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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.
[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.
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