A simple curve that often makes a good model for the variable costs of a company, as a function of the sales level x, has the form y 1 x 2 x 2 3 x 3 There is no constant term because fixed costs are not included a Give the design matrix and the parameter vector for the linear model that leads to a ...

A simple curve that often makes a good model for the variable costs of a company, as

Question:

A simple curve that often makes a good model for the variable costs of a company, as a function of the sales level x, has the form y = β₁x +β₂x²+ β₃x³.There is no constant term because fixed costs are not included.

a. Give the design matrix and the parameter vector for the linear model that leads to a least-squares fit of the equation above, with data (x₁; y₁)........(x_n; y_n).

b. Find the least-squares curve of the form above to fit the data (4,1.58),(6, 2.08),(8,2.5), (10,2.8),(12,3.1),(14,3.4),(16,3.8) and (18,4.32) with values in thousands. If possible, produce a graph that shows the data points and the graph of the cubic approximation.
c. If a machine learned the curve you found in (b), what output would it provide for an input of x = 9?

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