Question

Three independent group students took an exam. The data presented below represent the students’ test scores.

Group 1 Group 2 Group 3

95 79 65

98 76 68

80 60 70

77 88 87

99 69 66

90 66 72

Based on the post hoc (Tukey) test, which two groups have a significant difference at the 0.05 level ?

a. group 1 and group 2

b. group 1 and group 3

c. group 2 and group 3

d. both a and b

Answer 1

The Summary statistics are as follows

	Gr1	Gr2	Gr3
Total	539	438	428
n	6	6	6
Mean	89.833	73.000	71.333
Sum Of Squares	440.3534	504	327.3334
Variance	88.0707	100.8	65.467

The ANOVA table is as calculated below

Source	SS	DF	Mean Square	F
Between	1256.76	2	628.38	7.41
Within/Error	1271.69	15	84.78
Total	2528.45	17

SS error = Sum of squares = 440.3534 + 504 + 327.33 = 1271.69

Df error = N - k = 18 - 3 = 15

MS error = SS error / Df error = 1271.69 / 15 = 84.78

From Tukeys table, the critical value for k = 3, df = 15 is 3.140

If HSD is > critical value, then there is a significant difference

Group	Mean			Absolute Difference	Observed	Critical	Obs > Crit
1	89.83		M1 - M2	16.83	4.4773	3.14	Yes
2	73		M1 - M3	18.5	4.9215	3.14	Yes
3	71.33		M2 - M3	1.67	0.4443	3.14	No
Mserror	84.78
n	6
Sqrt(Msbetween/n)	3.759

Therefore Option d: Both a and b