Final answer:
Option C, indicating that the residuals are randomly dispersed across the values of xj, is the correct choice showing no violation of assumptions in residual plots. This dispersion suggests both homoscedasticity and independence which are key assumptions in regression analysis.
Step-by-step explanation:
Residual plots are a useful tool for diagnostic checking in regression analysis. These plots can help assess if a linear regression model satisfies certain assumptions. The question asks which option indicates that the assumption concerning the residuals is not violated. The correct answer is that the residuals are randomly dispersed across the values of xj (option C) which means that there is no violation of the assumptions of homoscedasticity (constant variance) and independence. The residuals should not show any discernible pattern when plotted against each predictor variable xj.
If they are randomly dispersed, it suggests that the variance of errors is constant (homoscedasticity) and that there is no clear relationship between the residuals and the predictor variable, indicating independence. To detect outliers and assess the quality of a regression model we can plot the residuals and check for points lying more than two standard deviations from the best-fit line. If outliers are identified it may be valuable to consider their impact on the regression model. A method to gauge the influence of outliers is to remove them and re-fit the model to see how the sum of the squared errors (SSE) and the correlation coefficient change.