Data from a randomized block design are shown in the following table. $Data from a randomized block design are shown in the following table. Using \alpha = 0.05,the appropriate decision for the treatments is ________. A) do not reject the null hypothesis \mu 1= \mu 2 = \mu 3 B) do not reject the null hypothesis \mu 1= \mu 2 = \mu 3= \mu 4 C) do not reject the null hypothesis \mu 1 \ge\mu 2 \ge \mu 3 \ge \mu 4 D) do not reject the null hypothesis \mu 1 \le\mu 2 \le \mu 3 E) do nothing$
Using $\alpha$ = 0.05,the appropriate decision for the treatments is ________.

Question

Data from a randomized block design are shown in the following table. $Data from a randomized block design are shown in the following table. Using \alpha = 0.05,the appropriate decision for the treatments is ________. A) do not reject the null hypothesis \mu 1= \mu 2 = \mu 3 B) do not reject the null hypothesis \mu 1= \mu 2 = \mu 3= \mu 4 C) do not reject the null hypothesis \mu 1 \ge\mu 2 \ge \mu 3 \ge \mu 4 D) do not reject the null hypothesis \mu 1 \le\mu 2 \le \mu 3 E) do nothing$
Using

\alpha

= 0.05,the appropriate decision for the treatments is ________.

Quizplus · Accepted Answer

To determine the appropriate decision for the treatments, we need to perform an ANOVA using F1F1F1S1αF1F1F10 = 0.05 significance level.
The ANOVA table is not provided, so we need to calculate the necessary values.
First, we calculate the mean of each treatment, the overall mean, and the sum of squares for treatments (SST).
Treatment 1: (35 + 41 + 45) / 3 = 40.33
Treatment 2: (46 + 50 + 51) / 3 = 49
Treatment 3: (30 + 34 + 37) / 3 = 33.67
Overall mean: (35 + 41 + 45 + 46 + 50 + 51 + 30 + 34 + 37) / 9 = 40
SST = [(35 - 40)^2 + (41 - 40)^2 + (45 - 40)^2 + (46 - 40)^2 + (50 - 40)^2 + (51 - 40)^2 + (30 - 40)^2 + (34 - 40)^2 + (37 - 40)^2]
= 880

Next, we calculate the sum of squares for error (SSE) and the total sum of squares (SSTO).
SSE = SS Block + SS Error
= ∑ ∑ (Xij - X..i)^2 + ∑ ∑ (Xij - Xij..)^2
= [(35-40.33)^2 + (41-40.33)^2 + (45-40.33)^2) + ((46-49)^2 + (50-49)^2 + (51-49)^2) + ((30-33.67)^2 + (34-33.67)^2 + (37-33.67)^2]
= 99.33 + 9 + 69.33
= 177.66
SSTO = SST + SSE = 880 + 177.66 = 1057.66

We then calculate the degrees of freedom for treatments (dfT), the degrees of freedom for error (dfE), and the total degrees of freedom (dfT0).
dfT = k - 1 = 3 - 1 = 2
dfE = N - k = 9 - 3 = 6
dfT0 = N - 1 = 9 - 1 = 8

Next, we calculate the mean square for treatments (MST) and the mean square for error (MSE).
MST = SST/dfT = 880/2 = 440
MSE = SSE/dfE = 177.66/6 = 29.61

Finally, we calculate the F-statistic.
F = MST/MSE = 440/29.61 = 14.84

Using an F-distribution table with dfT = 2 and dfE = 6, we find that the critical value for F1F1F1S1αF1F1F10 = 0.05 is 5.14.

Since our calculated F-value (14.84) is greater than the critical value (5.14), we reject the null hypothesis and conclude that there is a significant difference between at least two of the treatments.

The appropriate decision for the treatments is therefore to reject the null hypothesis and to conclude that there is a significant difference between at least two of the treatments. In choice B, this result is correctly summarized.

Data from a Randomized Block Design Are Shown in the Following