Answer :
To find the residual for a given point with coordinates [tex]\((1, 4)\)[/tex] on the line of best fit described by the equation [tex]\(y = 2x + 1.5\)[/tex], we can follow these steps:
1. Substitute [tex]\(x = 1\)[/tex] into the line equation [tex]\(y = 2x + 1.5\)[/tex] to find the predicted value of [tex]\(y\)[/tex].
[tex]\[ y_{\text{predicted}} = 2(1) + 1.5 = 2 + 1.5 = 3.5 \][/tex]
2. Compare the predicted [tex]\(y\)[/tex] to the actual [tex]\(y\)[/tex] value from the point [tex]\((1, 4)\)[/tex]. The actual [tex]\(y\)[/tex] value is 4.
3. 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]\[ \text{Residual} = 4 - 3.5 = 0.5 \][/tex]
Therefore, the residual for the point [tex]\((1, 4)\)[/tex] is [tex]\(0.5\)[/tex].
So, the correct answer is:
0.5
1. Substitute [tex]\(x = 1\)[/tex] into the line equation [tex]\(y = 2x + 1.5\)[/tex] to find the predicted value of [tex]\(y\)[/tex].
[tex]\[ y_{\text{predicted}} = 2(1) + 1.5 = 2 + 1.5 = 3.5 \][/tex]
2. Compare the predicted [tex]\(y\)[/tex] to the actual [tex]\(y\)[/tex] value from the point [tex]\((1, 4)\)[/tex]. The actual [tex]\(y\)[/tex] value is 4.
3. 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]\[ \text{Residual} = 4 - 3.5 = 0.5 \][/tex]
Therefore, the residual for the point [tex]\((1, 4)\)[/tex] is [tex]\(0.5\)[/tex].
So, the correct answer is:
0.5
