TRUE OR FALSE. The Pearson correlation between X1 and Y is r = 0.50. When a second variable, X2, is added to the regression equation, we obtain R2 = 0.60. Adding the second variable increases the variability that is predicted by the regression equation for the Y scores by 36% - 25% = 11%.
True
The statement is false. Let's break it down and explain why.
The information given states that the Pearson correlation coefficient (r) between X1 and Y is 0.50. This indicates a moderate positive linear relationship between X1 and Y.
When a second variable, X2, is added to the regression equation, we are told that we obtain an R-squared value (R2) of 0.60. R2 represents the proportion of the variance in the dependent variable (Y) that can be explained by the independent variables (X1 and X2) in the regression model. An R2 value of 0.60 means that 60% of the variability in Y can be accounted for by the combined effects of X1 and X2.
Now let's analyze the claim that adding the second variable increases the variability predicted by the regression equation for the Y scores by 36% - 25% = 11%. This claim is incorrect.
To understand why, we need to consider the relationship between the Pearson correlation coefficient (r) and R-squared (R2). The squared value of the Pearson correlation coefficient (r) is equal to R-squared (R2). In this case, the squared value of 0.50 is 0.25. Therefore, the original R-squared value for the relationship between X1 and Y is 0.25 or 25%.
When the second variable, X2, is added to the regression equation, the R-squared value increases to 0.60 or 60%. Therefore, the increase in R-squared is 60% - 25% = 35%. This means that adding X2 accounts for an additional 35% of the variability in Y beyond what can be explained by X1 alone.
So, the correct statement would be that adding the second variable increases the variability predicted by the regression equation for the Y scores by 35%, not 11% as claimed.