Jazmine's graphing calculator provided a trend line represented by the equation \(y = -0.69x + 17.22\) and a correlation coefficient of \(-0.9153\). Here are the conclusions Jazmine can make based on this information:
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Negative Correlation: The correlation coefficient (\(r = -0.9153\)) indicates a strong negative correlation between the variables in the dataset. This means that as one variable increases, the other variable tends to decrease.
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Strength of the Linear Model: The value of the correlation coefficient ranges from -1 to +1. A value close to -1 indicates a very strong negative linear relationship. Since \(-0.9153\) is significantly close to \(-1\), it suggests that the linear model is a good fit for the data.
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Appropriateness of Linear Model: Given the strong negative correlation and the linear equation, Jazmine can conclude that the linear model is appropriate for her data. The trend line effectively captures the relationship between the variables.
However, she should also consider:
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Data Points: If the data points do not visibly follow a linear trend (for example, if they show a nonlinear pattern or contain outliers), it could affect the conclusion. Analyzing the scatter plot would help confirm the appropriateness of the linear model visually.
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Residual Analysis: Jazmine could perform a residual analysis to check for any patterns in the residuals. If residuals are randomly dispersed, it supports the model's appropriateness. If there are patterns, it may indicate that a nonlinear model might be more suitable.
In summary, Jazmine can conclude that her linear model is appropriate based on the strong negative correlation, but she should also verify the model by checking the scatter plot and residuals.