Final answer:
Correlated observations in residual plots are indicated by consistent or structured patterns in residuals over time, with option A being correct. Positive and consistently negative residuals suggest correlations that can violate the necessary assumptions for proper statistical analysis.
Step-by-step explanation:
When examining residual plots sequentially over time for correlated observations, various patterns in the residuals can indicate potential issues with the correlation of the data. Violations of the assumption that observations are uncorrelated are indicated when you observe a structured pattern in the sequence of residuals. If the residuals display no clear pattern, that suggests the observations are independent of one another as assumed in many statistical models.
Option A, which suggests a violation is indicated when positive residuals are shown consistently over time and negative residuals are shown consistently over time, is correct. This indicates a potential problem of correlated errors. If residuals were purely random, we would expect them to fluctuate above and below the line without structure.
Residual analysis is a critical part of regression modelling, and it's important to assess for outliers and influential data points. This involves plotting the residuals and looking for points that lie more than two standard deviations from the mean. When there is a consistent sign in the residuals over time, either positive or negative, this could be indicative of a problem such as an autocorrelation or a model that doesn't adequately capture the underlying trend or structure of the data.
By using a scatter plot with lines drawn two standard deviations above and below a best-fit line, outliers can be visually identified. Data points lying outside these boundaries may be potential outliers. Alternatively, outlier detection can be conducted numerically by calculating each residual and comparing it to twice the standard deviation. Both methods aim to identify points that differ significantly from the predicted values based on the regression model.
Ultimately, the correct option that indicates violations in the sequence of residuals is A. Positive and consistently negative residuals over time suggest that observations may be correlated, violating the assumption required for certain types of statistical analysis.