Statistics question please....

Question

Some colleges use SAT scores to predict freshmen GPA.A freshmen college class has a mean SAT score of 1217 with a standard deviation of 79. After one semester their GPA was 2.88 with a standard deviation of 54. The correlation between SAT and GPA was.49. How much of a variation in GPA can be explained by SAT scores?

Answer 1

To determine how much of the variation in GPA can be explained by SAT scores, we need to calculate the coefficient of determination, also known as R-squared (R²). The coefficient of determination represents the proportion of the variance in the dependent variable (GPA) that is predictable from the independent variable (SAT scores).

Let's solve this step by step:

1. We're given the correlation coefficient (r) between SAT and GPA: 0.49

2. The coefficient of determination (R²) is simply the square of the correlation coefficient:

R² = r²

R² = (0.49)²

R² = 0.2401

3. Convert this to a percentage:

0.2401 * 100 = 24.01%

Therefore, approximately 24.01% of the variation in GPA can be explained by SAT scores.

This means that about 24% of the differences in freshman GPAs can be accounted for by differences in their SAT scores. The remaining 76% of the variation is due to other factors not captured by SAT scores, such as study habits, motivation, course difficulty, or other personal or environmental factors.