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Simple linear regression commonly uses a least squares approach, where the linear model attempts to minimize the residual distance between the actual points and the predicted line. Consequently, its residual is defined as:
$$e_i = y_i -\hat{y}_i$$

where $y_i$ represents the actual ith response value and $\hat{y}_i$ represents the ith response value of the linear model. 

Least Squares Approach

A simple linear model is defined by the equation $\hat{y} = 1.5x + 4$. For an observed data point where the actual response value ($y$) is 12 when the predictor value ($x$) is 5, what is the residual?

Calculating a Residual

Evaluate the proposed linear model's fit to the data provided in the case study. In your evaluation, calculate the difference between each actual response value and the value predicted by the model. Based on these differences, is this model a good representation of the data according to the principle of finding a line that is as close as possible to all data points? Justify your answer.

Evaluating a Linear Model's Fit

The Residual Sum of Squares (RSS) is the sum of the squared residuals (differences between predicted and actual values) across all data points. It is defined by the formula:

$$RSS = e_1^2 + e_2^2 + \\dots + e_n^2$$

Alternatively, in simple linear regression, it can be written as:

$$RSS = \\sum_{i=1}^n (y_i - \\hat{\\beta}_0 - \\hat{\\beta}_1 x_i)^2$$

where $$\\hat{\\beta}_0$$ and $$\\hat{\\beta}_1$$ are the estimated model parameters. A smaller RSS indicates that the regression model fits the data more closely.

Residual Sum of Squares (RSS)

A data scientist is building a simple linear model for a set of data points. Four potential lines are generated to fit the data. According to the least squares approach, which of the following lines would be considered the best fit for the data?

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