A Harvard University biologist used multiple regression to model the swimming speed of an American eel. (Proceedings of the Royal Society, B, Dec. 2004.) Steady swimming speed, y (body lengths per second), was modeled as a function of the quantitative variables body wave speed, x₁ (body lengths per second), tail amplitude deviation, x₂ (body lengths), and tail velocity deviation, x₃ (body lengths per second). 

a. Write a first-order model for E(y) as a function of the three independent variables.

b. Give an interpretation of the value of β₁ in the model, part a.

c. Write a model for E(y) as a function of the three independent variables that hypothesizes an interaction between body wave speed, x₁, and tail amplitude deviation, x₂.

d. In terms of the β’s in the model, part c, what is the change in E(y) for every 1-unit increase in tail velocity deviation, x₃, for fixed values of x₁ and x₂?
e. In terms of the β’s in the model, part c, what is the change in E(y) for every 1-unit increase in tail amplitude deviation, x₂, for fixed values of x₁ and x₃?