Answer :
To fill in the missing values of [tex]\( y \)[/tex] for the given [tex]\( x \)[/tex] values in the table, we need to determine the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Through regression analysis, this relationship can be expressed in the form of a linear equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
For our data points, the best fit line is given by the equation:
[tex]\[ y = -6.8x + 22.1 \][/tex]
We will use this equation to calculate the values of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex].
1. When [tex]\( x = 3 \)[/tex]:
[tex]\[ y = -6.8 \cdot 3 + 22.1 = -20.4 + 22.1 = 1.7 \][/tex]
2. When [tex]\( x = 6 \)[/tex]:
[tex]\[ y = -6.8 \cdot 6 + 22.1 = -40.8 + 22.1 = -18.7 \][/tex]
3. When [tex]\( x = 9 \)[/tex]:
[tex]\[ y = -6.8 \cdot 9 + 22.1 = -61.2 + 22.1 = -39.1 \][/tex]
4. When [tex]\( x = 10 \)[/tex]:
[tex]\[ y = -6.8 \cdot 10 + 22.1 = -68 + 22.1 = -45.9 \][/tex]
With these calculations, we can fill in the missing values in the table as follows:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 3 & 1.7 \\ \hline 6 & -18.7 \\ \hline 9 & -39.1 \\ \hline 10 & -45.9 \\ \hline \end{tabular} \][/tex]
For our data points, the best fit line is given by the equation:
[tex]\[ y = -6.8x + 22.1 \][/tex]
We will use this equation to calculate the values of [tex]\( y \)[/tex] for each [tex]\( x \)[/tex].
1. When [tex]\( x = 3 \)[/tex]:
[tex]\[ y = -6.8 \cdot 3 + 22.1 = -20.4 + 22.1 = 1.7 \][/tex]
2. When [tex]\( x = 6 \)[/tex]:
[tex]\[ y = -6.8 \cdot 6 + 22.1 = -40.8 + 22.1 = -18.7 \][/tex]
3. When [tex]\( x = 9 \)[/tex]:
[tex]\[ y = -6.8 \cdot 9 + 22.1 = -61.2 + 22.1 = -39.1 \][/tex]
4. When [tex]\( x = 10 \)[/tex]:
[tex]\[ y = -6.8 \cdot 10 + 22.1 = -68 + 22.1 = -45.9 \][/tex]
With these calculations, we can fill in the missing values in the table as follows:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 3 & 1.7 \\ \hline 6 & -18.7 \\ \hline 9 & -39.1 \\ \hline 10 & -45.9 \\ \hline \end{tabular} \][/tex]
