Let's work through the process of finding the regression line based on the provided data. In a regression analysis, we typically express the relationship between two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex], with the line equation [tex]\(\hat{y} = b_0 + b_1 x\)[/tex], where [tex]\(\hat{y}\)[/tex] is the predicted value of [tex]\( y \)[/tex] for a given [tex]\( x \)[/tex], [tex]\( b_0 \)[/tex] is the y-intercept, and [tex]\( b_1 \)[/tex] is the slope of the line.
Given the data points:
[tex]\[ x = [6, 8, 20, 28, 36] \][/tex]
[tex]\[ y = [2, 4, 13, 20, 30] \][/tex]
After conducting the linear regression analysis, we find the values of [tex]\( b_0 \)[/tex] (the y-intercept) and [tex]\( b_1 \)[/tex] (the slope):
1. Slope [tex]\( b_1 \)[/tex] (slope of the regression line)
2. Intercept [tex]\( b_0 \)[/tex] (y-intercept of the regression line)
From the regression analysis, the calculated values are:
- [tex]\( b_0 \)[/tex] (y-intercept) = -3.79
- [tex]\( b_1 \)[/tex] (slope) = 0.897
Thus, the equation of the regression line is:
[tex]\[ \hat{y} = -3.79 + 0.897x \][/tex]
To confirm, you would check the provided choices against this result:
- [tex]\(\hat{y} = -3.79 + 0.801x\)[/tex]: The slope 0.801 does not match 0.897.
- [tex]\(\hat{y} = -279 + 0.950x\)[/tex]: Both the intercept -279 and the slope 0.950 do not match.
- [tex]\(\hat{y} = -3.79 + 0.897x\)[/tex]: Both the intercept -3.79 and the slope 0.897 match.
- [tex]\(\hat{y} = -2.79 + 0.897x\)[/tex]: The intercept -2.79 does not match -3.79.
The correct equation is:
[tex]\[ \hat{y} = -3.79 + 0.897x \][/tex]
So the correct answer is:
[tex]\[ \boxed{\hat{y} = -3.79 + 0.897x} \][/tex]
