Answer:
(a) The students whose SAT scores were 12000 has a GPA of 2.23.
(b) The students whose SAT scores were 2400 has a GPA of 1.27.
Step-by-step explanation:
The data provided is for the student's SAT scores and their GPAs.
To estimate the GPA of a students based on his\her SAT scores we need to form a regression line where GPA will be the dependent variable and SAT scores will be the independent variable.
The general form of the a regression equation is:
[tex]y=\alpha +\beta x[/tex]
The formula for α and β are:
[tex]\alpha = \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}[/tex]
[tex]\beta = \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}[/tex]
Consider the table below.
Compute the value of α and β as follows:
[tex]\alpha = \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 36.96 \cdot 34721037 - 20655 \cdot 60898.47}{ 13 \cdot 34721037 - 20655^2} \approx 1.028 \\ \\\beta = \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 13 \cdot 60898.47 - 20655 \cdot 36.96 }{ 13 \cdot 34721037 - \left( 20655 \right)^2} \approx 0.001\end{aligned}[/tex]
The regression of GPA based on SAT scores is:
[tex]y=1.028+0.001x[/tex]
(a)
Compute the GPA of a students whose SAT scores were 12000 as follows:
[tex]y=1.028+0.001x[/tex]
[tex]=1.028+(0.001\times 12000)\\=1.028+1.2\\=2.228\\\approx2.23[/tex]
Thus, the students whose SAT scores were 12000 has a GPA of 2.23.
(b)
Compute the GPA of a students whose SAT scores were 2400 as follows:
[tex]y=1.028+0.001x[/tex]
[tex]=1.028+(0.001\times 2400)\\=1.028+0.24\\=1.268\\\approx1.27[/tex]
Thus, the students whose SAT scores were 2400 has a GPA of 1.27.
