Question

How well does a student's Verbal SAT score (on an 800 -point scale) predict future college grade point average (on a four-point scale)? Computer output for this regression analysis is shown, using the data in StudentSurvey: The regression equation is \(\mathrm{GPA}=2.03+0.00189\) VerbalSAT Analysis of Variance \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Regression } & 1 & 6.8029 & 6.8029 & 48.84 & 0.000 \\ \text { Residual Error } & 343 & 47.7760 & 0.1393 & & \\ \text { Total } & 344 & 54.5788 & & & \end{array}\) (a) What is the predicted grade point average of a student who receives a 550 on the Verbal SAT exam? (b) Use the information in the ANOVA table to determine the number of students included in the dataset. (c) Use the information in the ANOVA table to compute and interpret \(R^{2}\). (d) Is the linear model effective at predicting grade point average? Use information from the computer output and state the conclusion in context.

Short answer

a) The predicted GPA of a student who received a 550 on the Verbal SAT would be 3.0695. b) The dataset included 345 students. c) The \(R^{2}\) value is 0.1247, meaning about 12.47% of the GPA variance can be explained by SAT scores. d) The model is statistically significant, evidenced by the low P-value and high F-value; however, the \(R^{2}\) value suggests the model may not greatly predict GPA accurately using only SAT scores.

Step-by-step solution

Predict the GPA

Substitute the `Verbal SAT` value of `550` into the regression formula \(\mathrm{GPA}=2.03+0.00189\) VerbalSAT: the grade point average (GPA) is \(2.03+0.00189*550\).

Determine the number of students

The residual degrees of freedom (DF) is the number of data points minus 2 (one for the regression and one for the constant term). The ANOVA Table shows residual DF as `343`, so the number of students in the dataset is `343+2=345`.

Calculate and interpret \(R^{2}\)

\(\mathrm{R^{2}} = \frac{\mathrm{SS Regression}}{\mathrm{Total SS}} = \frac{6.8029}{54.5788}\). \(R^{2}\) represents the proportion of the variance for the dependent variable (GPA) that's explained by the independent variable (Verbal SAT). A higher \(R^{2}\) indicates a stronger relation between GPA and SAT scores.

Evaluate the model's effectiveness

The model's effectiveness can be evaluated based on the F value, the P-value, and the \(R^{2}\) value. A large F value (48.84), and a very small P-value (0.000) suggest that the regression model is statistically significant. However, while the \(R^{2}\) value indicates a certain degree of correlation, further evaluations need to be made to determine how precisely the SAT scores can predict GPA.

Key concepts

SAT Score Prediction

When it comes to understanding how well a standardized test score like the SAT can predict a student's future academic performance, regression analysis is a valuable tool. In our exercise, the analysis aims to predict college grade point average (GPA) based on a student's Verbal SAT scores. A regression equation, such as \( \text{GPA} = 2.03 + 0.00189 \times \text{VerbalSAT} \), represents the relationship, with the coefficients indicating how much GPA is expected to increase with each point increase in Verbal SAT scores.

To use the equation for prediction, you simply substitute the specific SAT score into the equation. For example, with a Verbal SAT score of 550, the predicted GPA would be calculated as \( 2.03 + 0.00189 \times 550 \). This equates to a GPA prediction based on the provided regression model. Remember that this model assumes a linear relationship between SAT scores and GPA and does not account for other potential factors that might influence a student's GPA.

Grade Point Average (GPA)

Understanding GPA is essential when discussing academic performance. The Grade Point Average is a standardized method of assessing a student's academic achievement. It is often on a four-point scale, where 4.0 represents an 'A' average. In the context of our regression analysis exercise, GPA serves as the dependent variable, meaning it's the outcome we're trying to predict based on the independent variable (Verbal SAT scores).

When interpreting the data from an educational perspective, it's important to note that while the SAT score is a significant predictor, it's only one of many factors that can contribute to a student's GPA. Other variables, like study habits, course difficulty, or extracurricular activities, are also critical but are not included in the regression model.

ANOVA Table

An Analysis of Variance (ANOVA) table is a fundamental tool in statistics used to analyze the differences among group means in a sample. In the realm of regression, the ANOVA table breaks down the variance in the dependent variable into components attributable to the model (regression) and the residual variance (error).

This breakdown includes degrees of freedom (DF), sum of squares (SS), mean square (MS), the F statistic (F), and the significance level (P). In simple terms, the ANOVA table helps us determine how well our model explains the variation in the data. If the P-value is sufficiently low (commonly less than 0.05), we can reject the null hypothesis that the regression coefficient is zero, suggesting the model has statistical significance.

R-Squared Interpretation

The \( R^2 \) value, or coefficient of determination, is a statistical measure of how close the data are to the fitted regression line. It is the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

In the given exercise, we calculate \( R^2 \) by dividing the regression SS by the total SS. A higher \( R^2 \) usually indicates a better fit for the model to the data. However, it's essential to understand that while it tells us the proportion of variance explained by the model, it does not tell us about the accuracy of the predictions or whether the model is appropriate for the data. For instance, a high \( R^2 \) value in the context of GPA prediction might look promising, but we must also consider the practical significance and the real-world applicability when advising students or predicting academic outcomes.