Consider the following ANOVA experiments. (Give your answers correct to two decimal places.)
(a) Determine the critical region and critical value that are used in the classical approach for testing the null hypothesis Ho: ì1 = ì2 = ì3 = ì4, with n = 23 and á = 0.025.
F
(b) Determine the critical region and critical value that are used in the classical approach for testing the null hypothesis Ho: ì1 = ì2 = ì3 = ì4 = ì5, with n = 22 and á = 0.025.
F
(c) Determine the critical region and critical value that are used in the classical approach for testing the null hypothesis Ho: ì1 = ì2 = ì3, with n = 17 and á = 0.025.
F
You will need to determine "degrees of freedom between" and "degrees of freedom within" before checking an ANOVA table using alpha level.
Note: k = number of levels.
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To calculate df within:
a) You have 4 levels. Your sample size is 23 Your alpha level is 0.025
df between = k - 1 = 4 - 1 = 3
df within = N - k = 23-4 = 19
Checking the table using 0.025 alpha level using the above degrees of freedom, I see critical value of 3.90
I'll let you try the rest.
To determine the critical region and critical value for an ANOVA experiment, you need two pieces of information: the degrees of freedom and the significance level (α).
(a) To test the null hypothesis Ho: μ1 = μ2 = μ3 = μ4, with n = 23 and α = 0.025:
- The degrees of freedom for the numerator (between groups) is k - 1 = 4 - 1 = 3.
- The degrees of freedom for the denominator (within groups) is N - k = 23 - 4 = 19.
Using a critical value table for the F-distribution at α = 0.025, the critical F-value can be found by intersecting the degrees of freedom in the numerator and denominator.
For a 3 degrees of freedom in the numerator and 19 degrees of freedom in the denominator, the critical F-value is approximately 3.10.
Thus, the critical region is F > 3.10, where F is the calculated F-statistic.
(b) To test the null hypothesis Ho: μ1 = μ2 = μ3 = μ4 = μ5, with n = 22 and α = 0.025:
- The degrees of freedom for the numerator (between groups) is k - 1 = 5 - 1 = 4.
- The degrees of freedom for the denominator (within groups) is N - k = 22 - 5 = 17.
Using the same critical value table, for 4 degrees of freedom in the numerator and 17 degrees of freedom in the denominator, the critical F-value is approximately 3.49.
Therefore, the critical region is F > 3.49.
(c) To test the null hypothesis Ho: μ1 = μ2 = μ3, with n = 17 and α = 0.025:
- The degrees of freedom for the numerator (between groups) is k - 1 = 3 - 1 = 2.
- The degrees of freedom for the denominator (within groups) is N - k = 17 - 3 = 14.
Using the same critical value table, for 2 degrees of freedom in the numerator and 14 degrees of freedom in the denominator, the critical F-value is approximately 3.81.
Hence, the critical region is F > 3.81.
Remember that the critical region represents the range of values of the F-statistic that would lead to rejecting the null hypothesis.