FUNDAMENTALS OF STATISTICS
Assignment 21
1) In a one-way ANOVA, what is the difference between the among groups variance MSA and
the within-groups variance MSW?
The among-groups variance MSA represents variation among the means of the different
groups. The within groups-variance MSW measures variation within each group.
2) What are the distinguishing features of the completely randomized design and two-factor
factorial designs?
The completely randomized design evaluates one factor of interest, in which sample
observations are randomly and independently drawn. The randomized block design also
evaluates one factor of interest, but sample observations are divided into blocks according to
common characteristics to reduce within group variation. The two-factor factorial design
evaluates two factors of interest and the interaction between these two factors.
3) What are the assumptions of ANOVA?
The major assumptions of ANOVA are randomness and independence, normality, and
homogeneity of variance.
4) Under what conditions should you use the one-way ANOVA F test to examine possible
differences among the means of c independent populations?
If the populations are approximately normally distributed and the variances of the groups
are approximately equal, you select the one-way ANOVA F test to examine possible differences
among the means of c independent populations.
5) When and how should you use multiple comparison procedures for evaluating pairwise
combinations of the group means?
When the ANOVA has indicated that at least one of the groups has a different population
mean than the others, you should use multiple comparison procedures for evaluating pairwise

combinations of the group means. In such cases, the Tukey-Kramer procedure should be used to
compare all pairs of means.
6) What is the difference between the one-way ANOVA F test and the Levene test?
The one-way ANOVA F test for a completely randomized design is used to test for the
existence of treatment effect of the treatment variable on the mean level of the dependent
variable, while the Levene test is used to test whether the amounts of variation of the dependent
variable are the same across the different categories of the treatment variable.
7) Under what conditions should you use the two-way ANOVA F test to examine possible
differences among the means of each factor in a factorial design?
You should use the two-way ANOVA F test to examine possible differences among the
means of each factor in a factorial design when there are two factors of interest that are to be
studied and more than one observation can be obtained for each treatment combination (to
measure the interaction of the two factors).
8) What is meant by the concept of interaction in a two-factor factorial design?
Interaction measures the difference in the effect of one variable for the different levels of
the second factor. If there is no interaction, any difference between levels of one factor will be
the same at each level of the second factor.
9) How can you determine whether there is an interaction in the two factors factorial design?
You can obtain the interaction effect and carry out an F test for its significance. In
addition, you can develop a plot of the response for each level of one factor at each level of a
second factor.
10) An experiment has a single factor with five groups and seven values in each group.
(a) 𝑑𝑓𝐴 = 𝑐 − 1 = 5 − 1 = 4
(b) 𝑑𝑓𝑊 = 𝑛 − 𝑐 = 35 − 5 = 30
(c) 𝑑𝑓𝑇 = 𝑛 − 1 = 35 − 1 = 34

11) Consider a two-factor factorial design with three levels for factor A, three levels for factor
B, and four replicates in each of the nine cells.
a. How many degrees of freedom are there in determining factor A variation and factor B
variation?
(a) 𝑑𝑓𝐴 = 𝑟 − 1 = 3 − 1 = 2
(b) 𝑑𝑓𝐵 = 𝑐 − 1 = 3 − 1 = 2
b. How many degrees of freedom are there in determining the interaction variation?
(a) 𝑑𝑓𝐴𝐵 = (𝑟 − 1)(𝑐 − 1) = (3 − 1)(3 − 1) = 4
c. How many degrees of freedom are there in determining random variation?
(a) 𝑑𝑓𝐸 = 𝑟𝑐(𝑛 − 1) = 3 𝑥 3 𝑥(4 − 1) = 27
d. How many degrees of freedom are there in determining the total variation?
(a) 𝑑𝑓𝑇 = 𝑛 − 1 = 35
References
Textbook
Levine, D. M., Stephan, D. F., Szabat, K. A. (2021). Analysis of Variance. Statistics for
Managers Using Microsoft Excel, 11, 352-388.