Forty eight subjects were randomly assigned to one of 4 treatment conditions in a study of different methods of coping with statistics exam anxiety. After the experiment, 12 subjects remained in group 1, 8 in group 2, 10 in group 3, and 10 in group 4. The mean anxiety score of group 1 was found to be 5.5 points higher than the mean of group 2, 3.5 points higher than the mean of group 3, 5.3 points higher than the mean of group 4, and 3.3 points higher than the grand mean. Interestingly, the sum of squares error was exactly 12.0 times larger than its degrees of freedom.
Did the different treatments result in significantly different amounts of anxiety? (á.01)
ANS: F = 5.83, Reject Ho
You might be able to fill in values using an ANOVA table to determine your F-ratio, since you can determine degrees of freedom and sum of squares error from what you are given in the problem.
To determine if the different treatments resulted in significantly different amounts of anxiety, we can use an analysis of variance (ANOVA) test.
First, let's calculate the degrees of freedom. In this problem, we have 4 treatment conditions, so we have 4 - 1 = 3 degrees of freedom for the between-group variation (treatments). We also have a total of 48 subjects, so we have 48 - 1 = 47 degrees of freedom for the within-group variation (error).
Next, let's calculate the sum of squares error. We are given that the sum of squares error is exactly 12.0 times larger than its degrees of freedom. Therefore, the sum of squares error is 12.0 * 47 = 564.0
Now, we can calculate the sum of squares between (treatments). We know that the mean anxiety score of group 1 is 5.5 points higher than the mean of group 2, 3.5 points higher than the mean of group 3, 5.3 points higher than the mean of group 4, and 3.3 points higher than the grand mean. Since we have 48 subjects in total, the sum of squares between can be calculated as follows:
Sum of squares between = (12 * (5.5^2) + 8 * (3.5^2) + 10 * (5.3^2) + 10 * (3.3^2)) = 467.62
Next, we can calculate the mean square between by dividing the sum of squares between by its degrees of freedom (3):
Mean square between = 467.62 / 3 = 155.87
Finally, we can calculate the F-ratio by dividing the mean square between by the mean square error:
F-ratio = Mean square between / Mean square error = 155.87 / (564.0 / 47) = 5.83
To determine if the result is significant at the alpha level of 0.01, we compare the calculated F-ratio to the critical F-value at the corresponding degrees of freedom. In this case, with 3 degrees of freedom for between-group variation and 47 degrees of freedom for error, the critical F-value at alpha = 0.01 is approximately 3.59.
Since the calculated F-ratio (5.83) is greater than the critical F-value (3.59), we reject the null hypothesis (Ho) and conclude that the different treatments did result in significantly different amounts of anxiety at the alpha level of 0.01.
Thus, the answer is "F = 5.83, Reject Ho."