Question

Let grad be a dummy variable for whether a student-athlete at a large university graduates in five years. Let \(h s G P A\) and \(S A T\) be high school grade point average and SAT score, respectively. Let study be the number of hours spent per week in an organized study hall. Suppose that, using data on 420 student-athletes, the following logit model is obtained: $$\widehat{\mathbf{P}}(g r a d=1 | h s G P A, S A T, s t u d y)=\Lambda(-1.17+.24 h s G P A+.00058 S A T+.073 \text { study })$$, where \(\Lambda(z)=\exp (z) /[1+\exp (z)]\) is the logit function. Holding \(h s G P A\) fixed at 3.0 and \(S A T\) fixed at \(1,200,\) compute the estimated difference in the graduation probability for someone who spent 10 hours per week in study hall and someone who spent 5 hours per week.

Short answer

The graduation probability increases by 0.068 when study hours increase from 5 to 10.

Step-by-step solution

Identify the fixed values

We are given that \( hsGPA = 3.0 \) and \( SAT = 1200 \). Define these as constants in the logit model and substitute these values.

Substitute fixed values into the equation

Substituting \( hsGPA = 3.0 \) and \( SAT = 1200 \) into the equation gives:\[\widehat{P}(grad = 1) = \Lambda(-1.17 + 0.24(3.0) + 0.00058(1200) + 0.073 \cdot study)\] This simplifies to:\[\widehat{P}(grad = 1) = \Lambda(0.55 + 0.073 \cdot study)\]

Compute probability with 10 hours of study

Plug \( study = 10 \) into the simplified equation: \[\widehat{P}(grad = 1) = \Lambda(0.55 + 0.073 \times 10) = \Lambda(1.28)\] Calculate the logistic function:\[\widehat{P}(grad = 1) = \frac{e^{1.28}}{1 + e^{1.28}} \approx 0.782\]

Compute probability with 5 hours of study

Plug \( study = 5 \) into the simplified equation: \[\widehat{P}(grad = 1) = \Lambda(0.55 + 0.073 \times 5) = \Lambda(0.915)\] Calculate the logistic function:\[\widehat{P}(grad = 1) = \frac{e^{0.915}}{1 + e^{0.915}} \approx 0.714\]

Calculate the difference in probabilities

The difference in graduation probabilities for 10 hours versus 5 hours of study is:\[0.782 - 0.714 = 0.068\]This shows the increase in probability when increasing study hours from 5 to 10.

Key concepts

Understanding Student-Athlete Performance

Student-athlete performance refers to how well student-athletes perform academically while balancing their athletic commitments. In this context, the performance is measured by whether a student graduates within a certain timeframe. Many factors contribute to the academic success of student-athletes, including their high school GPA, SAT scores, and additional influences such as study habits.

It's crucial to understand that student-athletes often face unique challenges compared to their non-athlete peers. They juggle rigorous training schedules and competitive pressures alongside their academic responsibilities. Thus, evaluating their performance requires considering both their academic outputs and the level of their athletic participation.

In studies like the one described, researchers use logistic regression to understand the likelihood of graduation within a set period, given several predictors or explanatory variables. This gives educational institutions insights into what can help improve academic achievement among student-athletes, which is valuable for developing support programs and resources tailored to their needs.

Role of a Dummy Variable in Research

In statistical modeling, particularly in logistic regression, a dummy variable is a numerical variable used to represent subgroups in the data. It takes the value of 0 or 1 to indicate the absence or presence of some categorical effect.

In this exercise, 'grad', the variable representing whether a student-athlete graduates in five years, is a dummy variable. This variable is crucial because it simplifies the analysis by converting a categorical outcome (graduate or not) into a form that can be easily used in a mathematical model.

Dummy variables are handy tools in regression models, as they allow researchers to include categorical data and analyze effects that might not otherwise be captured by numerical data alone. Consequently, they enhance the model's ability to reflect real-world scenarios and can be invaluable in producing meaningful predictions.

The Importance of High School GPA

High school GPA is a standard measure used to evaluate a student's academic abilities before attending college or university. In our logistic regression model, high school GPA is one of the crucial predictors for a student-athlete's likelihood to graduate.

High school GPA effectively reflects a student's consistent academic performance over many years, making it a reliable indicator of their ability to handle college-level coursework. It is often considered alongside standardized tests like the SAT to form a comprehensive view of a student's capabilities.

In the context of this logistic regression analysis, holding the high school GPA constant helps isolate the effects of other variables, such as the number of study hours. This approach allows for a more precise examination of how each factor contributes to graduation likelihood, by reducing potential variations and focusing on the specific elements under study.

SAT Score as a Predictor

The SAT score is another critical factor in predicting student success, and is widely used across the United States for college admissions. It measures a student's readiness for college by assessing skills in math, reading, and writing.

In the described logistic regression model, the SAT score is one of the predictors for the probability of a student-athlete graduating in five years. Alongside high school GPA, it offers a measure of academic preparedness, which can greatly influence how well a student copes with the challenges of higher education.

SAT scores provide a standardized benchmark that complements the high school GPA, allowing researchers to account for different educational backgrounds and systems. By including SAT scores in the regression model, the analysis can better control for academic preparedness, offering a clearer picture of the role other factors like study habits play in predicting graduation outcomes.

• Adding SAT scores helps adjust for differences in educational quality between schools.
• It provides a controlled insight into the non-academic factors affecting student-athlete performance.