For a regression equation with a positive slope, if a certain X value is above the mean for the X scores, then the predicted Y value will be above the mean for the Y scores.
since slope is ∆y/∆x, that would have to be true.
To understand why a regression equation with a positive slope behaves in this way, let's break it down step by step.
1. Regression equation: In a regression analysis, we attempt to model the relationship between a dependent variable (Y) and one or more independent variables (X). The regression equation provides a mathematical representation of this relationship.
2. Positive slope: A positive slope means that as the X variable increases, the Y variable also increases. This indicates a positive correlation between X and Y, implying that as the X values increase, the predicted Y values will also increase.
3. X value above the mean: When a certain X value is above the mean for the X scores, it means that this particular X value is greater than the average value of all the X values. This suggests that the X value is relatively large compared to others in the dataset, indicating the presence of a bigger or higher value of X.
4. Predicted Y value: Based on the regression equation, when we have a certain X value, we can use the equation to predict the corresponding Y value. The predicted Y value represents the expected value of Y based on the given X value.
5. Above mean for Y scores: If the X value is above the mean for the X scores and the regression equation has a positive slope, then the predicted Y value will also be above the mean for the Y scores. This is because the positive slope indicates that as X increases, Y also increases. Therefore, a larger-than-average X value will result in a larger-than-average predicted Y value.
In summary, for a regression equation with a positive slope, if a certain X value is above the mean for the X scores, the predicted Y value will be above the mean for the Y scores due to the positive relationship between X and Y.