For some time Eagle Boys pizza chain in Australia has claimed that its pizzas are generally larger than those of Domino's, the biggest pizza chain in the country. For example, advertising material on the company website (http://www.eagleboys.com.au) in 2012 proclaimed "Eagle Boys pizzas have been found to be on average a whole slice larger than Domino's pizzas". To investigate, a group of students in a university Statistics class purchased five "12 inch" pizzas of each of three crust types (deep pan, classic/mid and thin) from both Eagle Boys and Domino's, and measured the diameters of each (in cm). Suppose the data were as given below:



Part a)
which graphical method would best illustrate main effects and possible interactions? Create such a plot for your data using the command interaction.plot in R.

A. A plot of means
B. Two histograms
C. A pie chart
D. A bar chart
E. A stem-and-leaf plot


Part b)
The most general model for the data assumes that the diameter of the $k$th pizza of type $j$ from company $i$ is



in which $\mu$ is the expected value of $\bar{y}$, the overall mean across all the data, $\alpha_i$ is the effect of the $i$th company, $i=1,2$, $\beta_j$ is the effect of the $j$th crust type, $j=1,2,3$, $(\alpha\beta)_{ij}$ is the company-crust interaction effect, and $\epsilon$ is an independent error, assumed to be from the $N(0,\sigma)$ distribution for some $\sigma$.

Fit the above model to your data using R. Hence, compute the ANOVA table below, providing the requested values (i) and (ii) to 3 decimal places:



Value for $\text{(i)}$


Value for $\text{(ii)}$


Part c)
Based on the analysis above, which model would you select for the pizza diameters?

A. $y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon$
B. $y_{ijk} = \mu + \alpha_i + \beta_j + \epsilon$
C. $y_{ijk} = \mu + \alpha_i + \epsilon$
D. $y_{ijk} = \mu + \beta_j + (\alpha\beta)_{ij} + \epsilon$
E. $y_{ijk} = \mu + \beta_j + \epsilon$
F. $y_{ijk} = \mu + (\alpha\beta)_{ij} + \epsilon$
G. $y_{ijk} = \mu + \epsilon$


Part d)
Do the data allow us to determine the truth of the claim made on Eagle Boys' website? Select the best answer from the following:

A. Yes, since clearly the Eagle Boys' pizzas are bigger than those from Domino's.
B. Yes, since the analysis shows that Eagle Boys' pizzas are at least a slice bigger than those from Domino's.
C. No, since there is no evidence that the Eagle Boys' pizzas are bigger than those from Domino's.
D. No, since there is no way to quantify what 'a whole slice larger' means.
E. Both (c) and (d).