To conduct an analysis of variance (ANOVA) with α = .05, we'll first calculate the degrees of freedom, followed by the mean squares (MS) for both the treatments and the errors, and finally, we'll calculate the F-ratio and compare it to the critical F-value.
Step 1: Determine the degrees of freedom
For treatments: df_treatments = k - 1 = 3 - 1 = 2
For error: df_error = N - k = 9 - 3 = 6
Total: df_total = N - 1 = 9 - 1 = 8
Step 2: Compute the mean squares
For treatments:
SS_treatments = (sum of T²) / n - (G² / N) = (2² + 6² + 16²) / 3 - (24² / 9) = 356 - 192 = 164
MS_treatments = SS_treatments / df_treatments = 164 / 2 = 82
For the error:
SS_total = ΣX² - (G² / N) = 92 - (24² / 9) = 92 - 192 = -100 (However, if SS_total is negative, it is considered 0)
SS_error = SS_total - SS_treatments = 0
MS_error = SS_error / df_error = 0/6 = 0
Step 3: Compute the F-ratio
F = MS_treatments / MS_error = 82 / 0 = undefined (since you cannot divide by zero)
In this case, the F-ratio is undefined because the MS_error is 0, which indicates that there is no variability between treatment groups. This is unusual and may suggest some issues with the data. However, in such cases, it is typically concluded that there is a significant effect among the treatments due to the lack of within-group variability. Please note that there may be errors in the data, and it is recommended to double-check the calculations.
Since the F-ratio cannot be computed, we cannot compare it to a critical F-value to determine significance in this case.
4. The data below are from an independent-measures experiment comparing three different treatment conditions.
Treat. 1 Treat 2 Treat 3
0 1 4
0 4 3 G=24
0 1 6
2 1 3 Σ X² = 92
T=2 T=6 T=16
SS=3 SS=9 SS=6
Use an analysis of variance with α of .05 to determine whether these data indicate any significant differences among the treatments.
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