A researcher begins with a sample of 60 subjects and randomly assigns 12 to each of 5 treatment conditions. Based on his reading of other studies in the area, the researcher predicts that group 3 will obtain a significantly different mean from group 1, and that group 4 will obtain a significantly higher mean than group 1.
a)Given the following data (note that some subjects have dropped out) are the researcher's predictions supported with á .01?
1 2 3 4 5
n = 8 10 10 12 12
X-bar = 43.5 45.2 61.8 77.3 66.7
Si = 5.6 6.3 5.4 8.7 7.5
b)After collecting and inspecting the data, the researcher decides to perform three more tests. Specifically, he wishes to see whether X-bar3 is significantly different from X-bar2; whether X-bar5 is significantly different from X-bar2; and whether X-bar4 is significantly greater than X-bar2. Perform whichever statistical analyses are necessary and, if appropriate, use the procedure that allows each comparison to be tested with á .01.
Answer:
a) 3 vs. 1, t = 5.53, Reject Ho
4 vs. 1, t = 10.608, Reject Ho
b) F = 43.696, Reject Ho
3 vs. 2, q = 7.583, Reject Ho
5 vs. 2, q = 9.822, Reject Ho
4 vs. 2, no test valid
If you do an ANOVA test on your data, there are different ways to approach this.
You will need to determine mean squares between and within to find the F-ratio.
Using the data you listed, I found the F-ratio = 43.696 using the following method.
Mean squares within:
[7(5.6)^2 + 9(6.3)^2 + 9(5.4)^2 + 11(8.7)^2 + 11(7.5)^2]/[(8 + 10 + 10 + 12 + 12) - 5] = (2290.51)/(47) = 48.734
Mean squares between:
8(43.5) + 10(45.2) + 10(61.8) + 12(77.3) + 12(66.7) = 3146 -->first step
8(43.5)^2 + 10(45.2)^2 + 10(61.8)^2 + 12(77.3)^2 + 12(66.7)^2 = 198850.96 -->second step
[198850.96 - (3146)^2/52]/(5 - 1) = 2129.49 -->final step (Note: 52 = total number of subjects; 5 = total number of groups)
To find F-ratio:
(2129.49)/(48.734) = 43.696
To compare two means, you may want to use multiple comparison procedures to answer your other questions. (Note: the term "within" may be listed as "error" in some texts.)
I hope this will help.
To perform the multiple comparison procedures mentioned in part b) of the question, you will need to use the q-test (otherwise known as the Tukey test). This test allows you to compare multiple group means while controlling for the overall familywise error rate.
Here are the steps to perform the q-test for the given comparisons:
1) Calculate the standard error between group means (SEm) using the formula:
SEm = sqrt((MSW)/n)
where MSW is the mean squares within (error mean square) value obtained from the ANOVA test, and n is the number of subjects per group (in this case, 8, 10, or 12).
2) Calculate the q-value for each comparison using the formula:
q = (X-bar1 - X-bar2)/SEm
where X-bar1 and X-bar2 are the means of the two groups being compared.
3) Compare the calculated q-value to the critical q-value based on the significance level (á) of 0.01.
If the calculated q-value is greater than the critical q-value, then the means are significantly different. Otherwise, there is not enough evidence to conclude a significant difference between the means.
Now, let's apply these steps to the given comparisons:
1) X-bar3 vs. X-bar2:
SEm = sqrt((48.734)/10) = 2.206
q = (61.8 - 45.2)/2.206 = 7.583
Compare q-value (7.583) to the critical q-value (obtained from a q-table or calculator) at á = 0.01 (for 50 degrees of freedom in the denominator, since the total number of subjects is 52). If the q-value is greater than the critical q-value, we can reject the null hypothesis.
2) X-bar5 vs. X-bar2:
SEm = sqrt((48.734)/10) = 2.206
q = (66.7 - 45.2)/2.206 = 9.822
Compare q-value (9.822) to the critical q-value at á = 0.01. If the q-value is greater than the critical q-value, we can reject the null hypothesis.
3) X-bar4 vs. X-bar2:
Since there is no valid test for this comparison, we cannot make any conclusions.
Based on the above calculations, the researcher can conclude that X-bar3 is significantly different from X-bar2, X-bar5 is significantly different from X-bar2, but there is not enough evidence to conclude a significant difference between X-bar4 and X-bar2.
I hope this clarifies the steps and answers your question.