Question

The article "Cost-Effectiveness in Public Education" (Chance [1995]: \(38-41\) ) reported that for a regression of \(y=\) average SAT score on \(x=\) expenditure per pupil, based on data from \(n=44\) New Jersey school districts, \(a=766, b=0.015, r^{2}=.160\), and \(s_{e}=53.7\) a. One observation in the sample was \((9900,893)\). What average SAT score would you predict for this district, and what is the corresponding residual? b. Interpret the value of \(s_{e}\). c. How effectively do you think the least-squares line summarizes the relationship between \(x\) and \(y ?\) Explain your reasoning.

Short answer

Predicted SAT score for a district with an expenditure per pupil of $9900 is \(y = 766 + 0.015(9900)\). The corresponding residual would be \(y - y'\). The standard error of 53.7 indicates the average error in estimating the SAT score from the expenditure per pupil. The r^2 value of 0.160 implies that 16% of the variation in the SAT scores is explained by the linear relationship between SAT score and expenditure per pupil, suggesting that the regression line may not effectively summarize the relationship between expenditure per pupil and SAT score.

Step-by-step solution

Predict the SAT Score

To predict a SAT score for a district with an expenditure per pupil of $9900, we use the equation of the regression line: \(y = a + bx\). Substituting the given values a = 766, b = 0.015, and x = 9900, we obtain \(y = 766 + 0.015(9900)\).

Calculate the Residual

The residual (y - y') is the difference between the actual y-value (given as 893 in the sample observation) and the predicted y-value you calculated in step 1. This gives the amount by which the observed SAT score differs from the predicted SAT score.

Interpret the Standard Error

The standard error (s_{e} = 53.7) measures the standard deviation of the residuals, or the average amount that the SAT scores deviate from the regression line. This indicates the typical error in estimating the y-value of the regression line at a given x.

Evaluate the Effectiveness of the Least-Squares Line

The coefficient of determination (r^{2} = 0.160) gives us the proportion of the total variation in the SAT scores that can be explained by the linear relationship between SAT score and expenditure per pupil. This is a key measure to assess the effectiveness of the line.

Key concepts

SAT score prediction

Predicting SAT scores based on various factors is an essential application of simple linear regression in education. Regression helps educators and policy makers understand the relationship between educational inputs, like school expenditures, and outputs, like test scores. In the provided exercise, the prediction is made using the regression equation \( y = a + bx \), where \( a \) is the y-intercept and \( b \) is the slope of the line.

This equation represents the expected performance based on funding, and by substituting the district's expenditure per pupil and the established coefficients, a predicted SAT score is obtained. This predictive capability is vital for allocating resources efficiently and setting realistic performance expectations.

Regression analysis in education

Regression analysis offers powerful insights into the educational field, enabling stakeholders to assess and model the relationships between different educational variables. This statistical tool involves identifying how a dependent variable, like average SAT scores, is affected by one or more independent variables, such as school spending.

The process involves fitting a linear model to observed data and using it to predict or infer relationships. By analyzing these relationships, educators can formulate strategies to improve student outcomes, justify budget allocations, and measure the impact of educational programs.

Residual calculation

Residuals are differences between observed values and values predicted by a regression model. They are crucial in diagnosing the fit of the model. In the textbook's exercise, the residual for a specific data point is found by subtracting the predicted SAT score from the actual score.

Mathematically, it is expressed as \( e = y - \hat{y} \), where \( y \) is the actual value and \( \hat{y} \) is the predicted value. A small residual indicates that the predicted value is close to the actual value. Larger residuals suggest potential outliers or that the model may not capture all pertinent variables influencing the outcome.

Standard error

The standard error in regression analysis, denoted as \( s_e \) in our exercise, quantifies the precision of the regression model. It measures the average distance that the observed values fall from the regression line. A lower standard error implies that the model's predictions are more precise and that data points tend to be closer to the line.

In educational settings, understanding standard error helps evaluate the reliability of the predictions. A high standard error could indicate that the model may not be the best fit for the data, or that there is high variability in the data that isn't explained by the model.

Coefficient of determination

The coefficient of determination, represented as \( r^2 \) in statistical formulas, is a critical value that indicates the proportion of variability in the dependent variable that can be explained by the independent variable. In this context, it tells us how well expenditures per pupil explain the variation in SAT scores across districts.

An \( r^2 \) value of 0.160 means that 16% of the variability in SAT scores can be accounted for by the funding model. While this leaves a significant portion of the variance unexplained, it still provides valuable insights. It can also signify that other factors may significantly impact SAT scores beyond school funding. Understanding this helps educators recognize the limitations of funding as a single intervention for academic achievement.