Jes used GeoGebra to find a linear model (trend line) of the data collected for their science project. They calculated the following residuals for their data 0.26, 0.93, 0.5489, and 0.659. What conclusion can Jes make about the appropriateness of the linear model based on the residuals?(1 point)

Question

Responses

The linear model does not accurately represent the data since all residuals are positive.
The linear model does not accurately represent the data since all residuals are positive.

The linear model accurately represents the data since all residuals are less than 1.
The linear model accurately represents the data since all residuals are less than 1.

The linear model does not accurately represent the data since all data points are less than 1.
The linear model does not accurately represent the data since all data points are less than 1.

The linear model accurately represents the data since all residuals are positive.

Answer 1

To evaluate the appropriateness of the linear model based on the given residuals, we first need to understand what residuals are. Residuals are the differences between the observed values and the values predicted by the linear model. They help us assess how well the model fits the data.

The residuals provided are:

• 0.26
• 0.93
• 0.5489
• 0.659

Analysis of Residuals:

1. All residuals are positive: This indicates that the linear model tends to underestimate the observed values. However, just having all positive residuals doesn't automatically mean that the model is invalid; it suggests that it may not fit the data perfectly.

2. Magnitude of residuals: All residuals are less than 1, suggesting that the predictions from the model are relatively close to the actual values. However, this alone does not confirm the accuracy of the linear model; it may simply indicate that the errors are small but consistent.

Conclusion Options:

1. The linear model does not accurately represent the data since all residuals are positive.

   • This could be partially true because all positive residuals indicate underestimation, but it does not definitively mean the model is inaccurate.

2. The linear model accurately represents the data since all residuals are less than 1.

   • This response has some merit, as smaller residuals indicate a closer fit, but it is misleading to say the model is "accurate" based purely on this.

3. The linear model does not accurately represent the data since all data points are less than 1.

   • This statement is incorrect as it evaluates the data points and not the residuals.

4. The linear model accurately represents the data since all residuals are positive.

   • This is not accurate; being positive indicates a consistent underestimation rather than accuracy.

Best Conclusion:

Given the nature of residuals, the best conclusion Jes can make is likely:

The linear model does not accurately represent the data since all residuals are positive.

This captures the essence that while the residuals are relatively small, their being consistently positive indicates that the model underestimates the actual values, suggesting that improvements may be necessary.