# Homework Assignment for Lesson 9

### Homework Assignment

1. A survey of health awareness in households results in an "index'' for each household. This survey is made in 3 states, 3 cities within each state, and sampling 5 households within each of cities. The data can be found on ANGEL as "HealthData.xslx''.

1. First specify a model in which State and City are both fixed effects. Write out an appropriate model, fit the model in SAS, and report the ANOVA table. Clearly specify the null hypotheses, test statistics, and distributions of the test statistics for State and City.
2. Now specify a model in which State and City are both random effects. Write out an appropriate model, fit the model in SAS, and report the ANOVA table. Clearly specify the null hypotheses, test statistics, and distributions of the test statistics for State and City.
3. Now specify a model in which State is a fixed effect and City is a random effect. Write out an appropriate model, fit the model in SAS, and report the ANOVA table. Clearly specify the null hypotheses, test statistics, and distributions of the test statistics for State and City.
4. Which of these three models would be most appropriate if the researchers want their analysis to have a nation-wide scope of inference, and they randomly chose the 3 cities within each state from all large cities present in the state?

2. For the following descriptions of experiments, write out an appropriate model for the experiment. Clearly show which factors are nested and which are crossed, as well as which are fixed and which are random. It may be helpful (but is not required) to draw a diagram of the experiment.

1. An experiment was conducted to determine how much of the variation in measured manganese concentration in steel was due to operator variation. Ten steel samples were sliced from a steel billet. Each operator was asked to measure the manganese content of each sample twice. The measurements were done in a random order on a single day. There were four operators, who were regarded as representative of a large population of potential operators. The response variables are the 2 measurements of manganese content for each of the 4 operators on each of the 10 steel samples, giving 10 × 4 × 2 = 80 total measurements.
2. Fish were put at random into eight troughs of water. Two troughs were assigned to each of the four levels of the treatment factor “sulfamerazine” (0, 5,10, 15 grams per100 pounds of fish added to the diet per day). After 42 days, five fish were selected at random from each trough and the erythrocite count from the blood of each fish was measured in two different counting chambers, giving two measurements per fish. The response variables are the erythrocite counts, of which there are 2 × 5 × 8 = 80 total counts from the 5 fish pulled from each of the 8 troughs.