To find the equation of the regression line that passes through the points [tex]\((2, 10)\)[/tex] and [tex]\((7, 18)\)[/tex], we follow these steps:
1. Calculate the slope (m):
The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the given points [tex]\((x_1, y_1) = (2, 10)\)[/tex] and [tex]\((x_2, y_2) = (7, 18)\)[/tex], we get:
[tex]\[
m = \frac{18 - 10}{7 - 2} = \frac{8}{5} = 1.6
\][/tex]
2. Calculate the y-intercept (b):
The formula for the y-intercept [tex]\(b\)[/tex] of the line in the form [tex]\(y = mx + b\)[/tex] is:
[tex]\[
b = y_1 - m \cdot x_1
\][/tex]
Using the calculated slope [tex]\(m = 1.6\)[/tex] and the point [tex]\((2, 10)\)[/tex]:
[tex]\[
b = 10 - (1.6 \times 2) = 10 - 3.2 = 6.8
\][/tex]
3. Formulate the equation of the line:
Combining the slope and y-intercept, the equation of the regression line is:
[tex]\[
\hat{y} = 1.6 x + 6.8
\][/tex]
So, the equation of Jacob's regression line is:
[tex]\[
\hat{y} = 1.6 x + 6.8
\][/tex] Hence, the correct answer is the first option:
[tex]\[
\hat{y} = 1.6 x + 6.8
\][/tex]
