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

Calculate the sum of squared residuals (SSR) for each model and determine which model provides a better fit for the data.

A researcher collects the following data for light intensity ($I$) at varying distances ($x$) from a source:

$(1, 100), (2, 25), (3, 11.11), (4, 6.25), (5, 4)$

Two models are proposed to represent this data:

Model A: $I = \frac{100}{x^2}$

Model B: $I = 100e^{-1.1(x-1)}$

Compare the two models by computing the Sum of Squared Residuals (SSR) for each and determine which model provides a better fit.

The data points $(1, 3), (2, 8), (3, 20)$ and $(4, 50)$ are compared using two different models: $\text{Model A: } y = e^{1.1x}$ $\text{Model B: } y = 2x^2 + x$

Calculate the sum of squared residuals (SSR) for each model and determine which model provides a better fit.

Given the observed values $[2, 4, 6]$ and two models with the following predicted values:

• Model A: $[2.1, 3.9, 6.2]$
• Model B: $[1.9, 4.3, 5.8]$

Calculate the sum of squared residuals ($\text{SSR}$) for each model and state which model provides a better fit.

The following measurements $(x, y)$ are collected: $(0, 1)$, $(1, 2.7)$, $(2, 7.4)$, $(3, 20.1)$.

These data are to be modeled by an exponential function $y = a e^{bx}$ and a linear function $y = 6x - 1$.

(a) Fit the exponential model by linearizing the data using the transformation $Y = \ln y$, and find the values of $a$ and $b$.

(b) Compute the sum of squared residuals (SSR) for both the exponential model and the linear model.

(c) Determine which model provides a better fit to the data, justifying your answer.

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 sum of squared residuals ($SSR$) for each model and identifying which model provides the better fit.

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

For the set of data points $(0, 1), (1, 3), (2, 7), (3, 13), (4, 21)$, compare Model A: $y = 2^x + 1$ and Model B: $y = x^2 + x + 1$ by computing the sum of squared residuals (SSR) for each model. Determine which model fits the data better.

The monthly sales of a product over a five-month period are given by the coordinates $(t, y)$:

$(1, 1000), (2, 1050), (3, 1100), (4, 1150), (5, 1200)$

Two models are proposed to represent this data:

Model A: $y = 1000(1.05)^t$ (compound interest model) Model B: $y = 1000 + 50t$ (linear trend model)

where $y$ is the monthly sales and $t$ is the month number.

Calculate the Sum of Squared Residuals ($SSR$) for both models and determine which model provides a better fit for the data.

Two models are proposed to represent the value of the asset:

Model A (exponential): $V = 100(0.8)^t$ Model B (linear): $V = 100 - 20t$

Calculate the sum of squared residuals ($SSR$) for each model and determine which model provides a better fit for the data.