Question Type 1: Determining best fit model for a given set of data Exercises

Given the sum of squared residuals for three candidate models fitted to a data set are: SSR$_{\text{linear}}=150$, SSR$_{\text{quadratic}}=120$, SSR$_{\text{exponential}}=130$. Which model provides the best fit? Justify your answer.

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 SSR. (b) Fit an exponential model $y=A B^x$ by linearizing $\\ln y$ vs $x$ and compute SSR. (c) Determine which model is more appropriate.

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 $b$.

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 data $(x,y)$ are:

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

Investigate which model form—power $y=a x^b$ or exponential $y=A B^x$—fits the data better. Compute SSR for both fits.

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

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

Compare the models

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

by computing and comparing their SSR values. Which model fits better?

Radioactive material decays according to $y=A e^{kt}$. The activity measured at times $t$ (hours) is:

(0, 100), (1, 80), (2, 64), (3, 51.2).

(a) Fit the decay model and determine $A$ and $k$. (b) Compute the half‐life.

The following SSR values are obtained for four candidate models fitted to the same data:

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

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

A data set $(x,y)$ has 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 $b$.

After fitting a linear model and a cubic model to a given data set, you obtain SSR$_{\text{linear}}=50$ and SSR$_{\text{cubic}}=0$. 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 SSR values are SSR$_{\text{exp}}=120$ (2 parameters) and SSR$_{\text{cubic}}=110$ (4 parameters). Discuss which model you would choose, referring to SSR and model complexity.