To find the residual for the point [tex]\((2,5)\)[/tex] given the line of best fit [tex]\(y = 2.5x - 1.5\)[/tex], we need to follow these steps:
1. Substitute [tex]\(x = 2\)[/tex] into the line of best fit equation to find the predicted [tex]\(y\)[/tex]-value.
[tex]\[
y_{\text{predicted}} = 2.5 \cdot 2 - 1.5
\][/tex]
2. Calculate the predicted value:
[tex]\[
y_{\text{predicted}} = 5 - 1.5 = 3.5
\][/tex]
3. Compare the actual [tex]\(y\)[/tex]-value of the point [tex]\((2, 5)\)[/tex] to this predicted [tex]\(y\)[/tex]-value. The actual [tex]\(y\)[/tex]-value is 5.
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]
Substitute the values:
[tex]\[
\text{Residual} = 5 - 3.5 = 1.5
\][/tex]
Therefore, the residual for the point [tex]\((2, 5)\)[/tex] is [tex]\(1.5\)[/tex].
The correct answer is [tex]\( \boxed{1.5} \)[/tex].
