Sure, let's solve for the residual of the given point (2, 5) with respect to the line of best fit [tex]\( y = 2.5x - 1.5 \)[/tex].
1. Identify the given point: The point we are considering is [tex]\( (2, 5) \)[/tex].
- Here, [tex]\( x = 2 \)[/tex] and [tex]\( y_{\text{actual}} = 5 \)[/tex].
2. Calculate the expected [tex]\( y \)[/tex]-value on the line of best fit for [tex]\( x = 2 \)[/tex]:
[tex]\[
y_{\text{expected}} = 2.5 \times 2 - 1.5
\][/tex]
- First, multiply [tex]\( 2.5 \)[/tex] by [tex]\( 2 \)[/tex]:
[tex]\[
2.5 \times 2 = 5
\][/tex]
- Then, subtract [tex]\( 1.5 \)[/tex] from [tex]\( 5 \)[/tex]:
[tex]\[
5 - 1.5 = 3.5
\][/tex]
- Therefore, the expected [tex]\( y \)[/tex]-value ([tex]\( y_{\text{expected}} \)[/tex]) on the line of best fit when [tex]\( x = 2 \)[/tex] is [tex]\( 3.5 \)[/tex].
3. Calculate the residual: The residual is the difference between the actual [tex]\( y \)[/tex]-value and the expected [tex]\( y \)[/tex]-value.
[tex]\[
\text{residual} = y_{\text{actual}} - y_{\text{expected}}
\][/tex]
- Substitute the given and calculated values:
[tex]\[
\text{residual} = 5 - 3.5
\][/tex]
- Perform the subtraction:
[tex]\[
5 - 3.5 = 1.5
\][/tex]
Hence, the residual for the point [tex]\( (2, 5) \)[/tex] is [tex]\( 1.5 \)[/tex].
The correct answer is:
B. 1.5
