To find the residual for the point [tex]\((4, 7)\)[/tex] using the line of best fit equation [tex]\(y = 2.5x - 1.5\)[/tex], we need to follow these steps:
1. Identify the given point and the equation of the line of best fit.
- The point provided is [tex]\((4, 7)\)[/tex].
- The equation of the line of best fit is [tex]\(y = 2.5x - 1.5\)[/tex].
2. Calculate the predicted [tex]\(y\)[/tex]-value using the line of best fit.
- Substitute [tex]\(x = 4\)[/tex] into the equation [tex]\(y = 2.5x - 1.5\)[/tex].
[tex]\[
y_{\text{predicted}} = 2.5 \cdot 4 - 1.5
\][/tex]
[tex]\[
y_{\text{predicted}} = 10 - 1.5
\][/tex]
[tex]\[
y_{\text{predicted}} = 8.5
\][/tex]
3. Determine the actual [tex]\(y\)[/tex]-value from the given point.
- The actual [tex]\(y\)[/tex]-value is provided directly from the point, which is [tex]\(7\)[/tex].
4. Calculate the residual.
- The residual is the difference between the actual [tex]\(y\)[/tex]-value and the predicted [tex]\(y\)[/tex]-value.
[tex]\[
\text{Residual} = y_{\text{actual}} - y_{\text{predicted}}
\][/tex]
[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:
A. -1.5
