Question

A college dean is interested in the exam performance of students in a History course. After the final exam, students are randomly selected from three different section of the History course. What can be conclude with an α of 0.01? The data are below.

section 1	section 2	section 3
82 68 74 65 41 75 64 77 40 78 69	90 82 82 77 75 82 81 91 70 90 82	94 81 87 82 72 77 81 87 67 86 77

a) What is the appropriate test statistic?
---Select--- na one-way ANOVA within-subjects ANOVA two-way ANOVA

b) Compute the appropriate test statistic(s) to make a decision about H₀.
critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

c) Compute the corresponding effect size(s) and indicate magnitude(s).
η² =  ;  ---Select--- na trivial effect small effect medium effect large effect

d) Make an interpretation based on the results.

At least on section differs on the final exam.None of the sections differ on the final exam.    

e) Conduct Tukey's Post Hoc Test for the following comparisons:
1 vs. 3: difference =  ; significant:  ---Select--- Yes No
1 vs. 2: difference =  ; significant:  ---Select--- Yes No

f) Conduct Scheffe's Post Hoc Test for the following comparisons:
2 vs. 3: test statistic =  ; significant:  ---Select--- Yes No
1 vs. 3: test statistic =  ; significant:  ---Select--- Yes No

Answer 1

a) Appropriate test statistic = one-way ANOVA

b)

	Section 1	Section 2	Section 3	Total
Sum	733	902	891	2526
Count	11	11	11	33
Mean, Sum/n	66.6364	82	81
Sum of square, Ʃ(xᵢ-x̅)²	1980.5455	428	576

Number of treatment, k = 3

Total sample Size, N = 33

df(between) = k-1 = 2

df(within) = N-k = 30

df(total) = N-1 = 32

SS(between) = (Sum1)²/n1 + (Sum2)²/n2 + (Sum3)²/n3 - (Grand Sum)²/ N = 1625.6364

SS(within) = SS1 + SS2 + SS3 = 2984.5455

SS(total) = SS(between) + SS(within) = 4610.1818

MS(between) = SS(between)/df(between) = 812.8182

MS(within) = SS(within)/df(within) = 99.4848

Critical value Fc = F.INV.RT(0.01, 2, 30) = 5.390

Test statistic:

F = MS(between)/MS(within) = 8.1703

Decision: Reject H0

ANOVA
Source of Variation	SS	df	MS	F	Critical value
Between Groups	1625.6364	2	812.8182	8.1703	5.390
Within Groups	2984.5455	30	99.4848
Total	4610.1818	32

c) ɳ² = SS(between)/SS(total) = 1625.6364/4610.1818 = 0.3526

Effect size, large

d) Interpretation based on the results.

At least on section differs on the final exam.

e) Tukey's Post Hoc Test:

At α = 0.01, k = 3, N-K = 30, Q value = 4.45

Comparison	Absolute Diff. = |xi - xj|	Critical Range = Q* √(MSwithin/n)	Significant
x̅1-x̅3	14.3636	13.3827	Yes
x̅1-x̅2	15.3636	13.3827	Yes