To determine the residual value when \(x = 3\) for the given data and the line of best fit \(y = 2.69x - 7.95\), we will follow these steps:
1. Calculate the predicted value \(\hat{y}\):
- Substitute \(x = 3\) into the equation of the line of best fit.
[tex]\[
\hat{y} = 2.69 \cdot 3 - 7.95
\][/tex]
- Perform the multiplication and subtraction:
[tex]\[
\hat{y} = 8.07 - 7.95 = 0.12
\][/tex]
Thus, the predicted value \(\hat{y}\) when \(x = 3\) is \(0.12\).
2. Find the actual observed value \(y\):
- From the table, when \(x = 3\), the actual observed value is \(y = 1.0\).
3. Calculate the residual:
- The residual \( \text{residual} = y - \hat{y} \).
- Substitute the actual value \( y = 1.0 \) and the predicted value \( \hat{y} = 0.12 \):
[tex]\[
\text{residual} = 1.0 - 0.12 = 0.88
\][/tex]
Therefore, the residual value when [tex]\(x = 3\)[/tex] is [tex]\(0.88\)[/tex].
