Assignment:
Q1. Describe the similarities between an F-ratio and a t statistic.
Q2. Explain why you should use ANOVA instead of several t tests to evaluate mean differences when an experiment consists of three or more treatment conditions. Address what happens when you use a series of t test compared to just one ANOVA test.
Q3. The following data represent the results from an independent-measures study comparing three treatments.
a. Compute SS for the set of 3 treatment means (Use the three means as a set of n=3 scores and compute SS) Treat the means as if they are scores and compute their mean as you would calculate the mean for the scores. Then obtain the sum of squared deviations. Most students get confused about this. Just use the three means that are given, obtain the average for them, then obtain the deviations (the mean - the average) in each case. Then square these deviations and then add them up.
b. Using the result from part a, compute n(SSmeans). Note that this value is equal to SSbetween (see Equation 13.6).
c. Now, compute SSbetween with the computational formula using the T values (Equation 13.7). You should obtain the same result as in part b.
Treatment
I II III
n=10 n=10 n=10
M=2 M=3 M=7
T=20 T=30 T=70
Q4. For the preceding problem you should find that there are significant differences among the three treatments. The primary reason for the significance is that the mean for treatment I is substantially smaller than the means for the other two treatments. To create the following data, we started with the values from problem 7 and added 3 points to each score in treatment I. Recall that adding a constant causes the mean to change but has no influence on the variability of the sample. In the resulting data the mean differences are much smaller than those
I II III
n=6 n=6 n=6
M=4 M=5 M=6 N=18
T=24 T=30 T=36 G=90
Ss=30 SS=35 SS=40 ∑X2=567
a. Before you begin any calculations, predict how the change in the data should influence the outcome of the analysis. That is, how will the F-ratio and the value of η2 for these data compare with the values obtained in problem 7? Refer to the solution of Problem 7 at the end of the book and speak about the impact of the changes.
b. Use an ANOVA with α = .05 to determine whether there are any significant differences among the three treatment means. (Does your answer agree with your prediction in part a?)
c. Calculate η2 to measure the effect size for this study. (Does your answer agree with your prediction in part a?) B and C are straightforward.
Q5. For the preceding problem you should find that there are significant differences among the three treatments. One reason for the significance is that the sample variances are relatively small. To create the following data, we started with the values from problem 9 and increased the variability (the SS values) within each sample.
I II III
n=5 n=5 n=5
M=2 M=5 M=8 N=15
T=10 T=25 T=40 G=75
SS=64 SS=80 SS=96 ∑X2=705
a. Calculate the sample variance for each of the three samples. Describe how these sample variance compare with those from problem 9. Use the SS and df=n-1 to compute the variances of the three samples and compare
b. Predict how the increase in sample variance should influence the outcome of the analysis. That is, how will the F-ratio for these data compare with the value obtained in problem 9? Based on the comparison, how would the analysis change? Make your predictions according to the changes in the variances
c. Use an ANOVA with α= .05 to determine whether there are any significant differences among the three treatment means. (Does your answer agree with your prediction in part b?) Conduct the test and see if your prediction was accurate.
Q6. A researcher reports an F-ratio with df= 3,36 from an independent-measures research study.
a. How many treatment conditions were compared in the study?
b. what was the total number of participants in the study?
Q7. A pharmaceutical company has developed a drug that is expected to reduce hunger. To test the drug, three samples of rats are selected with n=10 in each sample. The first sample receives the drug every day. The second sample is given the drug once a week, and the third sample receives no drug at all. The dependent variable is the amount of food eaten by each rat over a 1-month period. These data are analyzed by an ANOVA, and the results are reported in the following summary table. (Hint: start with the df column). Gather the information you have above and some of the values given below to fill out the Table. Three samples means there are three treatments. There are 10 in each sample (Total number of participants is 30). It is easier to start with the df first. You can obtain MS within treatments using the relationship between the F and MS bet and MS within.
Source SS df MS
Between treatments ___ ___ 15 F=7.50
Within treatments ___ ___ ___
Total ___ ___
Q8. The following data were obtained from an independent-measures research study comparing three treatment conditions. Use an ANOVA with α = .05 to determine whether there are any significant mean difference among the treatments.
Treatment
I II III
2 5 7 N=14
5 2 3 G=42
0 1 6 ∑X2=182
1 2 4
2
2
T=12 T=10 T=20
SS=14 SS=9 SS=10
Q9. The structure of a two-factor study can be presented as a matrix with the levels of one factor determining the rows and the levels of the second factor determining the columns. With this structure in mind, describe the mean differences that are evaluated by each of the three hypothesis tests that make up a two-factor ANOVA.
For the data in the following matrix:
No treatment Treatment
Male M=5 M=3 Overall M =4
Female M=9 M=13 Overall M= 11
Overall M = 7 Overall M=8
a. Describe the mean difference that is the main effect for the treatment.
b. Describe the mean difference that is the main effect for gender.
c. Is there an interaction between gender and treatment? Explain your answer. Looking at the information above, does it appear that the treatment results depend on gender? Why?
Q10. If you have the 7th edition, please ignore these and refer to the text to see these problems and their solution at the end of the text. Note that odd-numbered problems are solved at the end of the book.
I II III
n=6 n=6 n=6
M=1 M=5 M=6 N=18
T=6 T=30 T=36 G=72
SS=30 SS=35 SS=40 ∑X2=477
I II III
n=5 n=5 n=5
M=2 M=5 M=8 N=15
T=10 T=25 T=40 G=75
SS=16 SS=20 SS=24 ∑X2=525