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Chapter 1: Problem 6

Use technology to compute the sum-ofsquares error (SSE) for the given set of data and linear models. Indicate which linear model gives the better fit. $$ (0,1),(1,1),(2,2) ; \quad \text { a. } y=0.4 x+1.1 \quad \text { b. } y=0.5 x+0.9 $$

Short Answer

Expert verified
We found the SSE values for both linear models: - For linear model (a) y=0.4x+1.1, SSE_a = 0.27 - For linear model (b) y=0.5x+0.9, SSE_b = 0.18 Since SSE_b is smaller than SSE_a, linear model (b) y=0.5x+0.9 provides a better fit for the given set of data points.

Step by step solution

01

Calculate the predicted values for each linear model

We have the given set of data points: (0,1); (1,1); (2,2) and two linear models: (a) y=0.4x+1.1, (b) y=0.5x+0.9 Using linear model (a), calculate the predicted values for each given point: - For point (0,1): \(y_a = 0.4(0) + 1.1 = 1.1\) - For point (1,1): \(y_a = 0.4(1) + 1.1 = 1.5\) - For point (2,2): \(y_a = 0.4(2) + 1.1 = 1.9\) Using linear model (b), calculate the predicted values for each given point: - For point (0,1): \(y_b = 0.5(0) + 0.9 = 0.9\) - For point (1,1): \(y_b = 0.5(1) + 0.9 = 1.4\) - For point (2,2): \(y_b = 0.5(2) + 0.9 = 1.9\)
02

Calculate SSE for both linear models

For each linear model, find the SSE using: \(SSE = \sum_{i=1}^n (y_i - \hat{y_i})^2\) For linear model (a): - SSE_a = \((1 - 1.1)^2 + (1 - 1.5)^2 + (2 - 1.9)^2\) = \((0.1)^2 + (-0.5)^2 + (0.1)^2\) = 0.01 + 0.25 + 0.01 = 0.27 For linear model (b): - SSE_b = \((1 - 0.9)^2 + (1 - 1.4)^2 + (2 - 1.9)^2\) = \((0.1)^2 + (-0.4)^2 + (0.1)^2\) = 0.01 + 0.16 + 0.01 = 0.18
03

Determine the better fit

We have calculated the SSE values for both linear models: - SSE_a = 0.27 - SSE_b = 0.18 Since SSE_b is smaller than SSE_a, linear model (b) y=0.5x+0.9 provides a better fit for the given set of data points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Regression
Linear regression is a fundamental statistical and machine learning technique used to model the relationship between a dependent variable and one or more independent variables. The idea is to find a linear equation, commonly of the form y = mx + c where y represents the dependent variable, m is the slope of the line, x is the independent variable, and c is the y-intercept.

The main goal of linear regression is to fit the best possible line that represents the trends in given data points, making it easy to predict or understand the outcome when the independent variables are known.

In the given exercise, we deal with a simple form of linear regression involving a single independent variable to predict the dependent variable. The models presented, y=0.4x+1.1 and y=0.5x+0.9, are candidates for the best fitting line to the data points provided, and we need to evaluate which one predicts the data more accurately.
Model Fitting
Model fitting in the context of linear regression involves adjusting the parameters of a chosen linear model to minimize the difference between the predicted and observed outcomes. This process requires a metric to determine how well a model fits the data. A common metric is the Sum-of-Squares Error (SSE), which quantifies the variance between predicted and actual points.

To fit a model effectively, we calculate the SSE for each candidate model and compare their values. The model with the lowest SSE is typically considered to provide the best fit because it has the smallest amount of error in its predictions compared to the others.

In our task at hand, the exercise involves model fitting by comparing two linear equations to see which one produces a smaller SSE, and hence, a closer fit to the actual data points.
SSE Calculation
SSE, or Sum-of-Squares Error, is a measure of the total squared differences between predicted values and the actual values in a dataset. It's an essential part of linear regression as it provides insight into how well the regression line fits the data.

The formula to calculate SSE is: \( SSE = \sum_{i=1}^n (y_i - \hat{y_i})^2 \), where \(y_i\) represents the actual data points and \(\hat{y_i}\) represents the predicted values from our regression model.

In the exercise provided, SSE is calculated for two linear models by using the respective formulas a and b. These computations give us the information we need to compare the models and determine which one offers the closer fit to our data by having the lowest SSE. The step-by-step calculation clearly shows that model b yields a lower SSE, indicating it is the better choice for fitting our dataset.

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