Chapter 3: Describing Relationships

What does an association between groups indicate in a distribution?

Differences in distribution indicate an association.

How do you display two quantitative variables graphically?

Use a scatter plot, with the explanatory variable on the x-axis and the response variable on the y-axis, labeling the axes with units.

What is correlation?

quantifies the strength and direction of a linear relationship, ranging from -1 (strong negative) to 1 (strong positive).

What does a residual represent?

The difference between the actual response value and the predicted value from the model.

How can residuals indicate model fit?

A residual plot with no pattern suggests the linear model is a good fit; a pattern suggests it is not.

What is the Least Squares Regression Line (LSRL)?

A linear model minimizing the sum of squared residuals, ensuring the smallest possible error.

What point does the LSRL always pass through?

(xˉ,yˉ)(\bar{x}, \bar{y})(xˉ,yˉ​), where xˉ\bar{x}xˉ and yˉ\bar{y}yˉ​ are the average x and y values.

How is the slope of the LSRL interpreted?

For every unit increase in the explanatory variable, the model predicts an average increase or decrease (equal to the slope) in the response variable.

How is the y-intercept of the LSRL interpreted?

When the explanatory variable is 0, the predicted value of the response variable is the y-intercept (though it may not always be meaningful in context).

What is the coefficient of determination (R²)?

It represents the proportion of variation in the response variable explained by the explanatory variable in the model.

What does high leverage indicate in a data point?

A point far from xˉ\bar{x}xˉ (mean of x values) that can strongly affect the slope and position of the LSRL.

What impact do outliers have in regression?

Outliers can significantly affect both correlation and the regression line.

What is extrapolation, and why is it risky?

Making predictions outside the data's range, which may be unreliable as trends might not continue.

How does the residual plot help assess a model?

It highlights possible trends in residuals, making it easier to evaluate model fit.

What is the purpose of summing squared residuals in LSRL?

It minimizes the sum of squared errors to find the line of best fit.