Question

The following data are from a completely randomized design. In the following calculations, use

α = 0.05.

	Treatment 1	Treatment 2	Treatment 3
	62	83	68
	46	71	55
	53	88	62
	39	70	47
xj	50	78	58
sj2	96.67	79.33	82.00

(a)

Use analysis of variance to test for a significant difference among the means of the three treatments.

State the null and alternative hypotheses.

H₀: At least two of the population means are equal.
H_a: At least two of the population means are different.

H₀: Not all the population means are equal.
H_a: μ₁ = μ₂ = μ₃    

H₀: μ₁ ≠ μ₂ ≠ μ₃
H_a: μ₁ = μ₂ = μ₃

H₀: μ₁ = μ₂ = μ₃
H_a: Not all the population means are equal.

H₀: μ₁ = μ₂ = μ₃
H_a: μ₁ ≠ μ₂ ≠ μ₃

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

State your conclusion.

Do not reject H₀. There is not sufficient evidence to conclude that the means of the three treatments are not equal.

Do not reject H₀. There is sufficient evidence to conclude that the means of the three treatments are not equal.    

Reject H₀. There is not sufficient evidence to conclude that the means of the three treatments are not equal.

Reject H₀. There is sufficient evidence to conclude that the means of the three treatments are not equal.

(b)

Use Fisher's LSD procedure to determine which means are different.

Find the value of LSD. (Round your answer to two decimal places.)

LSD =

Find the pairwise absolute difference between sample means for each pair of treatments.

x₂ − x₃₌

x₁ − x₂₌

x₁ − x₃=

Which treatment means differ significantly? (Select all that apply.)=

There is a significant difference between the means for treatments 1 and 2.

There is a significant difference between the means for treatments 1 and 3.

There is a significant difference between the means for treatments 2 and 3.

There are no significant differences.

Answer 1

Result:

(a)

Use analysis of variance to test for a significant difference among the means of the three treatments.

State the null and alternative hypotheses.

H0: μ1 = μ2 = μ3
Ha: Not all the population means are equal.

Find the value of the test statistic. (Round your answer to two decimal places.)

test statistic= 9.67

Find the p-value. (Round your answer to three decimal places.)

p-value = 0.006

State your conclusion.

Reject H0. There is sufficient evidence to conclude that the means of the three treatments are not equal.

(b)

Use Fisher's LSD procedure to determine which means are different.

Find the value of LSD. (Round your answer to two decimal places.)

critical t value at 9 df at 0.05 level = 2.262

LSD = 2.262*sqrt(86*(1/4+1/4))

=14.83

Find the pairwise absolute difference between sample means for each pair of treatments.

x2 − x3=20.0

x1 − x2= 28.0

x1 − x3= 8.0

Which treatment means differ significantly? (Select all that apply.)=

There is a significant difference between the means for treatments 1 and 2.

There is a significant difference between the means for treatments 2 and 3.

MINITAB used.

One-way ANOVA: Treatment 1, Treatment 2, Treatment 3

Method

Null hypothesis	All means are equal
Alternative hypothesis	Not all means are equal
Significance level	α = 0.05

Equal variances were assumed for the analysis.

Factor Information

Factor	Levels	Values
Factor	3	Treatment 1, Treatment 2, Treatment 3

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Factor	2	1664.0	832.00	9.67	0.006
Error	9	774.0	86.00
Total	11	2438.0

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
9.27362	68.25%	61.20%	43.56%

Means

Factor	N	Mean	StDev	95% CI
Treatment 1	4	50.00	9.83	(39.51, 60.49)
Treatment 2	4	78.00	8.91	(67.51, 88.49)
Treatment 3	4	58.00	9.06	(47.51, 68.49)

Pooled StDev = 9.27362

Fisher Pairwise Comparisons

Grouping Information Using the Fisher LSD Method and 95% Confidence

Factor	N	Mean	Grouping
Treatment 2	4	78.00	A
Treatment 3	4	58.00		B
Treatment 1	4	50.00		B

Means that do not share a letter are significantly different.

Fisher Individual Tests for Differences of Means

Difference of Levels	Difference of Means	SE of Difference	95% CI	T-Value	Adjusted P-Value
Treatment 2 - Treatment 1	28.00	6.56	(13.17, 42.83)	4.27	0.002
Treatment 3 - Treatment 1	8.00	6.56	(-6.83, 22.83)	1.22	0.253
Treatment 3 - Treatment 2	-20.00	6.56	(-34.83, -5.17)	-3.05	0.014

Simultaneous confidence level = 88.66%