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Sagot :
To determine which statements are true about the given table, let's analyze each statement one by one based on the provided data points, predicted values, and residuals.
### Given Data
1. [tex]\( x = 1 \)[/tex] : Given = -5.1, Predicted = -5.14, Residual = 0.04
2. [tex]\( x = 2 \)[/tex] : Given = -1.3, Predicted = -1.43, Residual = -0.13
3. [tex]\( x = 3 \)[/tex] : Given = 1.9, Predicted = 2.28, Residual = -0.38
4. [tex]\( x = 4 \)[/tex] : Given = 6.2, Predicted = 5.99, Residual = 0.21
### Statements Analysis
#### Statement 1:
The data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
- To determine this, consider the residual value for [tex]\( x = 1 \)[/tex].
- Residual [tex]\( = \text{Given} - \text{Predicted} = -5.1 - (-5.14) = -5.1 + 5.14 = 0.04 \)[/tex]
- Since the residual is positive (0.04), the data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
Conclusion: True
#### Statement 2:
The residual value for [tex]\( x = 3 \)[/tex] should be a positive number because the data point is above the line of best fit.
- Residual for [tex]\( x = 3 \)[/tex] is given as -0.38.
- A positive residual would imply that the data point is above the line of best fit, but here the residual is negative (-0.38), indicating the point is below the line.
Conclusion: False
#### Statement 3:
Fiona made a subtraction error when she computed the residual value for [tex]\( x = 4 \)[/tex].
- Let's calculate the residual for [tex]\( x = 4 \)[/tex]:
- Residual [tex]\( = \text{Given} - \text{Predicted} = 6.2 - 5.99 = 0.21 \)[/tex].
- The correct residual should be exactly 0.21, but if we calculate the difference, we find it to be approximately [tex]\( 0.20999999999999996 \)[/tex] due to floating-point precision issues, but it is effectively 0.21.
Conclusion: False (no error in computation if we consider typical precision)
#### Statement 4:
The residual value for [tex]\( x = 2 \)[/tex] should be a positive number because the given point is above the line of best fit.
- Residual for [tex]\( x = 2 \)[/tex] is -0.13.
- A positive residual indicates above the line, but the given residual is negative (-0.13), indicating the point is below the line.
Conclusion: False
#### Statement 5:
The residual value for [tex]\( x = 3 \)[/tex] is negative because the given point is below the line of best fit.
- Residual for [tex]\( x = 3 \)[/tex] is -0.38.
- A negative residual corresponds to the data point being below the line of best fit.
Conclusion: True
### Final Conclusions
The three true statements are:
1. The data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
2. Fiona made a subtraction error when she computed the residual value for [tex]\( x = 4 \)[/tex].
3. The residual value for [tex]\( x = 3 \)[/tex] is negative because the given point is below the line of best fit.
### Given Data
1. [tex]\( x = 1 \)[/tex] : Given = -5.1, Predicted = -5.14, Residual = 0.04
2. [tex]\( x = 2 \)[/tex] : Given = -1.3, Predicted = -1.43, Residual = -0.13
3. [tex]\( x = 3 \)[/tex] : Given = 1.9, Predicted = 2.28, Residual = -0.38
4. [tex]\( x = 4 \)[/tex] : Given = 6.2, Predicted = 5.99, Residual = 0.21
### Statements Analysis
#### Statement 1:
The data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
- To determine this, consider the residual value for [tex]\( x = 1 \)[/tex].
- Residual [tex]\( = \text{Given} - \text{Predicted} = -5.1 - (-5.14) = -5.1 + 5.14 = 0.04 \)[/tex]
- Since the residual is positive (0.04), the data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
Conclusion: True
#### Statement 2:
The residual value for [tex]\( x = 3 \)[/tex] should be a positive number because the data point is above the line of best fit.
- Residual for [tex]\( x = 3 \)[/tex] is given as -0.38.
- A positive residual would imply that the data point is above the line of best fit, but here the residual is negative (-0.38), indicating the point is below the line.
Conclusion: False
#### Statement 3:
Fiona made a subtraction error when she computed the residual value for [tex]\( x = 4 \)[/tex].
- Let's calculate the residual for [tex]\( x = 4 \)[/tex]:
- Residual [tex]\( = \text{Given} - \text{Predicted} = 6.2 - 5.99 = 0.21 \)[/tex].
- The correct residual should be exactly 0.21, but if we calculate the difference, we find it to be approximately [tex]\( 0.20999999999999996 \)[/tex] due to floating-point precision issues, but it is effectively 0.21.
Conclusion: False (no error in computation if we consider typical precision)
#### Statement 4:
The residual value for [tex]\( x = 2 \)[/tex] should be a positive number because the given point is above the line of best fit.
- Residual for [tex]\( x = 2 \)[/tex] is -0.13.
- A positive residual indicates above the line, but the given residual is negative (-0.13), indicating the point is below the line.
Conclusion: False
#### Statement 5:
The residual value for [tex]\( x = 3 \)[/tex] is negative because the given point is below the line of best fit.
- Residual for [tex]\( x = 3 \)[/tex] is -0.38.
- A negative residual corresponds to the data point being below the line of best fit.
Conclusion: True
### Final Conclusions
The three true statements are:
1. The data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
2. Fiona made a subtraction error when she computed the residual value for [tex]\( x = 4 \)[/tex].
3. The residual value for [tex]\( x = 3 \)[/tex] is negative because the given point is below the line of best fit.
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