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

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

A set of data points is given as $(1, 6)$, $(2, 20)$, $(3, 54)$, and $(4, 110)$.

Test the fit of these data points to the model $y = 2x^3 + bx^2 + cx + d$ by finding the values of $b$, $c$, and $d$ and checking for consistency.

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

A dataset $(x, y)$ consists of the points $(1, 3), (2, 17), (4, 129)$, and $(5, 314)$. It is suggested that the data follows a model of the form $y = ax^3$, where $a$ is a constant.

Determine the value of $a$ and check the fit of the model to the data.

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.

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.

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

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 $d$.

A researcher proposes the model $y = ax^3 + bx^2 + cx + 5$ to fit the data points $(1, 8), (2, 21), (3, 50)$, and $(4, 101)$.

Find the values of $a, b$, and $c$, and assess the fit of the model to the given data.

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.

A data set consists of the points $(0, 2)$, $(1, 4.5)$, $(2, 15.5)$, $(3, 40.5)$, and $(4, 85)$. It is proposed that the data follows a cubic model of the form $y = ax^3 + bx^2 + cx + d$.

Use the method of least squares (normal equations) to set up a system of linear equations to find the parameters $a, b, c$, and $d$.

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