The linear regression model for the data is:

$y = -613.582x + 8482.945$

where:

• Slope ($m$): -613.582
• Intercept ($b$): 8482.945
• $R^2$ (Coefficient of Determination): 0.976

Analysis of Model Fit

The high $R^2$ value of 0.976 suggests that the linear model explains about 97.6% of the variance in sales, indicating a strong fit to the data. However, the residual plot shows some variability, with certain points deviating from the zero line. Despite this, there is no obvious non-linear pattern in the residuals, suggesting that the linear model is still a reasonable approximation for the trend in sales over time.

Conclusion: The model is a good fit for this data, mainly supported by the high $R^2$ value, although some deviations in the residuals suggest that minor non-linear factors might exist.

Would you like further details on any part of this analysis?

Here are some related questions:

1. How would the model change if additional weeks were added to the data?
2. What does the slope indicate about the trend in sales?
3. How could we interpret the intercept value in this context?
4. How could non-linear regression be tested if linearity is questionable?
5. What impact does a low $R^2$ value have on model reliability?

Tip: Always examine both $R^2$ and the residual plot to evaluate model fit. While $R^2$ gives a measure of variance explained, the residual plot helps detect patterns that may indicate a need for a different model.