To find the equation of the line that best fits the given data points [tex]\((-2, -8)\)[/tex], [tex]\((-1, -5)\)[/tex], [tex]\( (0, -2)\)[/tex], [tex]\( (1, 1)\)[/tex], [tex]\( (2, 4)\)[/tex], and [tex]\( (3, 7)\)[/tex], we need to perform a linear regression analysis. The equation of the line is generally given by:
[tex]\[ y = mx + c \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( c \)[/tex] is the y-intercept of the line.
Given the data points, we first need to determine the slope ([tex]\( m \)[/tex]) and the y-intercept ([tex]\( c \)[/tex]).
The slope [tex]\( m \)[/tex] is calculated using the formula:
[tex]\[ m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \][/tex]
where [tex]\( \bar{x} \)[/tex] and [tex]\( \bar{y} \)[/tex] are the means of the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values, respectively.
The y-intercept [tex]\( c \)[/tex] is calculated using the formula:
[tex]\[ c = \bar{y} - m \cdot \bar{x} \][/tex]
From the analysis, we have found the following results:
1. The slope [tex]\( m \)[/tex] is calculated to be [tex]\( 3.0 \)[/tex].
2. The y-intercept [tex]\( c \)[/tex] is calculated to be [tex]\( -2.0 \)[/tex].
Therefore, the equation that best describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[ y = 3.0x - 2.0 \][/tex]
So, the value that belongs in the green box is [tex]\( 3.0 \)[/tex]. The complete equation is:
[tex]\[ y = 3.0x - 2.0 \][/tex]
