Certainly! Let's go through the problem step-by-step to understand how we arrive at the answer.
We are given that the equation for the line of best fit is [tex]\( y = 2x + 1.5 \)[/tex], and we are to find the residual for the given data point [tex]\((1, 4)\)[/tex].
### Step 1: Calculate the Predicted [tex]\( y \)[/tex] Value
First, we need to determine the predicted [tex]\( y \)[/tex] value based on the line of best fit for the given [tex]\( x \)[/tex]-coordinate. In this case, [tex]\( x = 1 \)[/tex].
Using the equation [tex]\( y = 2x + 1.5 \)[/tex]:
[tex]\[ y_{\text{predicted}} = 2(1) + 1.5 = 2 + 1.5 = 3.5 \][/tex]
### Step 2: Determine the Observed [tex]\( y \)[/tex] Value
The observed [tex]\( y \)[/tex]-value is given as part of the data point [tex]\((1, 4)\)[/tex], so:
[tex]\[ y_{\text{observed}} = 4 \][/tex]
### Step 3: Calculate the Residual
The residual is the difference between the observed [tex]\( y \)[/tex]-value and the predicted [tex]\( y \)[/tex]-value:
[tex]\[ \text{Residual} = y_{\text{observed}} - y_{\text{predicted}} = 4 - 3.5 = 0.5 \][/tex]
### Step 4: Conclusion
The residual, which is the difference between the observed value and the predicted value, is [tex]\( 0.5 \)[/tex].
Therefore, the correct answer is [tex]\(\boxed{0.5}\)[/tex].
