https://file.aiquickdraw.com/mathbot/file/1727104058032gwtowh5y.jpg

Give the equation for this new line and compare it with the line you found for the whole dataset, commenting on the effect of the outlier.

The equation of the best-fit line for the data points is:

\[
y = -0.44x + 1086.25
\]

This equation suggests a downward trend over time, where the value decreases by approximately 0.44 units for every year increase. The \( R^2 \) value, which is the coefficient of determination, is approximately 0.83. This indicates that 83% of the variation in the values is explained by the linear relationship with time.

### Comparison to a Line Without Outliers:
If we include an outlier (like the high value around 1885), the slope would likely become steeper, pulling the line upwards at the beginning, reducing the fit for more recent points. This would lower the \( R^2 \) value, making the line less representative of the overall data trend. By excluding the outlier, the slope is more moderate, and the fit better captures the long-term gradual decline.

Would you like further details or clarifications?

Here are 5 questions that can expand on this analysis:
1. How do you identify an outlier mathematically in a dataset?
2. How does the \( R^2 \) value influence the interpretation of linear models?
3. What are alternative models if the relationship is not linear?
4. Can outliers ever be informative in a dataset, rather than detrimental?
5. How do you calculate residuals for this linear regression?

**Tip:** Always visualize your data before performing regression to spot any outliers or trends that may not fit a linear model.

Understand the process of finding the best-fit line equation for a dataset, including the impact of outliers on the slope and the R² value. This analysis highlights key concepts in linear regression and data interpretation, suitable for students in Grades 9-12.

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