Take home problems for Exam 2

1. Medical researchers are studying the amount of sleep time that people get, and if it varies between gender and age groups. They collected data on subjects according to 6 age groups (1 = youngest, 6 = oldest) and had them record the average number of hours of sleep over time. Assume that age and gender are fixed effects. The data are:
 
age gender sleeptime 
1 M 8.8 
1 M 8.8 
1 M 8.7 
1 F 9.0 
1 F 9.1 
1 F 8.8 
2 M 8.1 
2 M 8.2 
2 M 7.8 
2 F 8.3 
2 F 8.5 
2 F 8.9 
3 M 8.0 
3 M 8.1 
3 M 8.0 
3 F 7.8 
3 F 8.1 
3 F 7.8 
4 M 7.8 
4 M 7.8 
4 M 7.8 
4 F 8.2 
4 F 8.4 
4 F 8.1 
5 M 8.0 
5 M 7.9 
5 M 8.0 
5 F 8.0 
5 F 8.1 
5 F 8.2 
6 M 8.5 
6 M 8.2 
6 M 8.8 
6 F 8.5 
6 F 8.8 
6 F 8.6
Write one or more sentences summarizing your conclusions for each 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.).

a) Name this design and give the model equation.
b) Conduct an analysis of variance (fill out the ANOVA table, including F statistics, and P values) to test the null hypotheses. If an interaction exists, use a profile plot to interpret the interaction. If no interaction exists, then calculate main effect means for each factor.
c) Examine a residual by predicted plot and a normal plot of the residuals to assess model assumptions.

2. Transportation researchers are examining how gasoline efficiency in cars varys among different companies (Makes), different types of vehicles (Categories), and between city and highway ratings (Locations). They had a computer program produce the following ANOVA table:

Source                  		DF    SS
Make                    		2     410.6
Category                		1     240.7
Location                		1     308.2
Make*Category           	2     410.1
Make*Location           	2     2.58
Category*Location       	1      1.5
Make*Category*Location  	2      1.75
Error                   		12    222.0
Total                   		23    1597.3
Treat the Make effect as random and the other two as fixed.

a) Calculate the correct F statistic value for the Make by Location interaction. Give the degrees of freedom for the F statistic.

b) Calculate the correct F statistic value for the main effect of Category. Give the degrees of freedom for the F statistic.

c) Calculate the correct F statistic value for the main effect of Make. Give the degrees of freedom for the F statistic.

Use one or more sentences to explain each result.


3. The following data are from a pilot study involving two factors.

The data:
B1 B2 B3
A1 5, 7 6, 8, 10 9
A2 7 8, 10 9, 7, 8

Calculate both the observed means and the least-squares means for both levels of factors A, and for the three levels of factor B. Calculate the standard errors of the least-squares means for factor A, level 1 and for factor B, level 2 by hand. You can use the computer to check your results, but you must have a hand calculation. You can use the MSE from the computer in your hand calculation.