Thirteen students entered the business program at Hillcrest College 2 years ago. The following table indicates what each student scored on the high school SAT math exam and their grade-point averages (GPAs) after students were in the Hillcrest program for 2 years.
Student | SAT Score | GPA |
A | 421 | 2.90 |
B | 377 | 2.93 |
C | 585 | 3.00 |
D | 690 | 3.45 |
E | 608 | 3.66 |
F | 390 | 2.88 |
G | 415 | 2.15 |
H | 481 | 2.53 |
I | 729 | 3.22 |
J | 501 | 1.99 |
K | 613 | 2.75 |
L | 709 | 3.90 |
M | 366 | 1.60 |
- Is there a meaningful relationship between SAT math scores and grades?
Ans:
Scatter Diagram:
From the 13 data points, there appears to be a positive relationship between the independent variable (SAT math scores) and the dependent variable (grades): As SAT math scores increases, grades (GPA) tend to be higher.
- If a student scores a 350, what do you think his or her GPA will be?
Ans:
Student | SAT Score | GPA | x^{2} | ^{y2} | xy |
A | 421 | 2.9 | 177241 | 8.41 | 1220.9 |
B | 377 | 2.93 | 142129 | 8.5849 | 1104.61 |
C | 585 | 3 | 342225 | 9 | 1755 |
D | 690 | 3.45 | 476100 | 11.9025 | 2380.5 |
E | 608 | 3.66 | 369664 | 13.3956 | 2225.28 |
F | 390 | 2.88 | 152100 | 8.2944 | 1123.2 |
G | 415 | 2.15 | 172225 | 4.6225 | 892.25 |
H | 481 | 2.53 | 231361 | 6.4009 | 1216.93 |
I | 729 | 3.22 | 531441 | 10.3684 | 2347.38 |
J | 501 | 1.99 | 251001 | 3.9601 | 996.99 |
K | 613 | 2.75 | 375769 | 7.5625 | 1685.75 |
L | 709 | 3.9 | 502681 | 15.21 | 2765.1 |
M | 366 | 1.6 | 133956 | 2.56 | 585.6 |
Σx/n | 529.62 | ||||
Σy/n | 2.84 | ||||
Σx2 | 3857893 | ||||
Σxy | 20299 | ||||
Σy2 | 110.2718 | ||||
b | 0.003 | ||||
a | 1.028 | ||||
r ( Correlation Coefficient) | 0.692 | ||||
coefficient of determination (r2) | 0.479 |
We can find a mathematical equation by using the least-squares regression approach:
The estimated regression equation, therefore, is:
Y = 1.028 + 0.003 *X
Putting a value of x= 350 in the regression equation will give
y = 1.028 + 0.003 *350 = 2.078
If a student scores a 350, his or her GPA = 2.078
- c) What about a student who scores 800?
Y = 1.028 + 0.003 *X
Putting a value of x= 800 in the regression equation will give
y = 1.028 + 0.003 *800 = 3.428
If a student scores a 800, his or her GPA = 3.428