To determine the [tex]\( y \)[/tex]-intercept of a linear function given by the data points, let's walk through the process of performing a linear regression. This technique helps us find the line of best fit for the given data. The linear equation can be represented in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
The provided data points are:
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 8.50 \)[/tex]
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 15.25 \)[/tex]
- When [tex]\( x = 8 \)[/tex], [tex]\( y = 22.00 \)[/tex]
- When [tex]\( x = 12 \)[/tex], [tex]\( y = 31.00 \)[/tex]
To find the equation of the line, we need to determine the slope ([tex]\( m \)[/tex]) and the [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]):
1. Calculate the slope [tex]\( m \)[/tex]:
The slope [tex]\( m \)[/tex] of the line can be found using the formula:
[tex]\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \][/tex]
where [tex]\( n \)[/tex] is the number of data points.
2. Calculate the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]:
Once we have the slope, we can find the [tex]\( y \)[/tex]-intercept using the formula:
[tex]\[ b = \frac{\sum y - m (\sum x)}{n} \][/tex]
Given the calculations, we find:
- The slope, [tex]\( m \)[/tex], is 2.25.
- The [tex]\( y \)[/tex]-intercept, [tex]\( b \)[/tex], is 4.00.
Therefore, the [tex]\( y \)[/tex]-intercept of the linear function is:
[tex]\[ b = 4.00 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{4.00} \][/tex]
