The graphs provided show residual plots for cubic, quadratic, and linear regression models based on the same data set. Residual plots are analyzed to determine which regression model fits the data best. Here's a step-by-step analysis:

Residual Analysis:

1. Cubic Residual Plot: The residuals in this plot appear to scatter randomly around zero without a clear pattern. This indicates that the cubic model may provide a good fit for the data.

2. Quadratic Residual Plot: The residuals show a noticeable pattern, curving upwards and downwards. This suggests that the quadratic model does not fully capture the data's variability.

3. Linear Residual Plot: The residuals in this plot display a distinct pattern, oscillating widely and systematically above and below zero. This indicates that the linear model is not a good fit for the data.

Conclusion:

The cubic model is the best option because its residuals show the least pattern and are randomly distributed around zero. This suggests the cubic regression captures the data's relationship more effectively than the other models.

Answer:

The correct choice is: A. The cubic model should be used.

Let me know if you need further details or clarifications!

Related Questions:

1. What are residuals, and why are they important in regression analysis?
2. How do you interpret patterns in residual plots for assessing model fit?
3. What are the differences between linear, quadratic, and cubic regression models?
4. When should we use higher-order regression models over simpler ones?
5. How can overfitting occur with higher-order regression models, and how is it identified?

Tip:

Always check for randomness in residual plots to assess if a regression model is appropriate for your data. Non-random patterns indicate a poor fit.