Question

The data in MEAPSINGLE were used to estimate the following equations relating school-level performance on a fourth-grade math test to socioeconomic characteristics of students attending school. The variable free, measured at the school level, is the percentage of students eligible for the federal free lunch program. The variable medinc is median income in the ZIP code, and \(p c t s g l e\) is percent of students not living with two parents (also measured at the ZIP code level). See also Computer Exercise \(\mathrm{C} 11\) in Chapter \(3 .\) $$\begin{aligned} \widehat{\text {math} 4} &=96.77-.833 \text { pctsgle} \\ &(1.60)(.071) \\ n &=299, R^{2}=.380 \end{aligned}$$ $$\begin{aligned} \widehat{\text {math} 4}=& 93.00-.275 \text { pctsgle }-.402 \text { free} \\ &(1.63)(.117) \\ n &=299, R^{2}=.459 \end{aligned}$$ $$\begin{aligned} \widehat{\text {math} 4}=& 24.49-.274 \text { pctsgle }-.422 \text { free}-.752 \text { lmedinc }+9.01 \text { lexppp} \\ &(59.24)(.161) \\ n=& 299, R^{2}=.472 \end{aligned}$$ $$\begin{aligned} \widehat{\text {math} 4} &=17.52-.259 \text { pctsgle }-.420 \text { free}+8.80 \text { lexppp} \\ &(32.25)(.117) \\ n &=299, R^{2}=.472 \end{aligned}$$ i. Interpret the coefficient on the variable \(p\) ctsgle in the first equation. Comment on what happens when free is added as an explanatory variable. ii. Does expenditure per pupil, entered in logarithmic form, have a statistically significant effect on performance? How big is the estimated effect? iii. If you had to choose among the four equations as your best estimate of the effect of \(p c t s g l e\) and obtain a \(95 \%\) confidence interval of \(\beta_{\text {ptegle}},\) which would you choose? Why?

Short answer

1. The effect of \(\text{pctsgle}\) reduces when controlling for \(\text{free}\). 2. Expenditure per pupil significantly affects performance with a 9.01 coefficient. 3. Choose the fourth equation; it is simpler and performs as well as more complex models.

Step-by-step solution

Understanding Coefficient Interpretation

In the first equation, the coefficient of \(\text{pctsgle}\) is \(-0.833\). This means that for each 1% increase in students not living with two parents, the math test score is expected to decrease by 0.833 points. When the variable \(\text{free}\) is added in the second equation, the \(\text{pctsgle}\) coefficient becomes \(-0.275\), indicating that the previous coefficient was capturing some effect of \(\text{free}\) on the math scores.

Analyzing Statistical Significance

The coefficient of \(\text{lexppp}\) in the equation \(\widehat{\text{math} 4} = 24.49 - 0.274 \text{ pctsgle} - 0.422 \text{ free} - 0.752 \text{ lmedinc} + 9.01 \text{ lexppp}\) is 9.01. This suggests that a 1% increase in expenditure per pupil, when entered in logarithmic form, increases the math test score significantly. However, to determine statistical significance, we must check if the coefficient is greater than twice its standard error, which in this case is approximately \(\frac{9.01}{0.161} = 55.96\), confirming it is statistically significant.

Selecting the Best Model

We should choose the equation with the highest \(R^2\) and relevant predictors. Equations three and four both have the same \(R^2\) of 0.472. The fourth equation excludes \(\text{lmedinc}\), giving us a simpler model with a high adjusted \(R^2\). Therefore, equation four is better given simplicity and marginally higher performance metrics.

Computing the Confidence Interval

Using equation four, the coefficient for \(\text{pctsgle}\) is \(-0.259\) with a standard error of \(0.117\). A \(95\%\) confidence interval is calculated as: \([-0.259 - 1.96\times0.117, -0.259 + 1.96\times0.117]\), which yields \([-0.488, -0.030]\).

Key concepts

Socioeconomic Factors

Socioeconomic factors play a crucial role in education economics. They refer to the conditions that shape individuals' social and economic experiences and opportunities. In the context of school performance, these factors often include elements such as family income, parental education level, and household composition. In the equations provided, the variables \(\text{pctsgle}\) and \(\text{free}\) are indicators of socioeconomic factors. \(\text{pctsgle}\) measures the percentage of students not living with two parents, indicating potential familial instability, which can affect students' academic achievement. The variable \(\text{free}\) represents the percentage of students eligible for the federal free lunch program, acting as a proxy for low-income households. The influence of these factors is pivotal in understanding disparities in educational outcomes. For example, a greater proportion of students from single-parent households or eligible for free lunches may face challenges such as reduced parental involvement in education or financial constraints, which can result in lower mathematics test scores.

School Performance

School performance is often assessed through student outcomes, particularly test scores, which serve as a measure of the effectiveness of educational systems and policies. Test scores in subjects like mathematics are used to gauge students' academic achievements and preparedness for future education. In the given study, performance is measured by math test scores of fourth-grade students. These scores are affected by various factors, including socioeconomic conditions, as shown in the statistical equations. As these scores are pivotal indicators of educational success, understanding the determinants of school performance can help in formulating better educational policies to improve student learning. Efforts to enhance school performance are typically focused on addressing the challenges posed by the socioeconomic backgrounds of students. By implementing supportive measures, schools can bridge performance gaps and foster a more equitable learning environment.

Expenditure Per Pupil

Expenditure per pupil is a critical factor in education economics, reflecting how much is invested in each student's education. This measure can include costs such as teachers' salaries, educational materials, and infrastructure maintenance. In many studies, there's a significant correlation between how much is spent per student and their academic outcomes. In the equations analyzed, expenditure was entered logarithmically as \(\text{lexppp}\) and showed a notable positive effect on math test scores. This implies that even a small increase in log-transformed spending can markedly improve student performance, emphasizing the importance of adequate funding in education. When schools have more resources, they can provide better facilities, hire qualified teachers, and offer additional learning support, which collectively enhance the educational experience and outcomes of students.

Mathematics Education

Mathematics education focuses on teaching and learning mathematics and is a fundamental component of a high-quality education. Math skills are essential for academic and occupational success, making math education a priority in educational systems worldwide. The performance on math tests, like those discussed in the equations, is a barometer for understanding how well students grasp mathematical concepts. Mathematics education is influenced by various factors, including the quality of instruction, curriculum design, and students' prior knowledge. Improving mathematics education involves several strategies, such as employing skilled educators, incorporating technology to enhance learning, and creating an engaging curriculum that relates math to real-world contexts. These efforts aim to build strong foundational skills in mathematics, enabling students to excel and apply these skills in their future careers.