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Sagot :
Certainly! Let's work through this step by step.
### Part A: Plotting the Residuals
Residuals are the differences between the observed values and the values predicted by a linear regression model. To plot the residuals:
1. Create a scatter plot: Each pair [tex]\((x_i, \text{Residual}_i)\)[/tex] will be a point on the plot.
2. Draw a horizontal line at [tex]\( y = 0 \)[/tex]: This reference line will help us evaluate how the residuals are distributed around zero.
Here are the `x` values and their corresponding residuals:
- [tex]\( x \)[/tex]: 1, 2, 3, 4, 5, 5, 6, 7, 8, 9
- Residuals: -2, 0, 0, 1, -3, 3, -4, -4, 4, 5
#### Steps to plot:
1. Draw the horizontal axis (x-axis) and the vertical axis (y-axis) on the plot.
2. Mark the x-values on the horizontal axis and mark residual values on the vertical axis.
3. Plot each residual as a dot at the corresponding x-coordinate.
4. Add a horizontal line at [tex]\( y = 0 \)[/tex] to serve as a reference line.
The resulting plot should look like this:
```
Residuals
5|
4|
3|
2|
1|
0|
-1|
-2|
-3|
-4|
-5|
+----+----+----+----+----+----+
1 2 3 4 5 6 7 8 9 x
```
### Part B: Assessing the Model
To assess how well the linear regression model fits the data, we need to visually inspect the residuals plot.
1. Random Distribution: A good linear regression model should have residuals scattered randomly around the horizontal line at [tex]\( y = 0 \)[/tex]. There should be no clear pattern or systematic structure.
2. Patterns or Trends: If the residuals show a pattern, like a systematic increase or decrease, this suggests the model might not be capturing some aspect of the underlying relationship in the data. Common patterns to look for include:
- Residuals increasing or decreasing with [tex]\( x \)[/tex]: Could indicate a non-linear relationship.
- Clustering: Residuals bunch up at certain [tex]\( x \)[/tex] values.
- Cyclic patterns: Regular oscillation in residuals might indicate missing variables or influences not captured by the model.
Analysis of the given residuals plot:
- The residuals for [tex]\( x = 1 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 3 \)[/tex] are pretty close to 0, suggesting the model fits well for these points.
- For [tex]\( x = 4 \)[/tex], the residual is 1, which is still close to 0.
- For [tex]\( x = 5 \)[/tex], the residuals are -3 and 3. This indicates a pair of points with significant differences from the predicted values.
- For [tex]\( x = 6 \)[/tex], the residual is -4 and it's quite far from 0.
- For [tex]\( x = 7 \)[/tex], the residual is still -4, staying far and unchanged.
- For [tex]\( x = 8 \)[/tex], the residual jumps to 4, which is quite significant.
- For [tex]\( x = 9 \)[/tex], the residual is 5, showing a substantial deviation as well.
Conclusion:
The residuals do not appear to be randomly distributed. There is a visible pattern indicating that the residuals increase and decrease systematically. For [tex]\( x \geq 5 \)[/tex], the residuals show larger deviations compared to those for smaller values of [tex]\( x \)[/tex].
This suggests that the linear regression model may not be the best fit for this data, as it appears not to capture the relationship sufficiently for higher values of [tex]\( x \)[/tex]. There may be a non-linear relationship or other factors not accounted for by the model.
### Part A: Plotting the Residuals
Residuals are the differences between the observed values and the values predicted by a linear regression model. To plot the residuals:
1. Create a scatter plot: Each pair [tex]\((x_i, \text{Residual}_i)\)[/tex] will be a point on the plot.
2. Draw a horizontal line at [tex]\( y = 0 \)[/tex]: This reference line will help us evaluate how the residuals are distributed around zero.
Here are the `x` values and their corresponding residuals:
- [tex]\( x \)[/tex]: 1, 2, 3, 4, 5, 5, 6, 7, 8, 9
- Residuals: -2, 0, 0, 1, -3, 3, -4, -4, 4, 5
#### Steps to plot:
1. Draw the horizontal axis (x-axis) and the vertical axis (y-axis) on the plot.
2. Mark the x-values on the horizontal axis and mark residual values on the vertical axis.
3. Plot each residual as a dot at the corresponding x-coordinate.
4. Add a horizontal line at [tex]\( y = 0 \)[/tex] to serve as a reference line.
The resulting plot should look like this:
```
Residuals
5|
4|
3|
2|
1|
0|
-1|
-2|
-3|
-4|
-5|
+----+----+----+----+----+----+
1 2 3 4 5 6 7 8 9 x
```
### Part B: Assessing the Model
To assess how well the linear regression model fits the data, we need to visually inspect the residuals plot.
1. Random Distribution: A good linear regression model should have residuals scattered randomly around the horizontal line at [tex]\( y = 0 \)[/tex]. There should be no clear pattern or systematic structure.
2. Patterns or Trends: If the residuals show a pattern, like a systematic increase or decrease, this suggests the model might not be capturing some aspect of the underlying relationship in the data. Common patterns to look for include:
- Residuals increasing or decreasing with [tex]\( x \)[/tex]: Could indicate a non-linear relationship.
- Clustering: Residuals bunch up at certain [tex]\( x \)[/tex] values.
- Cyclic patterns: Regular oscillation in residuals might indicate missing variables or influences not captured by the model.
Analysis of the given residuals plot:
- The residuals for [tex]\( x = 1 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 3 \)[/tex] are pretty close to 0, suggesting the model fits well for these points.
- For [tex]\( x = 4 \)[/tex], the residual is 1, which is still close to 0.
- For [tex]\( x = 5 \)[/tex], the residuals are -3 and 3. This indicates a pair of points with significant differences from the predicted values.
- For [tex]\( x = 6 \)[/tex], the residual is -4 and it's quite far from 0.
- For [tex]\( x = 7 \)[/tex], the residual is still -4, staying far and unchanged.
- For [tex]\( x = 8 \)[/tex], the residual jumps to 4, which is quite significant.
- For [tex]\( x = 9 \)[/tex], the residual is 5, showing a substantial deviation as well.
Conclusion:
The residuals do not appear to be randomly distributed. There is a visible pattern indicating that the residuals increase and decrease systematically. For [tex]\( x \geq 5 \)[/tex], the residuals show larger deviations compared to those for smaller values of [tex]\( x \)[/tex].
This suggests that the linear regression model may not be the best fit for this data, as it appears not to capture the relationship sufficiently for higher values of [tex]\( x \)[/tex]. There may be a non-linear relationship or other factors not accounted for by the model.
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