Question

Using the data in GPA2 on 4,137 college students, the following equation was estimated by OLS: $$ \begin{aligned} \widehat{\text {colgpa}} &=1.392-.0135 \text { hsperc }+. .00148 \text { sat } \\\ n &=4.137, R^{2}=.273 \end{aligned} $$ where colgpa is measured on a four-point scale, hsperc is the percentile in the high school graduating class (defined so that, for example, hsperc \(=5\) means the top \(5 \%\) of the class), and sat is the combined math and verbal scores on the student achievement test. i. Why does it make sense for the coefficient on \(h s p e r c\) to be negative? ii. What is the predicted college GPA when hsperc \(=20\) and \(s a t=1,050 ?\) iii. Suppose that two high school graduates, A and B, graduated in the same percentile from high school, but Student A's SAT score was 140 points higher (about one standard deviation in the sample). What is the predicted difference in college GPA for these two students? Is the difference large? iv. Holding hsperc fixed, what difference in SAT scores leads to a predicted colgpa difference of \(.50,\) or one-half of a grade point? Comment on your answer.

Short answer

i. Higher `hsperc` indicates lower class ranking, inversely affecting GPA. ii. Predicted GPA is 2.676. iii. GPA difference for 140 SAT points is 0.2072; the impact is small. iv. A 338 SAT point difference leads to 0.50 GPA change, indicating importance of SAT scores.

Step-by-step solution

Understand Why Coefficient is Negative

The coefficient on `hsperc` is negative (-0.0135) in the equation. A higher value of `hsperc` means the student is ranked lower in their class (e.g., `hsperc` = 1 is at the top of the class, whereas `hsperc` = 100 is at the bottom). Students with higher rankings (lower `hsperc`) are generally expected to have better academic performance in college, thus a negative relationship between `hsperc` and `colgpa` makes sense.

Calculate Predicted GPA for Given hsperc and sat

To find the predicted college GPA for `hsperc = 20` and `sat = 1,050`, substitute these values into the equation:\[\text{colgpa} = 1.392 - 0.0135 \times 20 + 0.00148 \times 1050\]Calculate:\[-0.0135 \times 20 = -0.27\]\[0.00148 \times 1050 = 1.554\]\[\text{colgpa} = 1.392 - 0.27 + 1.554 = 2.676\]

Predict Difference in GPA with Different SAT Scores

To find the difference in GPA when SAT scores differ by 140 points, use the coefficient of `sat`: 0.00148. Multiply by 140 to find the difference:\[0.00148 \times 140 = 0.2072\]Thus, the predicted difference in college GPA between the two students is 0.2072. This difference is relatively small compared to a typical GPA scale.

Find SAT Score Difference for 0.50 GPA Change

We need to find the SAT score difference that results in a 0.50 change in GPA. Using the coefficient of `sat` (0.00148), solve for the change in SAT that satisfies:\[0.00148 \times \Delta \text{sat} = 0.50\]\[\Delta \text{sat} = \frac{0.50}{0.00148} \approx 337.84\]Thus, an increase in SAT score of approximately 338 points is needed to achieve a half-point GPA increase, reflecting the significant impact of SAT performance.

Key concepts

Ordinary Least Squares (OLS)

Ordinary Least Squares (OLS) is a statistical method used to estimate the relationships between variables. It is one of the simplest and most commonly used techniques in regression analysis. The fundamental aim of OLS is to minimize the sum of the squared differences between observed data points and the values predicted by the linear model.

Here's how it works:

• OLS estimates the parameters (coefficients) of a linear equation, which can describe the relationship between independent variables and a dependent variable.
• It assumes a linear relationship, meaning changes in the independent variables are proportional to changes in the dependent variable.
• The method applies to continuous data and is most effective when the assumptions of linearity, independence, homoscedasticity (constant variance of errors), and normality (errors are normally distributed) are met.

In our problem, OLS is used to predict college GPA based on high school percentile and standardized test scores, giving an equation that helps understand how these factors predict academic success in college.

High School Percentile

High School Percentile (hsperc) refers to a student's ranking within their high school graduating class, expressed as a percentile. It is an indicator of a student's academic standing relative to their peers.

The percentile system works as follows:

• If a student is in the 5th percentile, they are among the top 5% of their class. Similarly, being in the 50th percentile means they are at the median of their class ranking.
• A lower hsperc value indicates a better class rank, while a higher value implies a lower ranking.

In our regression analysis, the negative coefficient for hsperc implies that as a student's percentile rank increases (they're ranked lower compared to peers), their predicted college GPA decreases. This makes intuitive sense, as students with stronger high school performance are expected to continue their success in college.

Standardized Test Scores

Standardized Test Scores, often represented in studies as the SAT scores, are designed to evaluate a student's readiness for college. They offer a consistent metric to compare different students' academic talents across the nation.

Key points about SAT scores:

• The SAT measures skills in math, reading, and writing, providing a cumulative score.
• In our regression model, a higher SAT score positively correlates with a better predicted college GPA, reflected by the positive coefficient in the equation.
• Standardized test scores offer a straightforward quantitative way to assess expected college performance, complementing high school performance metrics like the high school percentile.

Our model uses the coefficient for SAT to show that for every additional SAT point, there's a predicted increase in the college GPA, indicating the importance of these scores in predictive modeling.

Predictive Modeling

Predictive Modeling involves using statistical techniques to predict future outcomes based on historical data. It's a tool used to understand patterns and relationships in datasets.

Here's how it applies:

• In the context of regression analysis, predictive modeling helps understand how different factors, like high school percentile and SAT scores, influence a student's projected college GPA.
• The model allows for the calculation of predicted GPA under various scenarios, providing insights into how changes in input variables might affect the outcome.
• It emphasizes the importance of each variable in forecasting the dependent variable, contributing to evidence-based decision-making.

By applying predictive modeling in educational settings, educators and policymakers can design strategies to improve student outcomes and allocate resources where they are most needed. In our example, understanding the significant impact of SAT scores and high school performance helps tweak admission criteria or academic interventions.