Question

explain why a data point above a linear model has positive residual.
for a given data point (x,y), the residual is the dropdown value of dropdown subtracted from the dropdown value of dropdown. the value of dropdown for a data point above a linear model is greater than the dropdown value, so the difference is positive.

Explanation:

Step1: Recall Residual Formula

The residual for a data point \((x,y)\) in a linear model is defined as \( \text{Residual} = y - \hat{y} \), where \( y \) is the observed (actual) value of the data point, and \( \hat{y} \) is the predicted value from the linear model.

Step2: Analyze Data Point Above Model

If a data point is above the linear model, the observed \( y \)-value (actual \( y \)) of the data point is greater than the predicted \( \hat{y} \)-value (the \( y \)-value on the linear model at that \( x \)).

Step3: Determine Residual Sign

Using the residual formula \( \text{Residual} = y - \hat{y} \), when \( y > \hat{y} \), subtracting \( \hat{y} \) from \( y \) will result in a positive number. So the residual is positive.

Answer:

For a given data point \((x,y)\), the residual is the \(\boldsymbol{\text{actual}}\) value of \(\boldsymbol{y}\) subtracted from the \(\boldsymbol{\text{predicted}}\) value of \(\boldsymbol{\hat{y}}\) (or more precisely, residual \(= y - \hat{y}\)). The value of \(\boldsymbol{y}\) (actual \(y\)) for a data point above a linear model is greater than the \(\boldsymbol{\hat{y}}\) (predicted \(y\)) value, so the difference \(y - \hat{y}\) is positive.

(To fill in the blanks in the original problem's sentence: "For a given data point \((x,y)\), the residual is the \(\boldsymbol{\text{actual}}\) value of \(\boldsymbol{y}\) subtracted from the \(\boldsymbol{\text{predicted}}\) value of \(\boldsymbol{\hat{y}}\). The value of \(\boldsymbol{y}\) for a data point above a linear model is greater than the \(\boldsymbol{\hat{y}}\) value, so the difference is positive.")