Question 1194764
**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.