To find the critical $F$-value for this one-way ANOVA, we need to know:

1. The significance level ($\alpha = 0.05$)
2. The degrees of freedom for the between-groups and within-groups variability.

Steps to Calculate Degrees of Freedom

1. Degrees of Freedom Between Groups (df$_\text{between}$)

• There are 3 groups (A, B, C), so $\text{df}_\text{between} = k - 1 = 3 - 1 = 2$.

2. Degrees of Freedom Within Groups (df$_\text{within}$)

• Each group has 4 observations, so $n = 4$ for each group.
• The total number of observations across all groups is $N = 4 \times 3 = 12$.
• Thus, $\text{df}_\text{within} = N - k = 12 - 3 = 9$.

Finding the Critical $F$-Value

With $\alpha = 0.05$, $\text{df}_\text{between} = 2$, and $\text{df}_\text{within} = 9$, we can refer to an $F$-distribution table or use statistical software.

Based on these values, the critical $F$-value is approximately 4.26, which corresponds to option B.

Final Answer:

The correct answer is B. 4.26

Would you like a deeper explanation or further assistance?

Here are some related questions to deepen your understanding:

1. What is the purpose of using a one-way ANOVA in statistics?
2. How do you calculate the F-ratio in a one-way ANOVA?
3. Why do we use a critical $F$-value, and how does it determine statistical significance?
4. What assumptions must be met to perform a one-way ANOVA?
5. How would the critical $F$-value change if we increased the number of groups?

Tip: When using the ANOVA table, remember that higher degrees of freedom generally result in smaller critical $F$-values for the same significance level.