Final answer:
The statement in question is false; the r² value in a linear regression model cannot decrease by adding more predictor variables, as it represents the proportion of variance explained which can only increase or remain the same with additional variables.
Step-by-step explanation:
The statement 'R2 can decrease as we add more predictor variables to the linear regression model' is false. In linear regression, the coefficient of determination, r², is a key metric that measures the proportion of variance in the dependent variable that can be predicted from the independent variable(s). The value of r² lies between 0 and 1, and it can only stay the same or increase as additional predictor variables are added to the model. This is because each new variable has the potential to explain additional variance in the dependent variable, thus potentially increasing r². However, it’s important to note that adding more variables to the model can lead to overfitting, which means the model might not generalize well to new, unseen data.
Additionally, when discussing the relationship between two variables, the correlation coefficient, r, is used to measure the strength and direction of a linear relationship between them. A strong negative linear relationship between the X and Y variables indicates that they could be good candidates for analysis with linear regression. It is worth noting that the sign of r reflects the direction of the association (positive or negative), but not the strength. The r value must be between -1 and +1. When expressed as a percent, r² represents the percentage of variance in the dependent variable explained by the regression line.
It's important to remember that a strong correlation, whether positive or negative, does not imply causation between the variables.