IB Mathematics Applications & Interpretation 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 $y=2x^3+b{x}^2+cx+d$ 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 $y=a{x}^3$, 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 $y=a{x}^3+b{x}^2+cx+d$. 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 $y=a{x}^3+b{x}^2+cx+d$, determine the coefficients $a$, $b$, $c$ and $d$.

A researcher proposes the model $y=a{x}^3+b{x}^2+cx+5$ 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 $y=a{x}^3+b{x}^2+cx+d$, 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 $y=a{x}^3+b{x}^2+cx+d$.

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.