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Sagot :
Let’s analyze each statement given the data and compute the necessary steps to confirm the validity of each statement.
1. The data point for [tex]\(x=1\)[/tex] is above the line of best fit.
- From the table: When [tex]\(x=1\)[/tex], the given value is [tex]\(-5.1\)[/tex] and the predicted value is [tex]\(-5.14\)[/tex].
- Since [tex]\(-5.1 > -5.14\)[/tex], the given value is indeed above the predicted value.
- Therefore, this statement is correct.
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.
- From the table: When [tex]\(x=3\)[/tex], the given value is [tex]\(1.9\)[/tex] and the predicted value is [tex]\(2.28\)[/tex].
- Since [tex]\(1.9 < 2.28\)[/tex], the given value is below the predicted value.
- Therefore, this statement is incorrect.
3. Fiona made a subtraction error when she computed the residual value for [tex]\(x=4\)[/tex].
- From the table: When [tex]\(x=4\)[/tex], the given value is [tex]\(6.2\)[/tex] and the predicted value is [tex]\(5.99\)[/tex]. The residual Fiona computed is [tex]\(0.21\)[/tex].
- Residual should be: [tex]\[ \text{Residual} = \text{Given} - \text{Predicted} = 6.2 - 5.99 = 0.21 \][/tex]
- Fiona's calculated residual is [tex]\(0.21\)[/tex], which matches the theoretically calculated one.
- Therefore, this statement is incorrect.
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.
- From the table: When [tex]\(x=2\)[/tex], the given value is [tex]\(-1.3\)[/tex] and the predicted value is [tex]\(-1.43\)[/tex].
- Since [tex]\(-1.3 > -1.43\)[/tex], the given value is above the predicted value.
- Therefore, the residual should indeed be positive.
- This means this statement is correct.
5. The residual value for [tex]\(x=3\)[/tex] is negative because the given point is below the line of best fit.
- From the table: When [tex]\(x=3\)[/tex], the given value is [tex]\(1.9\)[/tex] and the predicted value is [tex]\(2.28\)[/tex].
- Since [tex]\(1.9 < 2.28\)[/tex], the given value is below the predicted value.
- This indicates the residual is negative.
- Therefore, this statement is correct.
Based on this analysis, the three correct statements are:
- The data point for [tex]\(x=1\)[/tex] is above the line of best fit.
- The residual value for [tex]\(x=2\)[/tex] should be a positive number because the given point is above the line of best fit.
- The residual value for [tex]\(x=3\)[/tex] is negative because the given point is below the line of best fit.
1. The data point for [tex]\(x=1\)[/tex] is above the line of best fit.
- From the table: When [tex]\(x=1\)[/tex], the given value is [tex]\(-5.1\)[/tex] and the predicted value is [tex]\(-5.14\)[/tex].
- Since [tex]\(-5.1 > -5.14\)[/tex], the given value is indeed above the predicted value.
- Therefore, this statement is correct.
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.
- From the table: When [tex]\(x=3\)[/tex], the given value is [tex]\(1.9\)[/tex] and the predicted value is [tex]\(2.28\)[/tex].
- Since [tex]\(1.9 < 2.28\)[/tex], the given value is below the predicted value.
- Therefore, this statement is incorrect.
3. Fiona made a subtraction error when she computed the residual value for [tex]\(x=4\)[/tex].
- From the table: When [tex]\(x=4\)[/tex], the given value is [tex]\(6.2\)[/tex] and the predicted value is [tex]\(5.99\)[/tex]. The residual Fiona computed is [tex]\(0.21\)[/tex].
- Residual should be: [tex]\[ \text{Residual} = \text{Given} - \text{Predicted} = 6.2 - 5.99 = 0.21 \][/tex]
- Fiona's calculated residual is [tex]\(0.21\)[/tex], which matches the theoretically calculated one.
- Therefore, this statement is incorrect.
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.
- From the table: When [tex]\(x=2\)[/tex], the given value is [tex]\(-1.3\)[/tex] and the predicted value is [tex]\(-1.43\)[/tex].
- Since [tex]\(-1.3 > -1.43\)[/tex], the given value is above the predicted value.
- Therefore, the residual should indeed be positive.
- This means this statement is correct.
5. The residual value for [tex]\(x=3\)[/tex] is negative because the given point is below the line of best fit.
- From the table: When [tex]\(x=3\)[/tex], the given value is [tex]\(1.9\)[/tex] and the predicted value is [tex]\(2.28\)[/tex].
- Since [tex]\(1.9 < 2.28\)[/tex], the given value is below the predicted value.
- This indicates the residual is negative.
- Therefore, this statement is correct.
Based on this analysis, the three correct statements are:
- The data point for [tex]\(x=1\)[/tex] is above the line of best fit.
- The residual value for [tex]\(x=2\)[/tex] should be a positive number because the given point is above the line of best fit.
- 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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