Use the graphing calculator tool to plot the residual points.

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

Does the residual plot show that the line of best fit is appropriate for the data?

Yes, the points have no pattern.
Yes, the points are evenly distributed about the x-axis.
No, the points are in a linear pattern.
No, the points are in a curved pattern.

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.

Explore the analysis of residuals in linear regression to determine if the line of best fit is suitable for given data points. Includes a detailed examination of residual points (1, 0.86), (2, –0.25), (3, –1.66), (4, –2.34), (5, –4.1), and their implications. Suitable for advanced high school students.

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