Establish with argument what effects there are, and whether they are random or fixed. Use the "lmer" function to then fit your proposed model to the data. Provide residual plots and statistical conclusions regarding extrapolating this model to other data points.

2.Explain the mathematical meaning behind each for the input parameters to the "lmer" function.

Explain your reasoning and provide examples where necessary when answering the following questions:

Are all hierarchical models necessarily mixed models?

Are mixed models necessarily hierarchical?