First, let's understand what a residual value is. A residual value is the difference between an observed value and the predicted value given by the line of best fit. Mathematically, it's given by:
[tex]\[ \text{residual} = y_{\text{actual}} - y_{\text{predicted}} \][/tex]
Now, we are given the line of best fit equation:
[tex]\[ y = 2.78x - 4.4 \][/tex]
We need to find the predicted value of [tex]\( y \)[/tex] when [tex]\( x = 4 \)[/tex].
1. Substitute [tex]\( x = 4 \)[/tex] into the line of best fit equation to find the predicted value [tex]\( y_{\text{predicted}} \)[/tex]:
[tex]\[ y_{\text{predicted}} = 2.78(4) - 4.4 \][/tex]
Using the values provided for [tex]\( y_{\text{predicted}} \)[/tex]:
[tex]\[ y_{\text{predicted}} \approx 6.72 \][/tex]
Next, we use the actual value of [tex]\( y \)[/tex] from the table when [tex]\( x = 4 \)[/tex]:
[tex]\[ y_{\text{actual}} = 7.3 \][/tex]
2. Calculate the residual using the actual and predicted values when [tex]\( x = 4 \)[/tex]:
[tex]\[ \text{residual} = y_{\text{actual}} - y_{\text{predicted}} \][/tex]
Substitute the known values:
[tex]\[ \text{residual} = 7.3 - 6.72 \][/tex]
Using the values provided for the residual:
[tex]\[ \text{residual} \approx 0.58 \][/tex]
Therefore, the residual value when [tex]\( x = 4 \)[/tex] is [tex]\( \boxed{0.58} \)[/tex].
