Question

The following analysis was obtained using data in MEAP93, which contains school-level pass rates (as a percent) on a tenth-grade math test. i. The variable expend is expenditures per student, in dollars, and \(m a t h 10\) is the pass rate on the exam. The following simple regression relates \(m\) ath 10 to lexpend \(=\log (\text {expend})\) : $$\begin{aligned} \widehat{\text {math} 10} &=-69.34+11.16 \text { lexpend} \\ &(25.53) \quad(3.17) \\ n &=408, R^{2}=.0297 \end{aligned}$$ Interpret the coefficient on lexpend. In particular, if expend increases by \(10 \%,\) what is the estimated percentage point change in math10? What do you make of the large negative intercept estimate? (The minimum value of lexpend is 8.11 and its average value is \(8.37 .\) ) ii. Does the small \(R\) -squared in part (i) imply that spending is correlated with other factors affecting math10? Explain. Would you expect the \(R\) -squared to be much higher if expenditures were randomly assigned to schools- that is, independent of other school and student characteristics-rather than having the school districts determine spending? iii. When log of enrollment and the percent of students eligible for the federal free lunch program are included, the estimated equation becomes $$\begin{aligned} \widehat{\text {math} 10} &=-23.14+7.75 \text { lexpend}-1.26 \text { lenroll }-.324 \text { lnchprg} \\ &(24.99)(3.04) \\ n &=408, R^{2}=.1893 \end{aligned}$$ Comment on what happens to the coefficient on lexpend. Is the spending coefficient still statistically different from zero? iv. What do you make of the \(R\) -squared in part (iii)? What are some other factors that could be used to explain math10 (at the school level)?

Short answer

The coefficient on lexpend indicates a 1.063 percentage point increase in pass rate for a 10% spending increase. Factors beyond spending are crucial for explaining pass rates, shown by the model's R-squared values.

Step-by-step solution

Interpret the Coefficient on lexpend

The coefficient of 11.16 on lexpend in the regression equation indicates the expected change in the pass rate (math10) when log(expenses per student) increases by one unit. Since this is in a log-linear model, a 1% increase in expend translates to a 0.1116 percentage point increase in math10. To find the effect of a 10% increase, the change in math10 is 11.16 * log(1.10), which approximates to 11.16 * 0.0953 = 1.063 percentage points.

Address the Large Negative Intercept

The large negative intercept of -69.34 does not have a straightforward interpretation in this context as it theoreticially gives a pass rate when lexpend is zero, which is outside the range of observed data. This reflects either extrapolation beyond the data set or the coefficient simply serves to anchor the regression line within the observed range of lexpend values.

Discuss the Small R-squared for the Simple Regression

An R-squared of 0.0297 in part (i) suggests that only about 2.97% of the variation in math10 is explained by the log of expenditures alone. However, it does not necessarily imply that spending is uncorrelated with other factors that affect math10; it may indicate that expenditures alone are not sufficient to explain the variation in pass rates. If expenditures were randomly assigned, R-squared might be higher, since it would indicate the pure effect of spending without confounding factors.

Analyze the Updated Model with Additional Variables

In the revised regression, the coefficient on lexpend decreases to 7.75. The presence of additional variables likely controls for factors correlated with both expenditures and pass rates, isolating the effect of expenditures itself. With a t-statistic of about 7.75 / 3.04 ≈ 2.55, the coefficient is still statistically different from zero at a significant level (p < 0.05).

Evaluate the R-squared in the Updated Model

The R-squared increases to 0.1893, indicating that about 18.93% of the variation in math10 is explained by the model with more variables, suggesting a better fit. Factors like teacher quality, school infrastructure, parental involvement, and socioeconomic status may also explain variations in math10 pass rates.

Key concepts

Regression Analysis

Regression analysis is a powerful statistical technique used to understand the relationship between variables. In our example, the regression equation connects school-level test pass rates with the log of expenditures per student. This relationship allows us to estimate how changes in spending per student might influence math test pass rates.
The regression line, given by the equation, mathematically predicts the dependent variable (math10, or pass rate) based on the independent variable (lexpend, or log of expenditures).
Using the coefficient from the regression equation, we can interpret the impact of spending on test pass rates. For example, a 1% increase in expenditures leads to an approximate 0.1116 percentage point rise in pass rates.
An important thing to note is the concept of intercepts, which can sometimes appear non-intuitive, such as the negative intercept in our study. However, this mainly anchors the regression line within the observed data range without considerable practical interpretation when extreme values are unrealistic.
Such an analysis helps us predict outcomes, assess the strength of relationships, and provides a quantitative basis for decision-making.

Education Funding

Education funding plays a crucial role in determining the quality and resources available to students. In the context of the regression analysis, the focus is on how expenditures per student affect educational outcomes. Although spending is just one piece of the puzzle, it is often viewed as a critical component.
The reason spending is critical is simple: more resources typically enable better education materials, qualified teachers, and improved facilities, which can contribute to better student performance. However, the effect of funding on outcomes like math test pass rates is not always straightforward.
In the regression model, as discussed, funding per student is captured by expenditures. But the results show low R-squared values, indicating that spending by itself explains only a small portion of variation in student outcomes. This suggests a need for deeper analysis, considering other influencing factors beyond just expenditures.

Statistical Modeling

Statistical modeling involves creating mathematical representations of real-world processes to predict or describe outcomes. In our study, statistical models were used to determine how different factors like expenditure, enrollment, and percentage of students eligible for free lunch affect test pass rates.
The models help in understanding complex relationships by isolating the effect of one variable while accounting for others, showcasing the multifaceted nature of real-world educational data.
For instance, in our updated regression model with additional variables, the effect of spending becomes clearer as other influential factors are included. This more holistic model provides a better fit to the data, as evidenced by the increased R-squared value, suggesting the model explains more variability in the pass rates.
Statistical models are thus instrumental in refining our understanding and forming evidence-based strategies. By carefully selecting and incorporating relevant factors, these models can shed light on the underpinnings of educational success.

School Performance

School performance, often measured by student outcomes on standardized tests, is influenced by a variety of factors. In our exercise, the performance is linked to funding, student socio-economic backgrounds, and other school-specific characteristics.
Regression analysis offers insight, but school performance cannot solely be understood through funding. Other factors like teacher quality, class sizes, and student-teacher ratios need consideration. Furthermore, aspects like parental involvement and community support also play roles in shaping outcomes.
In the expanded model, including variables such as enrollment and free lunch eligibility, we get a better measure of school performance drivers. This demonstrates that school success is multifaceted, requiring a comprehensive approach to fully understand and improve it.
As a result, enhancing school performance involves addressing an array of factors holistically rather than focusing on a single influence, such as funding. Each school's unique context and challenges must be carefully considered when devising strategies for improvement.