Chapter 9.10, Problem 3E

In an experiment with t = 4 populations means, consider the four linear combinations of those means.

$\begin{array}{l}{l}_1={\mu }_1-3{\mu }_2+{\mu }_3+{\mu }_4\\ l_2={\mu }_1+{\mu }_2-2{\mu }_4\\ l_3={\mu }_1+{\mu }_2+{\mu }_3+{\mu }_4\\ l_4={\mu }_1+{\mu }_2-3{\mu }_3+{\mu }_4\end{array}$

1. a. Which of the four linear combinations are contrasts?
2. b. Which pairs of contrasts are orthogonal?
3. c. Suppose we have two contrasts:

$l_1={\mu }_1+{\mu }_2+{\mu }_3-3{\mu }_4\text{ and }{l}_2=\frac{1}{3}{\mu }_1+\frac{1}{3}{\mu }_2+\frac{1}{3}{\mu }_3-{\mu }_4$

Is testing H₀ : l₁ = 0 equivalent to testing H₀ : l₂ = 0? Justify your answer.