Bring to the in-class exam on October 17
Please remember that this is an exam, so you are to do your own work and not communicate with other students about the exam
1. How do rent costs vary by state and the number of bedrooms? To address
this question certain states were selected and two estimates of rent costs
were obtained for a number of bedrooms varying from 0 to 4. The data are
in this file. Here is some code to help
read the data into R and SAS.
Remember to show your work and comment on your results for each problem
and sub-problem.
a. Use boxplots to visualize potential effects of state and number of
bedrooms on rents. What do the plots suggest about group differences and
model assumptions?
b. Assume that these eight states are the only ones of interest. Conduct
an analysis of variance to test the null hypotheses of equality of rents
by state, by number of bedrooms, and of no interaction between the two
factors (show the ANOVA table, including sums of squares, mean squares, F
statistic, and P value). If there is no interaction, use Tukey multiple
comparison tests to compare levels of significant main effect factors. If
an interaction is found, use an interaction plot and simple main effect
tests to help interpret the interaction.
c. Examine a residual by predicted plot and a normal plot of the residuals
to assess model assumptions of equal variance and normality.
d. Use the Box-Cox procedure to find a recommended transformation of the
response. Would you reject the null hypothesis that the power parameter =
1?
e. Select a transformation of the wages, and repeat the ANOVA on the
transformed data (just parts b and c above). Do the results change? Which
analysis is more appropriate? What is your conclusion about the effects of
state and number of bedrooms on rent?
2. 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 5
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 (linked with
lecture 16), calculate the value of the F statistic for each effect and
also report the degrees of freedom for each F statistic. Show your formula
for the F statistic as well as its numerical value. If an exact F test
cannot be found, calculate a value for an approximate F value (you do not
have to calculate degrees of freedom for approximate F tests, and you do
not have to find P values for the tests).
Source
DF Sum of squares
Mean
square F
A
40.4
F = MSA/MS? = xx.xx on y and z df
B
6.21
C
1096.4
A*B
43.3
A*C
220.5
B*C
110.4
A*B*C
76.3
Error
104.7
Total
1698.3