Final answer:
The key assumption of no correlation between error terms and predictors in linear regression fails when important predictor variables are omitted, leading to omitted variable bias and potentially misleading conclusions. Outliers and influential points can especially affect the correlation coefficient and the slope of the regression line, necessitating careful analysis. Option B is the correct answer.
Step-by-step explanation:
The assumption that the error term is not correlated with the predictor variables in a linear regression model generally breaks down when important predictor variables are excluded from the model. Excluding relevant predictors can lead to omitted variable bias, where the estimations of the other coefficients become biased and inconsistent due to their correlation with the excluded variable. This can distort the standard errors and affect the results of hypothesis testing, potentially providing misleading conclusions about the relationships between variables.
Identifying outliers and influential points is also key in linear regression analysis. Outliers can be identified by measuring the distance from the best-fit line, typically considering any point farther than two standard deviations above or below as an outlier. Influential points significantly impact the slope and the correlation coefficient, r, making their identification crucial. Computer software can assist in regression diagnostics to understand their effects.
Moreover, understanding the correlation coefficient and its significance helps in determining the strength of the linear relationship. In the context of linear regression, we assess whether there is a significant linear relationship that justifies the use of linear regression and the calculation of the correlation coefficient. We cannot say one variable causes the other based solely on correlation.
The correct option, in this case, is B. When important predictor variables are excluded.