Answer :
To find the residual for the point [tex]\((4, 7)\)[/tex] given the line of best fit equation [tex]\(y = 2.5x - 1.5\)[/tex], follow these steps:
1. Identify the given point: The coordinates of the point given are [tex]\((4, 7)\)[/tex]. Here, [tex]\(x = 4\)[/tex] and [tex]\(y_{actual} = 7\)[/tex].
2. Substitute the x-coordinate into the line of best fit equation to find the predicted y-value:
[tex]\[ y_{predicted} = 2.5 \times x - 1.5 \][/tex]
Substitute [tex]\(x = 4\)[/tex]:
[tex]\[ y_{predicted} = 2.5 \times 4 - 1.5 \][/tex]
[tex]\[ y_{predicted} = 10 - 1.5 \][/tex]
[tex]\[ y_{predicted} = 8.5 \][/tex]
3. Calculate the residual: The residual is the difference between the actual y-value ([tex]\(y_{actual}\)[/tex]) and the predicted y-value ([tex]\(y_{predicted}\)[/tex]):
[tex]\[ \text{Residual} = y_{actual} - y_{predicted} \][/tex]
Substitute the actual and predicted values:
[tex]\[ \text{Residual} = 7 - 8.5 \][/tex]
[tex]\[ \text{Residual} = -1.5 \][/tex]
Therefore, the residual for the point [tex]\((4, 7)\)[/tex] is [tex]\(-1.5\)[/tex].
The correct answer is [tex]\(A\)[/tex] [tex]\(-1.5\)[/tex].
1. Identify the given point: The coordinates of the point given are [tex]\((4, 7)\)[/tex]. Here, [tex]\(x = 4\)[/tex] and [tex]\(y_{actual} = 7\)[/tex].
2. Substitute the x-coordinate into the line of best fit equation to find the predicted y-value:
[tex]\[ y_{predicted} = 2.5 \times x - 1.5 \][/tex]
Substitute [tex]\(x = 4\)[/tex]:
[tex]\[ y_{predicted} = 2.5 \times 4 - 1.5 \][/tex]
[tex]\[ y_{predicted} = 10 - 1.5 \][/tex]
[tex]\[ y_{predicted} = 8.5 \][/tex]
3. Calculate the residual: The residual is the difference between the actual y-value ([tex]\(y_{actual}\)[/tex]) and the predicted y-value ([tex]\(y_{predicted}\)[/tex]):
[tex]\[ \text{Residual} = y_{actual} - y_{predicted} \][/tex]
Substitute the actual and predicted values:
[tex]\[ \text{Residual} = 7 - 8.5 \][/tex]
[tex]\[ \text{Residual} = -1.5 \][/tex]
Therefore, the residual for the point [tex]\((4, 7)\)[/tex] is [tex]\(-1.5\)[/tex].
The correct answer is [tex]\(A\)[/tex] [tex]\(-1.5\)[/tex].
