# Homework Assignment for Lesson 2, pt 1

### Homework Assignment

1. Use SAS to randomly assign 10 experimental units to each of three treatments (1, 2, and 3). Then simulate responses for the 30 experimental units satisfying the one-way ANOVA model:$Y_{it}=\mu+\tau_i+\epsilon_{it},\quad i=1,2,\ldots,v \quad t=1,2,\ldots,r_i$$\epsilon_{it} \stackrel{iid}{\sim} N(0,\sigma^2)$ with $\mu=4.7$, $\sigma^2=4$, and treatment effects $\tau_1=-3$,$\tau_2=5$, and $\tau_3=-2$. Your solution should include your SAS code and a plot of the simulated values.
2. Consider the situation in Problem 1. The experimenter wants to consider a reduced model where $\tau_1=\tau_2=\tau_3=0$. Simulate responses for the 30 experimental units satisfying this reduced model. Compare boxplots of simulated responses under this reduced model with boxplots of simulated responses under the full model described in Problem 1 (where there are differences in the treatment effects).
3. Now explore what happens to data simulated from the model in Problem 1 when the error variance increases. Try multiple values for $\sigma^2$ and find a value of $\sigma^2$ for which you cannot see any noticeable difference in the boxplots of response values from the three treatments.
4. Under the model in Problem 1, what is the distribution of $Y_{23}$, the response from the 3rd experimental unit to receive treatment 2?
5. Under the model in Problem 1, what is the distribution of $\bar{Y}_{2\cdot}=\frac{1}{r_2}\sum_{t=1}^{r_2} Y_{2t}?$
6. Under the model in Problem 1, what is the distribution of the difference between an experimental unit receiving treatment 1 and an experimental unit receiving treatment 2?
7. Under model 1, what is the distribution of$\sum_{t=1}^{10} \left(\frac{Y_{1t}-1.7}{2}\right)^2?$