For each of the three descriptions below, say whether or not each of factorial, with replication, and balanced
apply or not, and specifically how you could tell.
a) An ANOVA is performed to see how effective different methods of studying are, and how much extra time
spent helps. Students studied by either primarily going over old exams, reading notes, or reading the text; and
they studied for either two, five, or ten hours. The forty-five students were assigned at random so that there
were five in each of the combinations of studying strategies and time.
b) An ANOVA is performed to see how effective different methods of studying are, and how much extra time
spent helps. Students studied by either primarily going over old exams, reading notes, or reading the text; and
they studied for either two, five, or ten hours. The sixteen students were assigned at random so that four spent
two hours using the text, four spent two hours using the notes, four spent two hours using the old exams, two
spent five hours using text, and two spent ten hours using the text.
c) An ANOVA is performed to see how effective different methods of studying are, and how much extra time
spent helps. Students were assigned to study by either primarily going over old exams, reading notes, or
reading the text; and they studied for either two, five, or ten hours. The nine students were assigned at random
so that there was one in each of the combinations of studying strategies and time.
2) Questions 2 uses the following setting – Seven different types of material (labeled A-G) were sent out to a
sample of 13 laboratories for stress testing (since different laboratories use different testing methods), with four
copies of each sample being sent to each lab. The PROC GLM code used was:
PROC GLM;
CLASS lab material;
MODEL stress = lab material lab*material;
RANDOM lab lab*material;
RUN;
And the output was:
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 90 322913.2482 3587.9250 177.01 <.0001
Error 273 5533.5800 20.2695
Corrected Total 363 328446.8282
Source DF Type III SS Mean Square F Value Pr > F
lab 12 30328.0547 2527.3379 124.69 <.0001
material 6 268778.0771 44796.3462 2210.03 <.0001
lab*material 72 23807.1165 330.6544 16.31 <.0001
Source Type III Expected Mean Square
lab Var(Error) + 4 Var(lab*material) + 28 Var(lab)
material Var(Error) + 4 Var(lab*material) + Q(material)
lab*material Var(Error) + 4 Var(lab*material)
a) What null hypothesis is being tested by the material line in the Type III table (the F-value of 2210.03).
Carefully state the null and alternate hypothesis and your conclusion at the =0.05 level.
b) Find the value of the F statistic for testing H0:
2
lab =0 against HA:
2
lab >0, and test that hypothesis at the
=0.05 level. (The critical value can be found using qf in R or a web-page for calculating F-critical values such
as http://www.biokin.com/tools/f-critical.html; similarly, the p-value could be found using pf in
R or an F distribution calculator on line. In any case, make sure you are using it correctly to get the probability
in the end of the distribution you want).