To find the equation of the line of best fit for the given data, we'll use a linear regression method. Here is the step-by-step solution:
1. Gather the Data Points:
[tex]\[
(4, 3), (6, 4), (8, 9), (11, 12), (13, 17)
\][/tex]
2. Determine the General Form of the Line of Best Fit Equation:
The equation of the line of best fit is generally written as:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Calculate the Slope (m) and Y-Intercept (b):
For these calculations, techniques from linear algebra or statistical methods are typically used to minimize the sum of squared differences between the observed values [tex]\( y \)[/tex] and the values predicted by the linear model.
4. Provide the Solution:
The slope (m) and y-intercept (b) have been found to be:
[tex]\[
m = 1.560
\][/tex]
[tex]\[
b = -4.105
\][/tex]
5. Form the Equation:
Substitute [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the general form of the linear equation:
[tex]\[
y = 1.560x - 4.105
\][/tex]
6. Round the Values:
Both the slope and y-intercept have already been rounded to three decimal places.
Therefore, the equation of the line of best fit is:
[tex]\[
y = 1.560x - 4.105
\][/tex]
Given the multiple-choice options:
A. [tex]\( y = -1.560x + 4.105 \)[/tex]
B. [tex]\( y = -4.105x + 1.560 \)[/tex]
C. [tex]\( y = 1.560x - 4.105 \)[/tex]
D. [tex]\( y = 4.105x - 1.560 \)[/tex]
The correct option is:
[tex]\[ \boxed{C} \][/tex]
