In a completely randomized design, there are two factors, A with two levels and B with three levels. Suppose the 6 treatment means are \[ \begin{aligned} &\mu_{11}=6, \mu_{12}=10, \mu_{13}=8 \\ &\mu_{21}=5, \mu_{22}=5, \mu_{23}=5 . \end{aligned} \] a.[3pts] Are there interaction effects? Why?
b.[3pts] Find \(\mu, \alpha_i, \beta_j\) and \((\alpha \beta)_{i j}, i=1,2, j=1,2,3\), such that \[ \mu_{i j}=\mu+\alpha_i+\beta_j+(\alpha \beta)_{i j} . \]
Consider the reaction time experiment described in Exercise 4 of Chapter 4.
a.[3pts] Write down the two-way complete model for the experiment. Remember to explain each term in the model.
b.[3pts] Find the sums of squares that are accounted for by the factors and their interactions, i.e., ssA, ssB and ssAB.
c.[5pts] Generate an interaction plot. Do you see an obvious interaction between the two factors? Carry out a formal hypothesis testing for the interaction.
d.[3pts] Test the hypothesis that different elapsed times have the same effects on the reaction time.
e.[3pts] Find a \(95 \%\) confidence interval for the difference between the average reaction time from the auditory cue and the average reaction time from the visual cue [Hint: this is to compare the two main effects of factor cue].
f.[3pts] Find an appropriate confidence interval for the difference between auditory cue and the visual cue, when the elapse time is 5 seconds.