To solve the problem of finding the residual for [tex]\( x = 5 \)[/tex] using the line of best fit [tex]\( y = 5.2x - 0.4 \)[/tex], we can follow these steps:
1. Identify the actual [tex]\( y \)[/tex] value for [tex]\( x = 5 \)[/tex]:
From the given table, when [tex]\( x = 5 \)[/tex], the actual [tex]\( y \)[/tex] value is [tex]\( y_{\text{actual}} = 24 \)[/tex].
2. Calculate the predicted [tex]\( y \)[/tex] value using the line of best fit:
Plug [tex]\( x = 5 \)[/tex] into the equation of the line of best fit, [tex]\( y = 5.2x - 0.4 \)[/tex]:
[tex]\[
y_{\text{predicted}} = 5.2 \cdot 5 - 0.4 = 26 - 0.4 = 25.6
\][/tex]
3. Calculate the residual:
The residual is the difference between the actual [tex]\( y \)[/tex] value and the predicted [tex]\( y \)[/tex] value. This can be expressed as:
[tex]\[
\text{Residual} = y_{\text{actual}} - y_{\text{predicted}}
\][/tex]
Substituting the values, we get:
[tex]\[
\text{Residual} = 24 - 25.6 = -1.6
\][/tex]
The residual for [tex]\( x = 5 \)[/tex] is [tex]\(-1.6\)[/tex]. Hence, the correct answer is:
[tex]\(-1.6\)[/tex].
