Linear Regression Model

The equation for the linear regression model is: $y = -4105.955x + 71397.727$ where:

• Slope ($m$) = -4105.955
• Intercept ($b$) = 71397.727

Goodness of Fit

The $R^2$ value (coefficient of determination) is approximately 0.989. This high $R^2$ value (close to 1) suggests that the linear model explains about 98.9% of the variability in sales over time, indicating a good fit.

Residual Plot Analysis

In the residual plot above, residuals (differences between actual and predicted values) are scattered around the horizontal line at zero but do not form a clear random pattern. Some of the residuals are positive, and some are negative, but they do not show a strong trend, which suggests the model captures most of the linear trend in the data. However, the lack of a random pattern could hint at slight non-linear behavior.

Conclusion

While the linear model is a good fit based on $R^2$, there may still be minor deviations that a more complex model could capture. However, for simplicity, the linear model is appropriate for this data set.

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Further Questions

1. How do we interpret the slope in the context of sales decline?
2. What does an $R^2$ value close to 1 imply in terms of model accuracy?
3. How would a non-linear model improve this fit, if at all?
4. How could we test the linear regression's assumptions in this scenario?
5. How might outliers impact the residual plot?

Tip:

A residual plot is essential to check for randomness; a non-random pattern may suggest a more suitable model than linear regression.