Exam 3 (all take home problems)
Please clearly separate your answers to these four questions. As usual, write
one or more sentences summarizing your conclusions for each part of the
following problems. Your summary should allow a person to understand your
results even if they do not know the software that you used (SAS, R, etc.).
Attach key parts of the computer output (ANOVA table listing, plots, not all of
the output) either alongside your written answers, or else in an appendix.
1. Dairy researchers wanted to test the effect of three diets on milk
production of cows. They conducted a replicated latin square experiment using
three 3 x 3 latin squares. Each latin square was blocked by cow and time period.
Over the replicated squares, different cows were used (nine total), but the same
three time periods were re-used. The data are in this file.
a) Write down the model equation.
b) Conduct an analysis of variance (fill out the ANOVA table, including sums
of squares, mean squares, F statistic for each source, and a P value for the
treatments. Be sure to include a line for total SS and df) to test the null
hypotheses of equality of diets on milk production.
c) If a treatment effect is found use Tukey’s method to find which means
differ.
d) Examine a residual by predicted plot and a normal plot of the residuals to
assess model assumptions.
2. For the second square of the replicated Latin square in problem 1, find the
efficiency of the Latin square design relative to a completely randomized
design.
3. Listed below are the sums of squares from a completely randomized factorial
experiment with three factors. Factors A and B are fixed effects with 3 and 2
levels, respectively. Factor C is a random effect with 3 levels. The design is
balanced with 2 replicates for each treatment combination.
a) Complete the ANOVA table including degrees of freedom, mean squares, and a
row for total ss and df.
b) Then, using a table of expected values of mean squares, calculate the value
of the F statistic for each effect and also report the degrees of freedom for
each F statistic. If an exact F test cannot be found, calculate a value for an
approximate F test (you do not have to calculate degrees of freedom for
approximate F tests).
Source Sum of squares
A 273.97
B 441.30
C
1555.25
AB 158.00
AC 1414.24
BC 60.06
ABC 76.51
Error 1768.39
4. Food researchers are investigating the effect of different recipes and baking
temperatures on the moistness of cookies. A single oven is available, so 20
trays will be used with 10 trays randomized to 330 degrees and the other 10
trays to 360 degrees. A standard cookie dough will be used, but either chocolate
chips, peanut butter chips, or pecans will be added to the dough. Each tray will
have 15 cookies with 5 of each type randomly arranged on the tray. All cookies
from the same recipe will have their moistness measured, and an average will be
taken of each recipe for each tray. The data are in
this file.
a) Name the design and write down the model equation.
b) Conduct an analysis of variance (fill out the ANOVA table, including sums of
squares, mean squares, F statistic for each source, and a P value for the
treatments. Be sure to include a line for total SS and df) to test the null
hypotheses of equality of temperature and recipes on moistness, and their
interaction.
c) If an interaction is found, use profile plots or tests to investigate. If no
interaction is found, investigate treatment effects using Tukey’s method to find
which means differ.
d) Examine a residual by predicted plot and a normal plot of the residuals to
assess model assumptions.