To determine if the line of best fit is appropriate for the given data points using a residual plot, we need to examine the pattern formed by the residuals. A residual plot is a scatter plot where the x-values are the independent variable values (in this case, 1, 2, 3, 4, 5), and the y-values are the residuals (in this case, 0.86, -0.25, -1.66, -2.34, -4.1).

Plotting the Residual Points

Here are the residual points given:

• (1, 0.86)
• (2, -0.25)
• (3, -1.66)
• (4, -2.34)
• (5, -4.1)

Analysis of the Residual Plot

Now, let's analyze the distribution of these points:

1. Pattern of the Points: As we examine the y-values of the residuals, we notice that they are decreasing in a non-linear fashion, suggesting that there is a potential curve in the distribution of residuals. This indicates that the residuals are not randomly scattered around the x-axis.

2. Even Distribution: The points do not appear to be evenly distributed about the x-axis; instead, they show a trend where the residuals become more negative as the x-value increases.

3. Curved Pattern: The residuals seem to follow a downward curve, showing that the error between the data points and the line of best fit increases with larger x-values. This suggests a curved pattern in the residuals, indicating that the model may not be fitting the data well.

Conclusion

Based on the observations from the residual plot:

• No, the points are in a curved pattern.

This curved pattern in the residuals implies that a linear model is not the best fit for the data. Instead, a different model (such as a quadratic or higher-degree polynomial model) might better capture the relationship between the variables.

Follow-up Questions:

1. How do you calculate the residuals for a set of data points?
2. What are the characteristics of a good residual plot for a linear regression model?
3. How can you use a residual plot to determine the appropriateness of a non-linear model?
4. What are other methods, besides residual plots, to evaluate the goodness of fit for a regression model?
5. How does increasing the degree of a polynomial regression affect the residual plot?

Tip:

Always check for patterns in the residual plot after fitting a model. Patterns like curves or systematic deviations suggest the need for a more complex model to better capture the underlying data structure.