To find the best predicted value for [tex]\( y \)[/tex] given [tex]\( x = 2.6 \)[/tex] using the least-squares regression line [tex]\(\hat{y} = -3.5x - 2.9\)[/tex], follow these steps:
1. Identify the regression equation and given [tex]\( x \)[/tex] value.
The regression equation is [tex]\(\hat{y} = -3.5x - 2.9\)[/tex]. We are given [tex]\( x = 2.6 \)[/tex].
2. Substitute the given [tex]\( x \)[/tex] value into the regression equation.
[tex]\[
\hat{y} = -3.5 \cdot 2.6 - 2.9
\][/tex]
3. Perform the multiplication and addition to solve for [tex]\(\hat{y}\)[/tex].
[tex]\[
-3.5 \cdot 2.6 = -9.1
\][/tex]
Then,
[tex]\[
\hat{y} = -9.1 - 2.9
\][/tex]
4. Complete the calculation.
[tex]\[
\hat{y} = -9.1 - 2.9 = -12.0
\][/tex]
Therefore, the best predicted value for [tex]\( y \)[/tex] when [tex]\( x = 2.6 \)[/tex] is [tex]\(\boxed{-12}\)[/tex].
