To determine the critical value given the parameters alpha [tex]\(\alpha = 0.05\)[/tex], [tex]\(N = 12\)[/tex], and [tex]\(k = 3\)[/tex], we need to follow these steps:
1. Determine Degrees of Freedom:
- The degrees of freedom (df) in this context can be found using the formula:
[tex]\[
\text{df} = N - k
\][/tex]
Substituting the given values:
[tex]\[
\text{df} = 12 - 3 = 9
\][/tex]
2. Identify the Appropriate Statistical Distribution:
- With the number of observations [tex]\(N\)[/tex] and the number of groups [tex]\(k\)[/tex], we typically use the F-distribution for such cases when dealing with variances. Given [tex]\(\alpha = 0.05\)[/tex], we want to find the critical F-value for the degrees of freedom associated with our scenario.
3. Look Up or Calculate the Critical Value:
- For an F-distribution with degrees of freedom [tex]\((d_1 = k-1 = 2, d_2 = N - k = 9)\)[/tex] at a significance level [tex]\(\alpha = 0.05\)[/tex]:
- Using standard F-tables or statistical software, the critical F-value for [tex]\(d_1 = 2\)[/tex] and [tex]\(d_2 = 9\)[/tex] at [tex]\(\alpha = 0.05\)[/tex] is approximately [tex]\(4.26\)[/tex].
4. Conclusion:
- Based on the given data and the appropriate statistical table, the critical value we are looking for is [tex]\(4.26\)[/tex].
So the correct answer is 4.26.
