To find the slope-intercept form of the equation of the line that best fits the given data points, we can use the method of linear regression. Linear regression will give us the best-fit line equation in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Here are the data points provided:
[tex]\[
\begin{array}{c|c}
X & Y \\
\hline
0.7 & 5 \\
0.9 & 6 \\
0.1 & 1 \\
0.3 & 1 \\
0.4 & 2 \\
0.8 & 5 \\
0.3 & 1 \\
0.9 & 6 \\
0.2 & 2 \\
1 & 7 \\
\end{array}
\][/tex]
Given these data points, we aim to calculate:
1. Slope (m): This measures the steepness of the line.
2. Intercept (b): This is the value of the y-coordinate when x is 0.
Step-by-Step Solution:
### 1. Calculate the Slope (m)
The slope [tex]\( m \)[/tex] can be interpreted as the change in [tex]\( Y \)[/tex] for a unit change in [tex]\( X \)[/tex]. Using the method of least squares, we find that the slope [tex]\( m \)[/tex] is approximately:
[tex]\[ m = 7.01 \][/tex]
### 2. Calculate the Intercept (b)
The intercept [tex]\( b \)[/tex] is the point where the line crosses the y-axis (i.e., when [tex]\( X = 0 \)[/tex]). From our calculations, the intercept [tex]\( b \)[/tex] is approximately:
[tex]\[ b = -0.33 \][/tex]
### 3. Form the Equation of the Line
Substituting the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the slope-intercept form, we get the equation of the best-fit line:
[tex]\[ y = 7.01x - 0.33 \][/tex]
In summary, the equation of the line that best fits the given data points is:
[tex]\[ y = 7.01x - 0.33 \][/tex]
