Question

Suppose the entering freshmen at a certain college have a mean combined SAT score of 1230, with a standard deviation of 127. In the first semester, these students attained a mean GPA of 2.66, with a standard deviation of 0.52. A scatterplot showed the association to be reasonably linear, and the correlation between SAT score and GPA was 0.45. What SAT score would you predict a freshman who attained a first-semester GPA of 2.1 would have gotten? Note that in this case, the explanatory variable is the student's GPA and the response variable is their SAT score.

The freshman is expected to have had an SAT score of ___
(Round to the nearest integer as needed.)

Answer 1

To predict the SAT score of a freshman with a first-semester GPA of 2.1, we can use linear regression based on the correlation between SAT score and GPA. The predicted SAT score would be 1230.

Step-by-step explanation:

To predict the SAT score of a freshman with a first-semester GPA of 2.1, we can use the linear regression equation based on the correlation between SAT score and GPA. The equation takes the form: SAT score = constant + slope × GPA. In this case, the constant is the mean SAT score of 1230, and the slope is calculated by multiplying the correlation coefficient (0.45) by the standard deviation of SAT scores (127) divided by the standard deviation of GPAs (0.52). Substituting the values into the equation, we get:

SAT score = 1230 + (0.45 × 127/0.52) × (2.1 - mean GPA)

Calculating the value gives:

SAT score ≈ 1230 + (0.872 × (2.1 - 2.66)) ≈ 1230 - 0.872 × 0.56 ≈ 1229.511

Rounding to the nearest integer, the predicted SAT score would be 1230.

Answer 2

To predict the SAT score of a freshman with a GPA of 2.1, we can use a linear regression equation. The slope and intercept of the equation can be calculated using the mean SAT score, mean GPA, standard deviation of SAT scores, and correlation between SAT score and GPA. By plugging in the GPA of 2.1, the predicted SAT score is approximately 1152.69.

Step-by-step explanation:

To predict the SAT score of a freshman who attained a first-semester GPA of 2.1, we can use the linear regression equation between SAT score and GPA. The equation is: SAT score = intercept + slope * GPA. The intercept and slope can be calculated using the mean SAT score, mean GPA, standard deviation of SAT scores, and the correlation between SAT score and GPA.

First, we need to find the slope: slope = correlation * (std deviation of SAT scores / std deviation of GPA). In this case, slope = 0.45 * (127 / 0.52) ≈ 110.625.

Next, we can find the intercept: intercept = mean SAT score - slope * mean GPA. Using the values given in the question, intercept = 1230 - 110.625 * 2.66 ≈ 919.354.

Now we can plug in the GPA of 2.1 into the linear regression equation to predict the SAT score: SAT score = 919.354 + 110.625 * 2.1 ≈ 1152.69.