Model Analysis

The linear regression model obtained is:

$y = 34115.682 \, x - 114126.455$

where:

• Slope $m = 34115.682$
• Intercept $b = -114126.455$
• $R^2 \approx 0.659$, indicating that about 65.9% of the variance in sales is explained by this linear model.

Fit Evaluation

The residual plot shows a clear pattern rather than random scatter, with residuals increasing substantially as weeks progress. This pattern indicates that the linear model is not a good fit for this data, as it does not capture the accelerating growth in sales over time. A linear model may not be suitable for data that appears to grow at an increasing rate, which suggests that a nonlinear model (such as exponential or quadratic) might better capture the underlying trend.

Conclusion

The linear model does not adequately fit the data due to the structured residuals and relatively low $R^2$ value. A different regression model, perhaps exponential or polynomial, should be considered.

Would you like further details on the regression calculations or exploring a nonlinear model?

Related Questions:

1. How do we interpret the $R^2$ value in regression analysis?
2. What other types of regression models could be used for data with accelerating growth?
3. How can we determine if a nonlinear model is a better fit for this data?
4. What is the significance of the residual plot in assessing model fit?
5. How would an exponential growth model be applied to this dataset?

Tip:

Always examine both the $R^2$ value and residual plot to determine the fit quality; patterns in residuals often signal the need for a different model type.