Alright class, let's go through a detailed, step-by-step solution to find the best-fit line for the given data points using linear regression. Here are the data points \( (x, y) \):
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline -2 & -7 \\
\hline -1 & -3 \\
\hline 0 & 1 \\
\hline 1 & 5 \\
\hline
\end{array}
\][/tex]
We want to find a linear relationship in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
### Step 1: Summing the necessary quantities
To find \( m \) and \( b \), we need to calculate the following sums:
- \( \sum x \)
- \( \sum y \)
- \( \sum xy \)
- \( \sum x^2 \)
Given data, we have calculated these sums as follows:
[tex]\[
\sum x = -2
\][/tex]
[tex]\[
\sum y = -4
\][/tex]
[tex]\[
\sum xy = 22
\][/tex]
[tex]\[
\sum x^2 = 6
\][/tex]
### Step 2: Calculate the slope \( m \)
The formula for the slope \( m \) is:
[tex]\[
m = \frac{n \sum (xy) - (\sum x)(\sum y)}{n \sum (x^2) - (\sum x)^2}
\][/tex]
Let's plug in the values:
[tex]\[
n = 4 \quad (\text{since there are 4 data points})
\][/tex]
[tex]\[
m = \frac{4 \cdot 22 - (-2) \cdot (-4)}{4 \cdot 6 - (-2)^2} = \frac{88 - 8}{24 - 4} = \frac{80}{20} = 4
\][/tex]
So, the slope \( m \) is \( 4 \).
### Step 3: Calculate the y-intercept \( b \)
The formula for the y-intercept \( b \) is:
[tex]\[
b = \frac{\sum y - m \sum x}{n}
\][/tex]
Using the values we have:
[tex]\[
b = \frac{-4 - 4 \cdot (-2)}{4} = \frac{-4 + 8}{4} = \frac{4}{4} = 1
\][/tex]
So, the y-intercept \( b \) is \( 1 \).
### Final Result
Therefore, the equation of the best-fit line is:
[tex]\[
y = 4x + 1
\][/tex]
Summarizing the key calculation results:
- \( \sum x = -2 \)
- \( \sum y = -4 \)
- \( \sum xy = 22 \)
- \( \sum x^2 = 6 \)
- \( m = 4 \)
- \( b = 1 \)
This completes our detailed solution for finding the best-fit line. The best-fit line for the given data points is [tex]\( y = 4x + 1 \)[/tex].
