Question

The article "Examined Life: What Stanley H. Kaplan Taught Us About the SAT" (The New Yorker [December 17, 2001]: \(86-92\) ) included a summary of findings regarding the use of SAT I scores, SAT II scores, and high school grade point average (GPA) to predict first-year college GPA. The article states that "among these, SAT II scores are the best predictor, explaining 16 percent of the variance in first-year college grades. GPA was second at 15.4 percent, and SAT I was last at 13.3 percent." a. If the data from this study were used to fit a least squares regression line with \(y=\) first-year college GPA and \(x=\) high school GPA, what would be the value of \(r^{2} ?\) b. The article stated that SAT II was the best predictor of first-year college grades. Do you think that predictions based on a least-squares line with \(y=\) first-year college GPA and \(x=\) SAT II score would be very accurate? Explain why or why not.

Short answer

a. The value of \( r^{2} \) when high school GPA is used as the predictor is 0.154. b. Predictions based on SAT II scores would not be very accurate, despite being the best predictor among SAT I scores, SAT II scores and high school GPA. The reason for this is SAT II explains only 16% of the variance in first-year college GPA, leaving 84% unpredicted.

Step-by-step solution

Understanding and computing \( r^{2} \)

The value of \( r^{2} \) (coefficient of determination) represents the proportion of variance in the dependent variable (in this case, first-year college GPA) that is predictable from the independent variable (here, high school GPA). Here, the article already states that the high school GPA explains 15.4 percent of the variance. So, \( r^{2} = 15.4 \% = 0.154 \).

Analyzing the effectiveness of SAT II scores as a predictor

Although the article informs SAT II scores are the best predictor among the three, explaining 16% of the variance, it's necessary to understand that being the 'best' in this collection does not mean it is very accurate. An \( r^{2} \) value of 0.16 implies that only 16% of the variance in the dependent variable is predictable from SAT II scores, leaving a large percentage (84%) of uncertainty.

Key concepts

Coefficient of Determination

The coefficient of determination, denoted as \( r^2 \), is a key statistical measure in regression analysis. It quantifies how well the independent variable(s) explain the variability of the dependent variable. In simpler terms, it tells us the percentage of the variation in the dependent variable that can be accounted for by the model.

For example, in the context of predicting first-year college GPA based on high school GPA, an \( r^2 \) value of 0.154 indicates that approximately 15.4% of the variation in college GPA can be explained by the high school GPA. The remaining 84.6% could be affected by other factors such as study habits, course difficulty, or test-taking skills. Hence, while a certain level of prediction is possible, there's still a large amount of variation that the high school GPA alone cannot account for in predicting college performance.

SAT II Scores

SAT II scores, also known as SAT Subject Test scores, are exams that measure knowledge in specific subjects. They are designed to demonstrate a student's proficiency in areas such as Math, Science, or Languages and are often used by colleges for admissions or placement purposes.

In the study referenced, SAT II scores were found to be slightly better than high school GPA at predicting first-year college GPA. An important takeaway is that while SAT II exams might be strong in assessing subject-specific knowledge, they only accounted for 16% of the variance in college GPA. Meaning, there's a multitude of other factors that contribute to a student's performance in college, indicating that SAT II scores are a piece, but not the whole, of the academic achievement puzzle.

Predictive Analysis

Predictive analysis involves using historical data to make informed guesses about the future. In the context of educational outcomes, it's the practice of using variables such as standardized test scores and high school GPA to forecast a student's future academic performance.

The effectiveness of these predictions is often limited by the accuracy and relevance of the data used. For example, using SAT II scores and high school GPA to predict college GPA is helpful but imperfect. Variables like motivation, time management, and the rigor of a student's high school curriculum aren't captured by these measures, which can lead to less accurate predictions.

First-Year College GPA

First-year college GPA is an important metric for students, as it can set the tone for their entire undergraduate experience. It is often the focus of predictive analysis because it can have a lasting impact on opportunities such as scholarships, internships, and even employment after graduation.

When using predictive analysis to estimate first-year college GPA, it's important to remember that the models, although beneficial, have limitations. They often do not consider variables such as personal circumstances, engagement with faculty and peers, and adaptation to college life, which can significantly influence a student's academic performance.

High School GPA

High school GPA is a cumulative indicator of a student's overall academic performance during their high school career. It reflects not only a student's ability to understand and apply concepts but also their work ethic and consistency.

While high school GPA was the second-best predictor in the study mentioned, according to the coefficient of determination value, it still leaves a large portion of variance unexplained. This highlights the complexity of academic success and suggests that additional factors such as personal traits, learning styles, and social influences also play significant roles in a student’s progress in college.