Sure! Let's go through the details step-by-step based on the given table:
[tex]\[
\begin{array}{|l|l|}
\hline
x & y \\
\hline
-3 & -12 \\
\hline
-1 & -7 \\
\hline
1 & -21 \\
\hline
3 & -51 \\
\hline
5 & -93 \\
\hline
7 & -142 \\
\hline
\end{array}
\][/tex]
### 1. Calculate the mean of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values
To find the mean, we add all the values and divide by the number of values.
[tex]\[
\text{Mean of } x = \frac{(-3) + (-1) + 1 + 3 + 5 + 7}{6} = \frac{12}{6} = 2.0
\][/tex]
[tex]\[
\text{Mean of } y = \frac{(-12) + (-7) + (-21) + (-51) + (-93) + (-142)}{6} = \frac{-326}{6} = -54.333333333333336
\][/tex]
### 2. Calculate the sum of the products of corresponding [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values
Each pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are multiplied together, and then these products are summed.
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
-3 \cdot (-12) = 36 \\
-1 \cdot (-7) = 7 \\
1 \cdot (-21) = -21 \\
3 \cdot (-51) = -153 \\
5 \cdot (-93) = -465 \\
7 \cdot (-142) = -994 \\
\end{array}
\][/tex]
Sum of the products:
[tex]\[
36 + 7 - 21 - 153 - 465 - 994 = -1590
\][/tex]
### 3. Calculate the sum of [tex]\( x \)[/tex] squared and [tex]\( y \)[/tex] squared values
Each [tex]\( x \)[/tex] and [tex]\( y \)[/tex] value is squared, then these squares are summed separately.
[tex]\[
\begin{array}{c|c}
x & x^2 \\
\hline
-3 & 9 \\
-1 & 1 \\
1 & 1 \\
3 & 9 \\
5 & 25 \\
7 & 49 \\
\end{array}
\][/tex]
Sum of [tex]\( x^2 \)[/tex]:
[tex]\[
9 + 1 + 1 + 9 + 25 + 49 = 94
\][/tex]
[tex]\[
\begin{array}{c|c}
y & y^2 \\
\hline
-12 & 144 \\
-7 & 49 \\
-21 & 441 \\
-51 & 2601 \\
-93 & 8649 \\
-142 & 20164 \\
\end{array}
\][/tex]
Sum of [tex]\( y^2 \)[/tex]:
[tex]\[
144 + 49 + 441 + 2601 + 8649 + 20164 = 32048
\][/tex]
### Summary:
- Mean of [tex]\( x \)[/tex]: [tex]\( 2.0 \)[/tex]
- Mean of [tex]\( y \)[/tex]: [tex]\( -54.333333333333336 \)[/tex]
- Sum of the products of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: [tex]\( -1590 \)[/tex]
- Sum of [tex]\( x^2 \)[/tex]: [tex]\( 94 \)[/tex]
- Sum of [tex]\( y^2 \)[/tex]: [tex]\( 32048 \)[/tex]
These are the solutions for the questions based on the given table.
