Question

the following ordered pairs give the entrance exam scores x and the grade-point averages y after 1 year of college for 10 students.find the equation of a line that models entrance exam scores and GPA.

Answer 1

If this is a linear function between x and y, the slope will be constant.

We can pick any two ordered pairs, like (75, 2.3) and (82, 3) and calculate the slope as:

`m=(y_2-y_1)/(x_2-x_1)=(3-2.3)/(82-75)=(0.7)/(7)=0.1`

We can write the point-slope form of the equation and rearrange to find the value of the y-intercept and the slope-intercept form of the equation:

`\begin{gathered} y-y_0=m(x-x_0) \\ y-3=0.1(x-82) \\ y=0.1x-0.1\cdot82+3 \\ y=0.1x-8.2+3 \\ y=0.1x-5.2 \end{gathered}`

We can test if the equation is correct with another point, like (65,2):

`\begin{gathered} y=0.1x-5.2 \\ y(65)=0.1(65)-5.2 \\ y(65)=6.5-5.2 \\ y(65)=1.3 \end{gathered}`

This is not a exact line, so we have to apply a regression model to find the approximate line that best represent this relationship:

Answer: the linear regression model that best represents the relation between x and y is y=0.0736x-3.0166.