To evaluate the appropriateness of the linear model given the trend line equation \( y = -0.69x + 17.22 \) and the correlation coefficient \( r = -0.9153 \), let's consider the following points:
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Correlation Coefficient:
- The correlation coefficient \( r \) ranges from -1 to 1. A value closer to -1 indicates a strong negative linear relationship between the variables, while a value closer to 1 indicates a strong positive linear relationship.
- In your case, since \( r = -0.9153 \), it indicates a strong negative correlation between the variables. This suggests that as \( x \) increases, \( y \) tends to decrease.
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Evaluating the Linear Fit:
- The trend line provides a linear equation that attempts to model the relationship between \( x \) and \( y \).
- You have some specific data points: (1, 16.88), (2, 15.27), and (4, 14.59). Let's check how these points fit into the linear model by calculating the predicted values at these \( x \) values.
For each \( x \) point:
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For \( x = 1 \): \[ y_{\text{pred}} = -0.69(1) + 17.22 = 16.53 \] Actual point is \( (1, 16.88) \), Predicted is \( 16.53 \) (Close)
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For \( x = 2 \): \[ y_{\text{pred}} = -0.69(2) + 17.22 = 15.84 \] Actual point is \( (2, 15.27) \), Predicted is \( 15.84 \) (Close)
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For \( x = 4 \): \[ y_{\text{pred}} = -0.69(4) + 17.22 = 14.86 \] Actual point is \( (4, 14.59) \), Predicted is \( 14.86 \) (Close)
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Conclusion:
- Given the strong negative correlation coefficient and the fact that the actual data points are relatively close to the predicted values from the trend line, it suggests that the linear model is appropriate for your data.
- The model effectively captures the overall trend of the data, and the predictions are fairly accurate given the data points presented.
In conclusion, the linear model appears to be a suitable representation of the relationship between \( x \) and \( y \) for the data you have provided.