In the context of residuals analysis, what does the independent observation assumption imply when examining a residuals plot?

• A. The residuals should form a funnel shape, indicating increasing variance.
• B. The residuals should all be close to $0$, indicating a perfect fit.
• C. The residuals should increase or decrease systematically as the fitted values increase.
• D. The residuals should display no discernible pattern or correlation with each other across the plot.

Answer: (D)

Verified step by step guidance

1. Understand that the independent observation assumption means each residual (error term) is independent of the others, with no relationship or correlation between them.
2. When examining a residuals plot, this assumption implies that residuals should be randomly scattered without any systematic pattern, trend, or clustering.
3. Recognize that if residuals show a pattern (such as increasing or decreasing trends, or a funnel shape), it suggests violations of independence or other assumptions like homoscedasticity.
4. Therefore, the residuals plot should not show residuals increasing or decreasing systematically, nor forming shapes like funnels, but rather a random scatter around zero.
5. Conclude that the correct interpretation of the independent observation assumption in residuals analysis is that residuals display no discernible pattern or correlation with each other across the plot.