Question

The following data represent the results from an independent-measures experiment comparing three treatment conditions. Use StatCrunch to conduct an analysis of variance with a = 0.05 to determine whether these data are sufficient to conclude that there are significant differences between the treatments. Treatment A Treatment B Treatment C 5 5 12 3 6 6 5 4 10 4 7 9 3 3 8 F-ratio = p-value = Conclusion: There is a significant difference between treatments These data do not provide evidence of a difference between the treatments The results obtained above were primarily due to the mean for the third treatment being noticeably different from the other two sample means. For the following data, the scores are the same as above except that the difference between treatments was reduced by moving the third treatment closer to the other two samples. In particular, 3 points have been subtracted from each score in the third sample. Before you begin the calculation, predict how the changes in the data should influence the outcome of the analysis. That is, how will the F-ratio for these data compare with the F-ratio from above?

Answer 1

Q1:

	Treatment A	Treatment B	Treatment C	Total
Sum	20	25	45	90
Count	5	5	5	15
Mean, Sum/n	4	5	9
Sum of square, Ʃ(xᵢ-x̅)²	4	10	20

Number of treatment, k = 3

Total sample Size, N = 15

df(between) = k-1 = 2

df(within) = N-k = 12

df(total) = N-1 = 14

SS(between) = (Sum1)²/n1 + (Sum2)²/n2 + (Sum3)²/n3 - (Grand Sum)²/ N = 70

SS(within) = SS1 + SS2 + SS3 = 34

SS(total) = SS(between) + SS(within) = 104

MS(between) = SS(between)/df(between) = 35

MS(within) = SS(within)/df(within) = 2.833333

F = MS(between)/MS(within) = 12.3529

p-value = F.DIST.RT(12.3529, 2, 12) = 0.0012

Conclusion:

There is a significant difference between treatments

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Q2:

	Treatment A	Treatment B	Treatment C	Total
Sum	20	25	30	75
Count	5	5	5	15
Mean, Sum/n	4	5	6
Sum of square, Ʃ(xᵢ-x̅)²	4	10	20

Number of treatment, k = 3

Total sample Size, N = 15

df(between) = k-1 = 2

df(within) = N-k = 12

df(total) = N-1 = 14

SS(between) = (Sum1)²/n1 + (Sum2)²/n2 + (Sum3)²/n3 - (Grand Sum)²/ N = 10

SS(within) = SS1 + SS2 + SS3 = 34

SS(total) = SS(between) + SS(within) = 44

MS(between) = SS(between)/df(between) = 5

MS(within) = SS(within)/df(within) = 2.833333

F = MS(between)/MS(within) = 1.7647

p-value = F.DIST.RT(1.7647, 2, 12) = 0.2129

Conclusion:

These data do not provide evidence of a difference between the treatments.