Statistics 3520
Assignment #5: Nov. 13, 2003
Due in class: Nov. 25, 2003

• All problem numbers are from the textbook Design and Analysis of Experiments, 5th Edition.

1. Consider the randomized complete block design model:
   
   $$y_{ij} = \mu + \tau_{i} + \beta_{j} + e_{ij}, \quad i = 1, \ldots, a, \quad j = 1, \dots, b$$
   
   
   where $e_{ij}$ are independent, $e_{ij} \sim \mbox{N}(0, \sigma^{2})$, and $\sum_{i} \tau_{i} = \sum_{j} \beta_{j} = 0$.

   Find the least squares estimators of $\mu$, $\tau_{i}$ and $\beta_{j}$, i.e. the estimators which minimize
   

   $$L = \sum_{i=1}^{a} \sum_{j = 1}^{b} (y_{ij} - \mu - \tau_{i} - \beta_{j})^{2}$$
   
   

2. Problem #4.32, p. 169.

3. Refer to the data in #4.29, p. 169, and complete the following:
   1. Does the hardwood concentration affect the strength of paper produced? Test at $\alpha = 0.05$, and make sure to state your conclusion in the context of the problem.

   2. Construct a set of three orthogonal contrasts for this problem.

4. Refer to the data in #5.7, p. 213, and complete the following:
   1. Construct by hand a plot of the treatment averages versus the treatment combinations (as done in class). What does this plot suggest about the effects of drill speed, feed rate, and their interaction?

   2. Test at $\alpha = 0.01$ if drill speed and feed rate interact in their effect on the thrust force.

   3. Is it appropriate to investigate main effects in this problem? Explain.

   4. Regardless of your answer in (c), test for the main effect of drill speed on the thrust force. Base your conclusion on your p-value.

5. For a two-factor factorial design with 3 levels of A, 4 levels of B, and 3 replicates per treatment, the following results were found:
   
   $$\bar{y}_{1 \cdot \cdot} = 20, \mbox{ } \bar{y}_{2 \cdot \... ...14, \mbox{ } \bar{y}_{3 \cdot \cdot} = 11, \mbox{ } MSE = 20$$
   
   
   Use the Bonferroni approach to find a 95% confidence interval to compare levels 1 and 3 of factor A. What is your conclusion?