Question

How does regression analysis decide which shape (plane or parabola) serves as the best decision rule to infer aggression from girth and ear wiggling?

1) By analyzing the relationship between girth and ear wiggling
2) By comparing the coefficients of the girth and ear wiggling variables
3) By fitting both a plane and a parabola to the data and selecting the one with the better fit
4) By using statistical tests to determine the significance of the relationship between girth and ear wiggling

Answer 1

Regression analysis uses statistical tests, including the correlation coefficient and significance tests, to determine if a linear or non-linear model best fits the relationship between two variables like girth and ear wiggling, based on the strength and significance of the relationship.

Step-by-step explanation:

Regression analysis decides the shape of the decision rule, whether it be a plane or a parabola, by using a combination of graphical analysis and statistical tests such as the calculation of the correlation coefficient and significance testing. When studying the relationship between two variables, like girth and ear wiggling in this case, the first step is often to create a scatter plot to visually assess the relationship. If the data points roughly align along a straight trajectory, a linear model may be suitable; conversely, if they form a curved pattern, a quadratic or higher-order polynomial model may be more appropriate.

Once the scatter plot is made, the next step is to calculate the least-squares regression line or curve, which is the line (or curve) that minimizes the sum of the squared residuals (the distances from the actual data points to the line). This provides a mathematical model for the relationship. To assess if the line is indeed the best-fit model, the correlation coefficient, 'r', is used to quantify the strength and direction of a linear relationship between two variables.

A high correlation coefficient (close to 1 or -1) indicates a strong linear relationship and the line as a good model, while a low correlation coefficient (close to 0) suggests a weak linear relationship. The significance of this correlation coefficient is then tested, often with a t-test, to determine if the observed relationship is statistically meaningful and not a result of random variation in the sample data. The final decision on the best model to use is based on these statistical assessments, which guide whether a plane (linear relationship) or a parabola (non-linear relationship) best represents the data.