Question 1177142
**1. Hypotheses**

* **Null Hypothesis (H0):** The means of all the groups are equal.
* **Alternative Hypothesis (H1):** At least one group mean is different from the others.

**2. Significance Level**

* α = 0.05

**3. Degrees of Freedom**

* df(between) = 2 (from the ANOVA table)
* df(within) = 17 (from the ANOVA table)

**4. F-statistic**

* F = 52.74 (from the ANOVA table)

**5. Critical F-value**

* Using an F-distribution table or calculator with α = 0.05, df1 = 2, and df2 = 17, we find the critical F-value to be approximately 3.59.

**6. Decision**

* Since the calculated F-statistic (52.74) is greater than the critical F-value (3.59), we reject the null hypothesis.

**7. Conclusion**

At a 5% significance level, there is enough evidence to conclude that there is a statistically significant difference between the means of at least two of the groups.

**In simpler terms:** The ANOVA test shows that it's very unlikely that the observed differences between the group means are just due to random chance. Therefore, we can confidently say that there's a real difference in the average values of those groups.

**Further Analysis**

While ANOVA tells us that there's a difference, it doesn't tell us *which* groups are different. To find that out, you would need to perform post-hoc tests (like Tukey's HSD or Bonferroni correction) to compare specific pairs of group means.