Sure, let's find the slope of the line of best fit for the given data. Here are the steps to find it:
1. List the Data Points:
The points given are [tex]\((-4, -32)\)[/tex], [tex]\((-2, -8)\)[/tex], [tex]\( (0, 10)\)[/tex], [tex]\( (2, 8)\)[/tex], and [tex]\( (4, 32)\)[/tex].
2. Calculate the Means:
First, calculate the mean of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
[tex]\[
\text{Mean of } x = \frac{-4 + (-2) + 0 + 2 + 4}{5} = \frac{0}{5} = 0.0
\][/tex]
[tex]\[
\text{Mean of } y = \frac{-32 + (-8) + 10 + 8 + 32}{5} = \frac{10}{5} = 2.0
\][/tex]
3. Calculate the Numerator of the Slope Formula:
The numerator is the sum of the products of the deviations of each [tex]\( x \)[/tex] and [tex]\( y \)[/tex] pair from their respective means.
[tex]\[
\sum (x_i - \text{mean}(x)) \cdot (y_i - \text{mean}(y))
= \sum (x_i - 0.0) \cdot (y_i - 2.0)
\][/tex]
Substituting the values:
[tex]\[
= (-4 - 0.0) \cdot (-32 - 2.0) + (-2 - 0.0) \cdot (-8 - 2.0) + (0 - 0.0) \cdot (10 - 2.0) + (2 - 0.0) \cdot (8 - 2.0) + (4 - 0.0) \cdot (32 - 2.0)
\][/tex]
[tex]\[
= -4 \cdot (-34) + (-2) \cdot (-10) + 0 \cdot 8 + 2 \cdot 6 + 4 \cdot 30
\][/tex]
Calculate each term:
[tex]\[
= 136 + 20 + 0 + 12 + 120 = 288.0
\][/tex]
4. Calculate the Denominator of the Slope Formula:
The denominator is the sum of the squares of the deviations of each [tex]\( x \)[/tex] from the mean of [tex]\( x \)[/tex].
[tex]\[
\sum (x_i - \text{mean}(x))^2 = \sum (x_i - 0.0)^2
\][/tex]
Substituting the values:
[tex]\[
= (-4 - 0)^2 + (-2 - 0)^2 + (0 - 0)^2 + (2 - 0)^2 + (4 - 0)^2
\][/tex]
[tex]\[
= 16 + 4 + 0 + 4 + 16 = 40.0
\][/tex]
5. Calculate the Slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] is the ratio of the numerator to the denominator.
[tex]\[
m = \frac{\text{numerator}}{\text{denominator}} = \frac{288.0}{40.0} = 7.2
\][/tex]
Therefore, the slope of the line of best fit for the given data is 7.2.
