**Student's t-Test for Matched Pairs **

**© 1998 by Dr. Thomas W. MacFarland -- All Rights Reserved**

************ stu_pair.doc ************ 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'