Question

The following is the R output for 3 different multiple regression models. In each model, we are predicting the weight of the bear from the predictor variables. Interpret the output and make a recommendation based on the model results.

Answer 1

The slope of a regression line indicates the change in the dependent variable for a unit change in the independent variable, while the y-intercept predicts the dependent variable value when independents are zero. The coefficient of determination explains the percentage of variability of the dependent variable explained by the independent variable. Outliers with large residuals can significantly affect model predictions, suggesting they may be influential points.

Step-by-step explanation:

The slope in a regression analysis represents the estimated change in the dependent variable for a one-unit change in the independent variable, assuming all other variables in the model are held constant. The y-intercept is the predicted value of the dependent variable when all independent variables are equal to zero. For example, in the third exam/final exam example, the regression equation is: ý = -173.51 + 4.83x. To calculate the predicted weight for someone 68 inches tall, we substitute x=68 into the equation, yielding a predicted weight of ý = -173.51 + (4.83 × 68).

The correlation coefficient represented by 'r' tells us the strength and direction of the linear relationship between two variables. The coefficient of determination, r², represents the proportion of variability in the dependent variable that is explained by the independent variable using the regression line. For instance, with a correlation coefficient of -0.56, the coefficient of determination is (-0.56)² = 0.3136, meaning 31.36% of the variability in fuel efficiency can be explained by body weight.

To interpret the coefficient of determination in context, if r² = .4397, approximately 44 percent of the variation in final exam grades can be explained by the variation in the third exam grades using the regression line. An outlier with a large residual may significantly change the slope and correlation, indicating its potential to be an influential point, which, if erroneously recorded, should be considered for deletion to improve the model quality.