"Graphical Analysis In Exercises 1–3, use the figure.

2. Describe the explained variation about a regression line in words and in symbols."

Verified step by step guidance

1. Step 1: Understand the context of explained variation in regression analysis. Explained variation refers to the portion of the total variation in the response variable (y) that is accounted for by the regression line.
2. Step 2: In words, explained variation measures how much of the difference between the observed values (y_i) and the mean of y (ȳ) is explained by the predicted values (ŷ_i) from the regression line.
3. Step 3: Symbolically, explained variation is represented as the sum of squared differences between the predicted values and the mean of y: \( \sum (\hat{y}_i - \bar{y})^2 \).
4. Step 4: This quantity contrasts with residual variation, which is the sum of squared differences between observed values and predicted values: \( \sum (y_i - \hat{y}_i)^2 \).
5. Step 5: The total variation in y is the sum of explained variation and residual variation, expressed as \( \sum (y_i - \bar{y})^2 = \sum (\hat{y}_i - \bar{y})^2 + \sum (y_i - \hat{y}_i)^2 \).

Key Concepts

Explained Variation

Explained variation measures how much of the total variation in the response variable (y) is accounted for by the regression line. It is the difference between the predicted value (ŷᵢ) and the mean of y (ȳ), showing how well the model explains the data.

Regression Line

The regression line represents the best linear fit to the data points, predicting the response variable y based on the explanatory variable x. It minimizes the sum of squared residuals and is used to estimate ŷᵢ, the predicted values of y.

Residuals

Residuals are the differences between observed values (yᵢ) and predicted values (ŷᵢ) from the regression line. They represent the unexplained variation or error in the model, indicating how far each data point is from the fitted line.