Final answer:
To show that residuals are uncorrelated with the predictor variable, one must demonstrate that the residuals, calculated as the difference between observed and predicted values, are randomly distributed about the regression line and not systematically related to the independent variable.
Step-by-step explanation:
To show that the residuals are uncorrelated with the predictor variable, one has to demonstrate that the residuals do not share a systematic relationship with the independent variable. Here's an approach to identify and handle outliers potentially affecting the correlation in a regression analysis:
- Scatter plot: Visualize the data to identify potential outliers.
- Best-fit line: Calculate the least-squares regression line, typically represented as ý = a + bx, where 'a' is the y-intercept and 'b' is the slope.
- Calculate the residuals: The difference between observed and predicted values (y - ý).
- Identify outliers: Compare each residual to twice the standard deviation, flagging data points with residuals beyond this range as potential outliers.
- Analyze the impact of removing outliers by recalculating the best-fit line and correlation coefficient, which should ideally result in a stronger correlation and smaller SSE (sum of the squared errors).
After this process, the residuals should ideally be random and centered around zero across the range of predictor values, indicating no correlation between residuals and predictors.