Question Type 3: Comparing two different models over the same data using SSR Exercises

Given observed values $[2,4,6]$ and two models with predicted values Model A: $[2.1,3.9,6.2]$, Model B: $[1.9,4.3,5.8]$, compute the sum of squared residuals (SSR) for each model and state which model provides a better fit.

For the data points $(0,3),(1,6),(2,11)$, compare the fit of Model A: $y=2x+3$ and Model B: $y=3e^{0.5x}$ by calculating the SSR for each model and indicating the better model.

An asset depreciates: at year $t$ its value is observed $(0,100),(1,80),(2,64),(3,51.2)$. Compare decay models: $y=100(0.8)^t$ and linear $y=100-20t$ by computing SSR for each.

Observed monthly sales are $(1,1000),(2,1050),(3,1100),(4,1150),(5,1200)$. Compare Model A: $y=1000(1.05)^t$ (compound interest) and Model B: $y=1000+50t$ (linear trend). Compute SSR for each and select the better model.

The data points $(1,2),(2,5),(3,15),(4,40)$ are modeled by Model A: $y=1.5e^{0.9x}$ and Model B: $y=0.2x^3$. Compute the SSR for each model and state which model fits best.

Light intensity vs distance data: $(1,100),(2,25),(3,11.11),(4,6.25),(5,4)$. Compare Model A: $I=100/x^2$ (inverse-square) and Model B: $I=100e^{-1.1(x-1)}$ by computing SSR for each.

For data $(0,1),(1,3),(2,7),(3,13),(4,21)$, compare Model A: $y=2^x+1$ and Model B: $y=x^2+ x+1$ by computing SSR for each model. Which model fits better?

The data $(1,3),(2,8),(3,20),(4,50)$ are compared by Model A: $y=e^{1.1x}$ and Model B: $y=2x^2+x$. Compute SSR for each and choose the better model.

The points $(1,2),(2,5),(3,11),(4,24),(5,55)$ are to be fitted by both an exponential model and a power-law model. (a) Fit $y=ae^{bx}$ by regressing $\ln y$ on $x$. (b) Fit $y=cx^d$ by regressing $\ln y$ on $\ln x$. Compute SSR in the original $y$ for both and determine which model is superior.

The measurements $(0,1),(1,2.7),(2,7.4),(3,20.1)$ are to be modeled by an exponential $y=ae^{bx}$ and a linear $y=6x-1$. Fit the exponential model by linearizing igl( ext{take }\\ln yigr), find $a$ and $b$, compute SSR for both models, and determine which is better.

A bacterial culture yields counts $(0,50),(1,120),(2,280),(3,600)$. Compare Model A: $N=50e^{1.2t}$ and Model B: $N=\displaystyle\frac{1000}{1+9e^{-t}}$ by computing SSR for each model and stating which fits better.