To analyze Jade's situation, we need to understand how residuals work. Residuals are the differences between the observed values and the values predicted by the model.
- Positive residuals indicate that the actual value is higher than the predicted value (the model underestimates).
- Negative residuals indicate that the actual value is lower than the predicted value (the model overestimates).
In Jade's case, she found the following residuals: 2.6, 9.3, -5.489, and -6.59. The residual values are a mix of positive and negative numbers.
Evaluating the Options:
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"They accurately represent the data since half of the residuals are positive and half are negative."
- This is misleading. While having both positive and negative residuals indicates that the model has both underestimations and overestimations, it does not inherently mean the model is accurate.
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"Accurately represents the data since the residual 2.6 means that the actual point is not far from the estimate."
- Not quite. While a residual of 2.6 does indicate a small error for that particular observation, the overall accuracy of the model cannot be concluded based purely on one residual.
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"Does not accurately represent the data since half are positive and half are negative."
- Again, this does not directly imply inaccuracy. The presence of both signs suggests some errors, but doesn’t necessarily indicate that the model is overall inaccurate.
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"Does not accurately represent the data since the residual 9.3 means that the actual point is far from the estimate."
- This is a valid concern. A large residual like 9.3 indicates that for that observation, the predicted value was significantly off from the actual value. Large residuals often suggest that the model may not be fitting the data well, which implies an issue with the linear model's appropriateness for the dataset.
Conclusion:
The most appropriate conclusion that Jade can draw based on the residuals is that the linear model does not accurately represent the data since the residual 9.3 means that the actual point is far from the estimate. Thus, the fourth option stands out as the best choice.