Final answer:
The assumption that the error term is not correlated with predictor variables in linear regression can break down when important predictors are omitted or when outliers and influential points are present, potentially distorting the model.
Step-by-step explanation:
The assumption that the error term is not correlated with the predictor variables is foundational in linear regression analysis. This assumption can break down notably when important predictor variables are excluded from the model. When there is a missing variable that correlates with both the dependent variable and one of the independent variables, the omitted variable can induce an apparent correlation between the error term and the included predictor, violating our assumption. Another situation where this assumption can break down is when outliers or influential points are present in the data set. Outliers can have a disproportionate effect on the slope of the regression line, and if they are not appropriately accounted for, they may lead to an error term that is correlated with predictor variables.
Moreover, influential points can change the slope of the regression line significantly when they are added or removed, challenging the robustness of the model. It is essential to identify and investigate potential outliers or influential points as part of the regression diagnostic process. If removing an outlier affects the correlation between variables or the fit of the line substantially it is an indication that the assumptions underlying the linear regression model might not hold well. Here, advanced tools like regression analysis and scatter plots can aid in detecting such points and assessing their impact.