Question

Thirteen students entered the undergraduate business program at Rollins College 2 years ago. The following table indicates what their grade-point averages (GPAs) were after being in the program for 2 year and what each student scored on the SAT exam (maximum of 2400) when he or she was in high school. Is there a meaningful relationship between grades and SAT scores? If a student scores a 12000 on the SAT, what do you think his or her GPA will be? What about a student who scores 2400?

Student/ SAT Score /GPA

A. 1263 2.90
B. 1131 2.93
C. 1755 3.00
D. 2070 3.45
E. 1824 3.66
F. 1170 2.88
G. 1245 2.15
H. 1443 2.53
I. 2187 3.22
J. 1503 1.99
K. 1839 2.75
L. 2127 3.90

Answer 1

(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.

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:

`y=\alpha +\beta x`

The formula for α and β are:

`\alpha = \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}`

`\beta = \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}`

Consider the table below.

Compute the value of α and β as follows:

`\alpha = \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = ( 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} = ( 13 \cdot 60898.47 - 20655 \cdot 36.96 )/( 13 \cdot 34721037 - \left( 20655 \right)^2) \approx 0.001\end{aligned}`

The regression of GPA based on SAT scores is:

`y=1.028+0.001x`

(a)

Compute the GPA of a students whose SAT scores were 12000 as follows:

`y=1.028+0.001x`

`=1.028+(0.001* 12000)\\=1.028+1.2\\=2.228\\\approx2.23`

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:

`y=1.028+0.001x`

`=1.028+(0.001* 2400)\\=1.028+0.24\\=1.268\\\approx1.27`

Thus, the students whose SAT scores were 2400 has a GPA of 1.27.