A state health department conducted an experiment to evaluate the reliability of assessing the level of contamination of e. coli in three food sources, meat, fruit, and vegetables. There are four unique methods for assessing e. coli€”M1, M2, M3, and M4€”and hundreds of laboratories that use one or more of these methods in the United States. For each of the methods of assessment, five laboratories are randomly selected to participate in the study. Forty containers are prepared for each food source by spiking the container with a known level of contamination of e. coli and then placing the container in a controlled climate for 3 weeks to allow the e. coli level to stabilize. Six containers, two of each of the three food sources, are then sent to each of the 20 laboratories selected for the study. The e. coli level (cfu/g), Yijkl, determined by the kth lab using assessment method j for the lth container of food source i is recorded for each of the 120 containers. The health department wants to compare the mean e. coli levels of the four assessment methods and their differences across the food sources. It also wants to determine if there are major differences in the mean e. coli determinations across the many laboratories in the United States.

a. Write a model that displays an appropriate relationship between a level of contamination, Yijkl, and its possible sources of variation. Include any restrictions on the parameters in your model and any distributional properties of the random variables in your model.
b. Do the necessary conditions for testing hypotheses and constructing confidence intervals appear to be satisfied? Justify your answers.
c. Construct an ANOVA table for this experiment. Make sure to include expected mean squares and the p-values for the F tests.
d. At the α = .05 level, which main effects and interaction effects are significant? Justify your answer by including the relevant p-values.
e. What are your overall conclusions about the differences in the four assessment methods?

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Assessment Method MI M2 Lab M3 Lab M4 Lab Lab Souce L L2 L3 L4 L5 L6 L7 L8 L9 L10 L1 L12 L13 L14 L15 L16 L17 LIS L19 L20 Meat 12.3 132 129 132 129 14.5 14.3 14.5 14.3 14.5 14.4 13.5 14.7 135 14.7 148 16.4 15.6 148 15.2 12.6 13.0 13.0 14.0 13.9 15,0 15.6 14.8 15.6 14.8 14.1 13.4 14.6 134 14.2 15.3 14.3 16.4 15.4 14.4 Fruit 13.2 14.4 12.9 14. 128 13.2 14.2 14.4 133 14.5 134 14.5 12.7 135 12.7 12.2 14.4 12.4 134 13.2 13.4 14.5 13.7 14.1 13.4 14.2 134 14.6 13.6 138 141 144 14.2 14.4 14.2 13.3 136 138 13.8 13.3 e 13.1 134 13.6 138 128 13.5 13.3 135 14.3 135 143 13.5 13.7 14.5 13.7 122 14.3 13.6 13.4 14.4 12.5 14.0 13.0 14.1 13.3 14.0 126 12.8 13,6 128 15. 154 152 154 15.2 13.3 139 12.7 13,9 14.1