Question Type 2: Determining the fit of a model for a given data set and a specific given model Exercises

Find the cubic polynomial that fits $(0,0)$, $(1,1)$, $(2,8)$, $(3,27)$ exactly, and comment on its significance.

For the points $(1,3)$, $(2,11)$, $(3,33)$ and $(4,83)$, verify whether they lie exactly on the cubic model $y = 2x^3 - x^2 + x + 1$.

Determine the residual sum of squares (RSS) for the cubic model $y = x^3 -2x^2 + x +1$ on data $(0,1)$, $(1,1)$, $(2,5)$, $(3,19)$.

Suppose a dataset $(x,y)$ with $x=1,2,4,5$ and $y=3,17,129,314$ is said to follow $y=ax^3$. Determine $a$ and check the fit.

For the model $y = 2x^3 - x^2 + 3x - 5$, calculate the residuals for the data points $(0,-5)$, $(1,-1)$, $(2,15)$ and comment on whether the model fits the data exactly.

A researcher proposes $y = ax^3 + bx^2 + cx +5$ to fit $(1,8)$, $(2,21)$, $(3,50)$, $(4,101)$. Find $a,b,c$ and assess the fit.

Given four observations $(x,y)$: $(1,2)$, $(2,17)$, $(3,56)$, $(4,129)$, determine if a unique cubic interpolant exists and describe how to find it.

A set of five points lies approximately on $y=ax^3+bx^2+cx+d$. Explain how you would assess the goodness of fit without computing full regression.

Given the points $(1,4)$, $(2,15)$, $(3,40)$, $(4,85)$ and the model $y = ax^3 + bx^2 + cx + d$, determine whether there is an exact fit and find the coefficients if it exists.

Given the data points $(1, -1)$, $(2, 5)$, $(3, 19)$ and $(4, 49)$ and the model $y = ax^3 + bx^2 + cx + d$, determine the coefficients $a$, $b$, $c$ and $d$.

Data: $(1,6)$, $(2,20)$, $(3,54)$, $(4,110)$. Test fit to $y = 2x^3 + bx^2 + cx + d$ by finding $b,c,d$ and checking consistency.

A data set has points $(0,2)$, $(1,4.5)$, $(2,15.5)$, $(3,40.5)$, $(4,85)$ and is proposed to follow $y = ax^3 + bx^2 + cx + d$. Use least squares (normal equations) to set up the system for $a,b,c,d$.