Sure! Let's address the problem step by step.
### Part A: Plot the Residuals
We are given the data points and their corresponding residuals. Here is how we can plot the residuals on a set of axes:
1. List the given values:
- [tex]\( x \)[/tex] values: [tex]\( [1, 2, 3, 4, 5, 5, 6, 7, 8, 9] \)[/tex]
- Residuals: [tex]\( [-2, 0, 0, 1, -3, 3, -4, -4, 4, 5] \)[/tex]
2. Create a plot:
- On the x-axis, mark the values from the [tex]\( x \)[/tex] list.
- On the y-axis, mark the corresponding residual values.
- Use dots or points to plot the residuals and a horizontal line at [tex]\( y = 0 \)[/tex] to denote the baseline.
```
^
|
5 | .
4 | .
3 | .
2 |
1 | .
| .--- .
-1 |
-2 | .
-3 | .
-4 | . .
|
+------------------------------------>
1 2 3 4 5 6 7 8 9
```
### Part B: Assessing the Linear Regression Model
To assess the linear regression model's fit using the residual plot, we need to look for patterns among the residuals. In a good linear regression model, the residuals should be randomly dispersed around the horizontal axis (y=0) without any obvious patterns.
Here is an analysis based on the residuals provided:
1. Pattern Observation:
- Residuals at [tex]\( x = 1, 6, 7 \)[/tex] are below the horizontal axis, indicating that the model underestimates the actual data values at these points.
- Residuals at [tex]\( x = 4, 8, 9 \)[/tex] are above the horizontal axis, which indicates that the model overestimates these data values.
- Notably, residuals appear to form distinctive clusters below and above the axis around specific [tex]\( x \)[/tex] values. For example, at [tex]\( x = 5 \)[/tex] there is both an overestimation and underestimation (residuals of 3 and -3).
2. Conclusion:
- The residuals are not completely random and show certain trends, including patterns and clusters.
- This non-randomness and the patterns observed indicate that the linear regression model may not be the best fit for the data. A better model might capture the relationship more accurately, suggesting that the relationship between other variables and [tex]\( x \)[/tex] might be non-linear or might need more complex modeling techniques.
The analysis concludes that the provided residual plot suggests that while a linear model has been tried, it does not fit the data well because the residuals display clear patterns rather than random scatter.
