https://file.aiquickdraw.com/mathbot/file/1728377396342cyruyktn.jpg

The image shows a question on "Least Squares" from Chapter 10, focused on residuals and regression lines. Let's break it down:

1. **Question a**: "What is a residual?"

From the multiple-choice options, the correct definition of a residual is:

**D.** "A residual is a value of \( y - \hat{y} \), which is the difference between an observed value of \( y \) and a predicted value of \( \hat{y} \)."  
This is the formal definition of a residual in regression analysis. It measures the vertical distance between an observed data point and the regression line (the predicted value).

2. **Question b**: "In what sense is the regression line the straight line that 'best' fits the points in a scatterplot?"

The regression line minimizes the sum of the squared residuals. In other words, it is the line for which the total of these squared differences between observed and predicted values is as small as possible. This is why it is often called the "line of best fit."

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Do you want any more details on these concepts or have further questions?

### 5 related questions:
1. What is the purpose of minimizing squared residuals in least squares regression?
2. How can outliers affect residuals and the regression line?
3. What is the difference between residuals and errors in a statistical context?
4. How are residual plots used to assess the quality of a regression model?
5. What assumptions must hold for residuals in linear regression to be valid?

### Tip:
When analyzing residuals, always check for patterns in a residual plot—patterns can indicate that the model may not be the best fit for the data.

What is a residual? In what sense is the regression line the straight line that 'best' fits the points in a scatterplot?

Explore the concept of residuals in least squares regression. Learn how the residual is the difference between observed and predicted values, and why the regression line 'best' fits the data by minimizing squared residuals. Suitable for advanced high school and undergraduate students.

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