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Sagot :
Let's analyze each statement in detail based on the given data and the information about the residuals from the table.
1. The data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
- Given value for [tex]\( x = 1 \)[/tex] is [tex]\(-5.1\)[/tex]
- Predicted value for [tex]\( x = 1 \)[/tex] is [tex]\(-5.14\)[/tex]
Since [tex]\(-5.1\)[/tex] (given value) is greater than [tex]\(-5.14\)[/tex] (predicted value), the data point is indeed above the line of best fit. Thus, this statement is True.
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.
- Given value for [tex]\( x = 3 \)[/tex] is [tex]\(1.9\)[/tex]
- Predicted value for [tex]\( x = 3 \)[/tex] is [tex]\(2.28\)[/tex]
- Residual value for [tex]\( x = 3 \)[/tex] is computed as [tex]\(1.9 - 2.28 = -0.38\)[/tex]
Since [tex]\(1.9\)[/tex] (given value) is less than [tex]\(2.28\)[/tex] (predicted value), the data point is below the line of best fit, and the residual should be negative. Thus, this statement is False.
3. Fiona made a subtraction error when she computed the residual value for [tex]\( x = 4 \)[/tex].
- Given value for [tex]\( x = 4 \)[/tex] is [tex]\(6.2\)[/tex]
- Predicted value for [tex]\( x = 4 \)[/tex] is [tex]\(5.99\)[/tex]
- Residual value for [tex]\( x = 4 \)[/tex] is computed as [tex]\(6.2 - 5.99 = 0.21\)[/tex]
Since Fiona's calculated residual value ([tex]\(0.21\)[/tex]) matches the computed residual value, there is no subtraction error. Thus, this statement is False.
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.
- Given value for [tex]\( x = 2 \)[/tex] is [tex]\(-1.3\)[/tex]
- Predicted value for [tex]\( x = 2 \)[/tex] is [tex]\(-1.43\)[/tex]
- Residual value for [tex]\( x = 2 \)[/tex] is computed as [tex]\(-1.3 - (-1.43) = 0.13\)[/tex]
Since [tex]\(-1.3\)[/tex] (given value) is greater than [tex]\(-1.43\)[/tex] (predicted value), the data point is above the line of best fit, and the residual should be positive. Thus, this statement is True.
5. The residual value for [tex]\( x = 3 \)[/tex] is negative because the given point is below the line of best fit.
- Given value for [tex]\( x = 3 \)[/tex] is [tex]\(1.9\)[/tex]
- Predicted value for [tex]\( x = 3 \)[/tex] is [tex]\(2.28\)[/tex]
- Residual value for [tex]\( x = 3 \)[/tex] is computed as [tex]\(1.9 - 2.28 = -0.38\)[/tex]
As mentioned earlier, since [tex]\(1.9\)[/tex] (given value) is less than [tex]\(2.28\)[/tex] (predicted value), the data point is below the line of best fit, and the residual is negative. Thus, this statement is True.
Therefore, the true statements are:
1. The data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
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.
5. 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.
- Given value for [tex]\( x = 1 \)[/tex] is [tex]\(-5.1\)[/tex]
- Predicted value for [tex]\( x = 1 \)[/tex] is [tex]\(-5.14\)[/tex]
Since [tex]\(-5.1\)[/tex] (given value) is greater than [tex]\(-5.14\)[/tex] (predicted value), the data point is indeed above the line of best fit. Thus, this statement is True.
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.
- Given value for [tex]\( x = 3 \)[/tex] is [tex]\(1.9\)[/tex]
- Predicted value for [tex]\( x = 3 \)[/tex] is [tex]\(2.28\)[/tex]
- Residual value for [tex]\( x = 3 \)[/tex] is computed as [tex]\(1.9 - 2.28 = -0.38\)[/tex]
Since [tex]\(1.9\)[/tex] (given value) is less than [tex]\(2.28\)[/tex] (predicted value), the data point is below the line of best fit, and the residual should be negative. Thus, this statement is False.
3. Fiona made a subtraction error when she computed the residual value for [tex]\( x = 4 \)[/tex].
- Given value for [tex]\( x = 4 \)[/tex] is [tex]\(6.2\)[/tex]
- Predicted value for [tex]\( x = 4 \)[/tex] is [tex]\(5.99\)[/tex]
- Residual value for [tex]\( x = 4 \)[/tex] is computed as [tex]\(6.2 - 5.99 = 0.21\)[/tex]
Since Fiona's calculated residual value ([tex]\(0.21\)[/tex]) matches the computed residual value, there is no subtraction error. Thus, this statement is False.
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.
- Given value for [tex]\( x = 2 \)[/tex] is [tex]\(-1.3\)[/tex]
- Predicted value for [tex]\( x = 2 \)[/tex] is [tex]\(-1.43\)[/tex]
- Residual value for [tex]\( x = 2 \)[/tex] is computed as [tex]\(-1.3 - (-1.43) = 0.13\)[/tex]
Since [tex]\(-1.3\)[/tex] (given value) is greater than [tex]\(-1.43\)[/tex] (predicted value), the data point is above the line of best fit, and the residual should be positive. Thus, this statement is True.
5. The residual value for [tex]\( x = 3 \)[/tex] is negative because the given point is below the line of best fit.
- Given value for [tex]\( x = 3 \)[/tex] is [tex]\(1.9\)[/tex]
- Predicted value for [tex]\( x = 3 \)[/tex] is [tex]\(2.28\)[/tex]
- Residual value for [tex]\( x = 3 \)[/tex] is computed as [tex]\(1.9 - 2.28 = -0.38\)[/tex]
As mentioned earlier, since [tex]\(1.9\)[/tex] (given value) is less than [tex]\(2.28\)[/tex] (predicted value), the data point is below the line of best fit, and the residual is negative. Thus, this statement is True.
Therefore, the true statements are:
1. The data point for [tex]\( x = 1 \)[/tex] is above the line of best fit.
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.
5. 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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