Based on the information provided about the trend line and the correlation coefficient, Jasmine can make the following conclusions about the appropriateness of the linear model she found:
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Slope Interpretation: The slope of the trend line is -0.694, which indicates that as the independent variable \( x \) increases, the dependent variable \( y \) decreases. This negative relationship suggests that there is an inverse correlation between the two variables.
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Correlation Coefficient: The correlation coefficient (denoted as \( r \)) is -0.9153. This value is close to -1, which indicates a strong negative linear correlation between the two variables. Generally, a correlation coefficient between -0.7 and -1.0 is considered to represent a strong relationship.
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Appropriateness of the Linear Model: Given the strong negative correlation (as indicated by the correlation coefficient) and the linear trend line, Jasmine can conclude that the linear model is appropriate for the data set. The strong correlation implies that the linear equation effectively captures the relationship between the variables.
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Other Considerations: While the strong correlation indicates that the linear model is appropriate, Jasmine should also check for other aspects like:
- The residuals of the model (to ensure they are randomly distributed) to confirm that a linear model is indeed suitable.
- If there are any outliers that could be affecting the correlation.
- The context of the data to ensure that a linear relationship makes sense.
In summary, Jasmine can conclude that the linear model is appropriate based on the strong negative correlation shown by the correlation coefficient and the linearity indicated by the trend line equation.