SOLUTION: The following is sample information. Test the hypothesis that the treatment means are equal. Use the 0.05 significance level.
Treatment 1 Treatment 2 Treatment 3
7
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Question 1194764: The following is sample information. Test the hypothesis that the treatment means are equal. Use the 0.05 significance level.
Treatment 1 Treatment 2 Treatment 3
7 7 4
6 10 5
8 4 5
6 7 5
a. State the null hypothesis and the alternate hypothesis.
H0:
(Click to select)
H1:
(Click to select)
b. What is the decision rule? (Round the final answer to 2 decimal places.)
Reject H0 if the test statistic is greater than
.
c. Compute SST, SSE, and SS total. (Round the final answers to 3 decimal places.)
SST =
SSE =
SS total =
d. Complete the ANOVA table. (Round the SS, MS, and F values to 3 decimal places.)
Source SS DF MS F
Treatment
Error
Total
e. State your decision regarding the null hypothesis.
(Click to select)
H0.
Answer by ElectricPavlov(122) (Show Source): You can put this solution on YOUR website!
**a) State the null hypothesis and the alternate hypothesis.**
* **H0:** μ1 = μ2 = μ3 (The mean values for all three treatments are equal)
* **H1:** At least one treatment mean is different from the others.
**b) What is the decision rule?**
* **Find the critical F-value:**
* Degrees of freedom between groups (df1): k - 1 = 3 - 1 = 2
* Degrees of freedom within groups (df2): N - k = 12 - 3 = 9
* Significance level (α) = 0.05
* Using an F-distribution table or statistical software, find the critical F-value.
* **F-critical ≈ 4.26**
* **Decision Rule:**
* Reject H0 if the calculated F-statistic is greater than 4.26.
**c) Compute SST, SSE, and SS total.**
* **Calculate Treatment Means:**
* Treatment 1: (7 + 6 + 8 + 6) / 4 = 6.75
* Treatment 2: (7 + 10 + 4 + 7) / 4 = 7
* Treatment 3: (4 + 5 + 5 + 5) / 4 = 4.75
* Grand Mean: (Sum of all observations) / Total number of observations = 5.5
* **Calculate Sum of Squares (SS):**
* **SST (Total Sum of Squares):** Σ(x_ij - x̄)²
* x_ij: Individual observation
* x̄: Grand mean
* SST = (7 - 5.5)² + (6 - 5.5)² + ... + (5 - 5.5)² = 26.75
* **SSB (Sum of Squares Between Groups):** Σ(n_i * (x̄_i - x̄)²)
* n_i: Number of observations in each group (4)
* x̄_i: Mean of each group
* SSB = 4 * (6.75 - 5.5)² + 4 * (7 - 5.5)² + 4 * (4.75 - 5.5)² = 12.5
* **SSE (Sum of Squares Error):** SST - SSB = 26.75 - 12.5 = 14.25
**d) Complete the ANOVA Table**
| Source | SS | DF | MS | F |
|---|---|---|---|
| Treatment | 12.500 | 2 | 6.250 | 4.167 |
| Error | 14.250 | 9 | 1.583 | |
| Total | 26.750 | 11 | | |
**e) State your decision regarding the null hypothesis.**
* Since the calculated F-statistic (4.167) is less than the critical F-value (4.26), we **fail to reject the null hypothesis (H0)**.
**Conclusion:**
There is not sufficient evidence at the 0.05 significance level to conclude that there are significant differences among the treatment means.
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