Question 1182694
Here's how to determine if the machines have equal output rates using an ANOVA test:

**1. State the Hypotheses:**

* **Null Hypothesis (H0):** The machines have equal output rates (μA = μB = μC).
* **Alternative Hypothesis (H1):** At least one machine has a different output rate.

**2. Significance Level:** α = 0.05

**3. Calculate the Sum of Squares:**

We need to calculate the Sum of Squares Between Groups (SSB) and the Sum of Squares Within Groups (SSW).

* **Calculate the Grand Mean:**
   Grand Mean = (Sum of all observations) / (Total number of observations)
   Grand Mean = (25+30+36+38+31 + 31+39+38+42+35 + 24+30+28+25+28) / 15
   Grand Mean = 480 / 15 = 32

* **Calculate SSB:**
   SSB = 5 * (Mean_A - Grand_Mean)² + 5 * (Mean_B - Grand_Mean)² + 5 * (Mean_C - Grand_Mean)²

   * Mean_A = (25+30+36+38+31)/5 = 32
   * Mean_B = (31+39+38+42+35)/5 = 37
   * Mean_C = (24+30+28+25+28)/5 = 27

   SSB = 5 * (32 - 32)² + 5 * (37 - 32)² + 5 * (27 - 32)²
   SSB = 0 + 5 * 25 + 5 * 25
   SSB = 250

* **Calculate SSW:**

   SSW = Σ(each observation - its group mean)²
   SSW = (25-32)² + (30-32)² + ... + (28-27)²  (add up the squared differences for all 15 observations)
   SSW = 49 + 4 + 16 + 36 + 1 + 1 + 49 + 36 + 100 + 9 + 4 + 4 + 1 + 9 + 1
   SSW = 310

* **Calculate SST (Total Sum of Squares):** SST = SSW + SSB = 310+250 = 560

**4. Degrees of Freedom:**

* df_between (degrees of freedom between groups) = Number of groups - 1 = 3 - 1 = 2
* df_within (degrees of freedom within groups) = Total number of observations - Number of groups = 15 - 3 = 12
* df_total = n-1 = 15-1 = 14

**5. Calculate Mean Square:**

* MSB (Mean Square Between) = SSB / df_between = 250 / 2 = 125
* MSW (Mean Square Within) = SSW / df_within = 310 / 12 ≈ 25.83

**6. Calculate the F-statistic:**

F = MSB / MSW = 125 / 25.83 ≈ 4.84

**7. Determine the Critical Value:**

Using an F-distribution table with α = 0.05, df_between = 2, and df_within = 12, the critical F-value is approximately 3.89.

**8. Decision:**

Our calculated F-statistic (4.84) is *greater than* the critical value (3.89). Therefore, we *reject* the null hypothesis.

**9. Conclusion:**

At the 5% level of significance, there *is* sufficient evidence to conclude that the machines do *not* have equal output rates. At least one machine has a different output rate.