IB Mathematics Applications & Interpretation Question Type 1: Determining Best Fit Model for a Given Set of Data Exercises

A set of data $(x,y)$ is given by:

$$(1,2),(2,4),(3,8),(4,16)$$

Investigate which model form—a power model $y=a{x}^b$ or an exponential model $y=A{B}^x$—fits the data better.

Calculate the sum of squared residuals (SSR) for both models to support your conclusion.

A data set $(x,y)$ consists of the following points:

$$(10,20),(20,80),(30,180),(40,320)$$

Assuming a power model $y=a{x}^b$, use linearization to find estimates of $a$ and $b$.

The data below are recorded for $x$ and $y$:

(1, 2), (2, 5), (3, 10), (4, 17).

Compare the models

(a) quadratic $y=a{x}^2+bx+c$, (b) exponential $y=A{B}^x$,

by computing and comparing their sum of squared residuals (SSR) values. Which model fits better?

A scatter plot of data $(x,y)$ shows that as $x$ doubles, $y$ roughly quadruples. Propose a model of the form $y=a{x}^b$, explain your reasoning, and describe briefly how you would estimate $a$ and $b$.

Given that the sum of squared residuals ($\text{SSR}$) for three candidate models fitted to a data set are ${\text{SSR}}_{\text{linear}}=150$, ${\text{SSR}}_{\text{quadratic}}=120$, and ${\text{SSR}}_{\text{exponential}}=130$.

State which model provides the best fit and justify your answer.

A bacterial culture grows according to an exponential law. The measured population at times $t$ (hours) is given:

$(0,100),(1,180),(2,324),(3,583)$

Find the exponential model $y=A{B}^t$ that best fits the data.

The following $\text{SSR}$ (sum of squared residuals) values are obtained for four candidate models fitted to the same data:

$${\text{SSR}}_{\text{linear}}=500,{\text{SSR}}_{\text{quadratic}}=200,{\text{SSR}}_{\text{cubic}}=180,{\text{SSR}}_{\text{exponential}}=190$$

Select the best model and justify your choice, mentioning any trade-offs.

The data set below is collected for variables $x$ and $y$:

$(1,3),(2,5),(3,7),(4,9)$.

(a) Fit a linear model $y=mx+b$ to these points and compute the sum of squared residuals (SSR).

(b) Fit an exponential model $y=A{B}^x$ by linearizing $\ln y$ vs $x$ and compute the SSR.

(c) Determine which model is more appropriate.

After fitting a linear model and a cubic model to a given data set, you obtain $SS{R}_{\text{linear}}=50$ and $SS{R}_{\text{cubic}}=0$. Should you choose the cubic model? Explain any additional considerations beyond $SSR$.

You fit both an exponential model and a cubic model to a data set. The sum of squared residuals ($SSR$) values are $SS{R}_{\text{exp}}=120$ (with 2 parameters) and $SS{R}_{\text{cubic}}=110$ (with 4 parameters).

Discuss which model you would choose, referring to $SSR$ and model complexity.