
16 16 61 78 -17
17 17 88 88 0
18 18 58 80 -22
19 19 73 70 3
20 20 82 83 -1
Continue? y
* NOTE * One or more variables are undefined.
MTB > ttest 0 'Differen'
TEST OF MU = 0.00 VS MU N.E. 0.00
N MEAN STDEV SE MEAN T P VALUE
Differen 20 -3.05 14.78 3.31 -0.92 0.37
MTB > #
MTB > # And just as you saw with SPSS, the t-statistic is
MTB > # equal to -0.92 and the p value is 0.37.
MTB > #
MTB > # There is no difference in final examination scores
MTB > # between the paired subjects/students.
MTB > stop
--------------------------
Disclaimer: All care was used to prepare the information in this
tutorial. Even so, the author does not and cannot guarantee the
accuracy of this information. The author disclaims any and all
injury that may come about from the use of this tutorial. As
always, students and all others should check with their advisor(s)
and/or other appropriate professionals for any and all assistance
on research design, analysis, selected levels of significance, and
interpretation of output file(s).
The author is entitled to exclusive distribution of this tutorial.
Readers have permission to print this tutorial for individual use,
provided that the copyright statement appears and that there is no
redistribution of this tutorial without permission.
Prepared 980316
Revised 980914
end-of-file 'stu_pair.ssi'