11.1
An experiment has a single factor with five groups and seven values in each group.
a. How many degrees of freedom are there in determining the among-group variation?
c–1=5–1=4
b. How many degrees of freedom are there in determining the within-group variation?
n – c = 35 – 5 = 30
c. How many degrees of freedom are there in determining the total variation?
n – 1 = 35 -1 = 34
11.2
You are working with the same experiment as in Problem 11.1.
a. If SSA = 60 and SST = 210, what is SSW?
SST – SSA = SSW, 210 – 60 = 150
b. What is MSA?
MSA = SSA/ (c – 1) = 60/4 = 15
c. What is MSW?
MSW = SSW/ (n – c) = 150/30 = 5
d. What is the value of FSTAT?
FSTAT = MSA/MSW = 15/3 = 3
11.3
You are working with the same experiment as in Problem 11.1 and 11.2.
a. Construct the ANOVA summary table and fill in all values in the table.
Source
Among groups
Within Groups
Total
Degrees of
Freedom
4
30
34
Sum of Squares
60
150
210
Mean Square
(Variance)
15
5
F
2.69
b. At the 0.05 level of significance, what is the upper-tail critical value from the F
distribution?
There are 4 degrees of freedom in the numerator and 30 degrees of freedom in the
denominator.
Fα, the uppertail critical value at the 0.05 level of significance, is 2.69.
c. State the decision rule for testing the null hypothesis that all five groups have equal
population means.
Reject H0 if FSTAT > 2.69; otherwise, do not reject H0.
d. What is your statistical decision?
Because FSTAT = 3 is greater than Fα = 2.69, then I reject the null hypothesis.
There is a significant difference in the population mean among the five groups.
11.4
Consider an experiment with three groups, with seven values in each group.
a. How many degrees of freedom are there in determining the among-group variation?
c–1=3–1=2
b. How many degrees of freedom are there in determining the within-group variation?
n – c = 21 – 3 = 18
c. How many degrees of freedom are there in determining the total variation?
n – 1 = 21 -1 = 20
11.5
Consider an experiment with four groups, with eight values in each. For the ANOVA
summary table below, fill in all missing results:
Source
Degrees of
Freedom
Sum of Squares
Mean Square
(Variance)
80
20
F
Among groups 3
240
2.95
Within Groups 28
560
Total
31
800
c=4
n = the number of values in each group times the number of groups = 8 x 4 = 32
DF Among groups = 4 – 1 = 3
DF Within groups = 32 – 4 = 28
DF Total = 32 – 1 = 31
SSA = (c – 1) (MSA) = 3 x 80 = 240
SST = SSA + SSW = 240 + 560 = 800
MSW = SSW/ (n – c) = 560/28 = 20