Question

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

α = 0.05.

	Treatment 1	Treatment 2	Treatment 3
	63	82	69
	46	72	54
	53	87	62
	46	63	47
xj	52	76	58
sj2	64.67	114.00	91.33

(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
H_a: Not all the population means are equal.

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: μ₁ ≠ μ₂ ≠ μ₃

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

t stat =

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

p-value =

State your conclusion.

Reject H₀. There is 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.    

Do not reject H₀. There is not 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.

(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.

x₁ − x_{2 =}

x₁ − x₃=

x2 − 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

	Treatment 1	Treatment 2	Treatment 3	Total
Sum	208	304	232	744
Count	4	4	4	12
Mean, Sum/n	52	76	58
Variance, s2	64.1667	114	91.3333

a) Null and Alternative Hypothesis:

Ho: µ1 = µ2 = µ3

H1: Not all the population means are equal.

Number of treatment, k = 3

Total sample Size, N = 12

df(between) = k-1 = 2

df(within) = N-k = 9

df(total) = N-1 = 11

SS(between) = (x̅1)²*n1 + (x̅2)²*n2 + (x̅3)²*n3 - (Grand Mean)²*N = 1248

SS(within) = (n1-1)*s1² + (n2-1)*s2² + (n3-1)*s3² = 810

SS(total) = SS(between) + SS(within) = 2058

MS(between) = SS(between)/df(between) = 624

MS(within) = SS(within)/df(within) = 90

F = MS(between)/MS(within) = 6.93

p-value = F.DIST.RT(6.9333, 2, 9) = 0.0151

conclusion.

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

b)

At α = 0.05, N-K = 9, t critical value, t_c =T.INV.2T(0.05, 9)2.262

LSD = t_c* √(2*MSW*/n) = 2.262√(2*90/4) = 15.18

Absolute difference:

x̅1-x̅2	24
x̅1-x̅3	6
x̅2-x̅3	18

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.