To find the residual for the point [tex]\((5, 1)\)[/tex] with a given line of best fit [tex]\(y = -0.2x + 1.7\)[/tex], follow these steps:
1. Substitute the [tex]\(x\)[/tex] value into the line of best fit equation to find the predicted [tex]\(y\)[/tex] value:
The given [tex]\(x\)[/tex] value is [tex]\(5\)[/tex]. Substitute this into the equation [tex]\(y = -0.2x + 1.7\)[/tex]:
[tex]\[
y_{\text{predicted}} = -0.2(5) + 1.7
\][/tex]
2. Calculate the predicted [tex]\(y\)[/tex] value:
[tex]\[
y_{\text{predicted}} = -1 + 1.7 = 0.7
\][/tex]
3. Determine the residual:
The residual is the difference between the observed [tex]\(y\)[/tex] value and the predicted [tex]\(y\)[/tex] value. The observed [tex]\(y\)[/tex] value is 1.
Residual = [tex]\(y_{\text{observed}} - y_{\text{predicted}}\)[/tex]:
[tex]\[
\text{Residual} = 1 - 0.7 = 0.3
\][/tex]
Therefore, the residual for the point [tex]\((5, 1)\)[/tex] is [tex]\(0.3\)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{0.3}
\][/tex]
