Question

arthur and tavon each calculated lines of best fit for their last five math tests. based on the residual plots, arthur states that his scores are more closely aligned to a linear model than tavon’s scores. make a mathematical argument to support or refute arthur’s claim.

arthur’s residual plot

tavon’s residual plot

____ model is the better of the two models. the points in his residual plot are generally __ the __ and are fairly evenly distributed above and below the x - axis, and the residual plot as a whole ____ a clear pattern.

Explanation:

Step1: Recall Residual Plot Concepts

A residual plot shows the difference between observed and predicted values (residuals) vs. the independent variable. For a good linear model, residuals should be randomly scattered around the x - axis (y = 0), with no clear pattern, and be close to the x - axis (small residual values).

Step2: Analyze Arthur's Residual Plot

Looking at Arthur's residual plot, the points are not close to the x - axis (some are far, like the one at y = 5) and seem to have a pattern (increasing as x increases).

Step3: Analyze Tavon's Residual Plot

Tavon's residual plot has points that are closer to the x - axis and seem to be more randomly distributed above and below the x - axis, with no strong pattern.

Step4: Evaluate the Claim

Since a good linear model has residuals close to the x - axis, randomly distributed, and no pattern, Tavon's model is better. Arthur's residual plot has points not close to the x - axis and a pattern, so his claim is refuted. The correct blanks: Tavon's model is the better of the two models. The points in his residual plot are generally closer to the x - axis and are fairly evenly distributed above and below the x - axis, and the residual plot as a whole does not show a clear pattern. Arthur's plot has points not close to the x - axis and a pattern, so his claim is wrong.

Answer:

To support or refute Arthur’s claim, we analyze residual plots:

• A good linear model has residuals randomly scattered near the \( x \) - axis (small residuals) with no clear pattern.
• Arthur’s residual plot: Points are not close to the \( x \) - axis (e.g., a point at \( y = 5 \)) and show a pattern (increasing with \( x \)).
• Tavon’s residual plot: Points are closer to the \( x \) - axis and randomly distributed (no strong pattern).

Thus, Tavon’s model is better. Arthur’s claim is refuted because his residuals are not close to the \( x \) - axis and show a pattern, while Tavon’s residuals are closer to the \( x \) - axis and lack a clear pattern.

For the blanks:

1. Tavon’s
2. closer to
3. \( x \) - axis
4. does not show