Question

The article "Cost-Effectiveness in Public Education" (Chance [1995]: \(38-41\) ) reported that, for a sample of \(n=44\) New Jersey school districts, a regression of \(y=\) average SAT score on \(x=\) expenditure per pupil (thousands of dollars) gave \(b=15.0\) and \(s_{b}=5.3\). a. Does the simple linear regression model specify a useful relationship between \(x\) and \(y\) ? b. Calculate and interpret a confidence interval for \(\beta\) based on a \(95 \%\) confidence level.

Short answer

The regression model suggests that there is some relationship between expenditure per student and average SAT score. The confidence interval for the true slope (β) at the 95% confidence level is approximated around \(15.0 \pm 10.71\), indicating that with 95% confidence, for each additional thousand dollars spent per student, the average SAT score increases between approximately 4.29 and 25.71 points.

Step-by-step solution

Determine the utility of the regression model

The usefulness of the regression model can be determined by examining the coefficient of the independent variable (b). Here, b = 15.0, which indicates that on average, SAT score increases by 15 points for each additional thousand dollars spent per student. Since b is not zero, it suggests that there is some relationship between the expenditure per student and average SAT score according to the sample data.

Formulate the null and the alternative hypothesis

In the context of a confidence interval for the true slope (β), the hypotheses become: \n Null hypothesis (H0): β = 0 (\(x\) has no effect on \(y\)). \n Alternative hypothesis (Ha): β ≠ 0 (\(x\) does have an effect on \(y\)).

Calculate the test statistic

The test statistic to be calculated with b and \(s_{b}\) is: \n \(t = \frac{b - 0}{s_{b}} = \frac{15.0 - 0}{5.3}\)

Construct the confidence interval

For a 95% confidence interval, the critical t-value (t*) falls at the 2.5 and 97.5 percentiles (if you consult a t-table with degree of freedom = n-2 = 42). Assuming t* is approximately 2.021, the confidence interval for β is given by: \n \(b \pm (t* × s_{b})\) which is: \n \(15.0 \pm (2.021 × 5.3)\)

Key concepts

SAT Score Analysis

The analysis of SAT scores in relation to educational spending is a significant aspect of evaluating the effectiveness of financial investments in education. In our exercise, a statistical model indicates that an average SAT score increase corresponds to higher expenditure per pupil.

This relationship is quantified by the regression coefficient, which suggests that for each thousand-dollar increase in spending per pupil, SAT scores increase on average by 15 points. If this model is accurate, it can provide valuable insights for policy makers and educators into how educational funding could enhance student academic performance.

However, it is essential to understand that a correlation does not necessarily imply causation. Factors such as teaching quality, school facilities, and socioeconomic background could also play significant roles in SAT performance. Thus, while SAT score analysis through linear regression can offer important clues, it should be considered alongside other educational metrics for a comprehensive understanding.

Educational Expenditure Impact

Exploring the impact of educational expenditure on student outcomes is crucial in determining how effectively money is being used in the public education system. In our problem, the coefficient of expenditure per pupil is positive, which could imply that increased spending has a favorable effect on SAT scores.

However, one must delve deeper into how this money is spent; whether it is allocated towards improved learning materials, smaller class sizes, teacher training, or other educational resources can have varied impacts on student achievement.

The linear regression analysis provides a simplified view that can guide further investigation into which aspects of educational spending are most beneficial for enhancing student performance. It's also pertinent to consider the diminishing returns at certain expenditure levels, showing that simply pouring more money into the system may not always yield proportionate improvements.

School District Performance Evaluation

Evaluating school district performance often involves looking at standardized test scores such as the SAT. The results from our regression model, which show a positive relationship between spending and test scores, could be one of many indicators used to assess a district's effectiveness.

School districts can use such analysis to pinpoint areas of strength and weakness. By understanding the regression coefficient's implications, districts can make data-informed decisions to improve educational processes and outcomes.

However, it is imperative to incorporate other performance metrics such as graduation rates, college enrollment rates, and student progress over time to ensure a comprehensive evaluation that captures diverse facets of educational success.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method of making decisions using data, whether from a controlled experiment or observational study. In the context of our SAT score analysis, hypothesis testing is employed to determine if the observed association between educational spending and SAT scores is statistically significant.

The null hypothesis (H0: β = 0) posits that there is no effect of spending on SAT scores, while the alternative hypothesis (Ha: β ≠ 0) suggests there is an effect. By calculating the test statistic (a t-value in our case) and comparing it against a critical value from the t-distribution, we assess the plausibility of the null hypothesis.

Here, constructing a confidence interval for the population parameter (β) also informs us about the range of values that are likely to contain the true effect of spending on SAT scores. If the confidence interval does not contain the value specified in the null hypothesis (in this case, zero), we have evidence against H0, indicating a potentially significant relationship between the variables.