Question

Given data from a completely randomized design experiment:

Treatment 1 = {3.8, 1.2, 4.1, 5.5, 2.3}

Treatment 2 = {5.4, 2.0, 4.8, 3.8}

Treatment 3 = {1.3, 0.7, 2.2}

a.) Calculate the treatment means and variances for each of the 3 treatments above.

b.) Use statistical software to complete the ANOVA table.

Source	df	SS	MS	F
Treatment
Error
Total

c.) In words, what is the null and alternative hypotheses for the ANOVA F-test?

d.) Test the null hypothesis that µ1=µ2=µ3against the alternative hypothesis that at least two means differ. Use α = .01.

e.) Explain in words what the ANOVA test tells us about the equality of treatment means?

Answer 1

Part a)

	Treatment 1	Treatment 2	Treatment 3
	3.8	5.4	1.3
	1.2	2	0.7
	4.1	4.8	2.2
	5.5	3.8
	2.3
Total(yi)	16.9	16	4.2
Averages y̅i	3.38	4	1.4
Treatment Effect	(3.38 - 3.0917 = 0.2883)	(4 - 3.0917 = 0.9083)	(1.4 - 3.0917 = -1.6917)

Part b)

	Sum of Squares	Degree of freedom	Mean Square	F0 = MST / MSE	P value
Treatment	12.3011	2	6.1506	2.9307	0.1047
Error	18.888	9	2.0987
Total	31.1891	11

Overall total = 37.1
Overall mean Y̅.. = 37.1 / 12 = 3.0917

Y̅ .. is overall mean
y̅i . is treatment mean
SS total = ΣΣ(Yij - & Y̅..)2 = 31.1891
SS treatment = ΣΣ(Yij - & y̅i.)2 = 12.3011
SS error = Σ(y̅i. - & Y̅..)2 = 18.888

MS treatment = ΣΣ(Yij - & y̅i.)2 / a - 1 = 6.1506
MS error = Σ(y̅i. - & Y̅..)2 / N - a = 2.0987

Part c)

To Test :-
H0 :- µ1 = µ2 = µ3 .....µn = 0 i.e There is no difference in the treatment means
H0 :- µ1 = µ2 = µ3 .....µn ≠ 0 i.e Some means are different

Part d)

Test Statistic :-
f = MS treatment / MS error = 2.9307

Test Criteria :-
Reject null hypothesis if f > f(α , a-1 , N-a )
Critical value f(0.01, 2 , 9 ) = 8.0215 (From F table)
Since 2.9307 < 8.0215, we fail to reject H0
conclusion = Treatment means are same

Decision based on P value
Reject null hypothesis if P value < α = 0.01
Since P value = 0.1047 > 0.01, hence we fail to reject the null hypothesis
conclusion :- Treatment means are same  

Part e)

There is sufficient evidence to conclude that the average treatment effect is same.