Question: Lipid oxidation will lead to the meat quality degradation during storage. Researchers have been investigating numerous way of improving the antioxidant activity of meat. This research treated meat sample with oregano and sage essential oil. Fifteen bovine meat samples were obtained from the local market and randomly assigned to three treatments (five samples for each of the three groups): homogenized with \(3 \%\) oregano essential oil, \(3 \%\) sage essential oil. and the control group with no essential oil added. Two proportions of \(3 \mathrm{~g}\) were taken from each meat sample of which one stay in raw and the other received heat treatment. Thiobarbituric acid (TBA) was used to measure the antioxidant ability for each meat portion. In total, there are 3 (essential oil treatment groups) \(\times\) 5 (samples per group) \(\times 2\) (portions per sample) \(=30\) measurements. The researchers are interested in investigating the essential oil treatment and heat treatment effects on the antioxidant ability of the meat samples.

1. [9 pts] Provide the appropriate model for this study. Clearly label and define all subscripts.
   Your answer for the model should be something like: \(y_{i j}=\mu+\tau_{i}+\epsilon_{i j}\) for \(i=1, \ldots, 8\) (make sure to clearly define the \(y_{i j}, \tau_{i}\), etc. in your model.)
2. Can you improve the power of the test for testing the essential oil effects without increasing

the sample size and \(\alpha ?\) If so, explain briefly how to increase the power. No calculation is needed.

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