Problem 7 - An Introduction to Mathematical Statistics and Its Applications

Question

The relationship between school funding and student performance continues to be a hotly debated political and philosophical issue. Typical of the data available are the following figures, showing the per-pupil expenditures and graduation rate for twenty-six randomly chosen districts in Massachusetts. Graph the data and superimpose the least squares line, \(y=a+b x\). What would you conclude about the \(x y\) relationship? Use the following sums: $$ \begin{array}{ll} \sum_{i=1}^{26} x_{i}=360 & \sum_{i=1}^{26} y_{i}=2,256.6 \\ \sum_{i=1}^{26} x_{i}^{2}=5,365.08 & \sum_{i=1}^{26} x_{i} y_{i}=31,402 \end{array} $$

Step-by-Step Solution

Answer

The equation of the least squares line is \(y = 43.27 + 0.78x\). This indicates a positive correlation between per-student expenditures (x) and graduation rates (y), meaning that on average, graduation rates increase with higher expenditures.

Step 1: Calculation of the Slope and Intercept

The formula for calculating the slope \(b\) of the least squares line is: \(b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^{2}) - (\sum x)^{2}}\). The formula for calculating the intercept \(a\) of the least squares line is: \(a = \frac{\sum y - b(\sum x)}{n}\), where \(n\) is the total number of observations. Now we just have to substitute given values: \(b = \frac{26(31402) - (360 * 2256.6)}{26 * 5365.08 - 360^{2}} = 0.78\) and \(a = \frac{2256.6 - 0.78 * 360}{26} = 43.27\). So, the equation of the least squares line is \(y = 43.27 + 0.78x\).

Step 2: Analysis of the XY Relationship

To explore the relationship between x (expenditures) and y (graduation rates), let's look at the equation \(y = 43.27 + 0.78x\). The positive value of \(b (0.78)\) indicates that there's a positive correlation between expenditures and graduation rates, meaning that on average, with every increase in expenditures, graduation rates also increase.

Step 3: Visualization of the XY Relationship

Finally, to visualize this relationship, a scatter plot of the data points can be created with expenditures along the x-axis and graduation rates along the y-axis. The least squares line \(y = 43.27 + 0.78x\) can then be superimposed onto this scatterplot. Points that lie close to the line confirm the positive relationship between x and y.

Key Concepts

Scatter Plot

A scatter plot is a type of graph that represents individual pieces of data as points on a two-dimensional chart. Each point is determined by a pair of numbers corresponding to two variables; the first variable is plotted on the x-axis (horizontal), and the second on the y-axis (vertical). Scatter plots are particularly useful for observing and showing the relationship between two variables.

One way to interpret a scatter plot is by observing the general direction of the points. If the points seem to rise together, this suggests a positive relationship—meaning as one variable increases, so does the other. Conversely, if the points seem to fall together, there is likely a negative relationship. If there is no discernible pattern and the points are scattered randomly, the variables may not be related.

In our exercise, plotting expenditures against graduation rates would reveal the nature of their relationship. The purpose of this plot is to visually assess how well the variables correlate with each other, which is the first step in many data analysis tasks.

Correlation

Correlation, in the context of statistics, measures the strength of a linear relationship between two quantitative variables. It gives us an indicator of how one variable may change in response to another. If the correlation is high and positive, it indicates that the variables tend to increase together. If the correlation is negative, it suggests that as one variable increases, the other tends to decrease.

The correlation coefficient, often denoted as \( r \), can take on values from \( -1 \) to \( 1 \). A coefficient close to \( 1 \) signifies a strong positive correlation, close to \( -1 \) indicates a strong negative correlation, and around \( 0 \) suggests little or no linear correlation.

In the given problem, the positive slope in the least squares line suggests a positive correlation between school expenditures and graduation rates. However, to assess how strong this correlation is, we would look for the correlation coefficient, which goes beyond what is provided in the step-by-step solution but is nonetheless a crucial part of understanding the relationship between the two variables.

Linear Regression

Linear regression is a method used to model the relationship between a dependent variable and one (simple regression) or more (multiple regression) independent variables by fitting a linear equation to observed data. The least squares line, or the line of best fit, is the result of this method, minimizing the sum of the squares of the differences between the observed values and those predicted by the line.

The general form of the linear regression equation is \( y = a + bx \), where \( y \) is the dependent variable, \( x \) is the independent variable, \( b \) is the slope of the line, and \( a \) is the y-intercept. The slope indicates the change in \( y \) for a one-unit change in \( x \), while the intercept is the predicted value of \( y \) when \( x \) equals zero.

In our scenario, the calculation of the slope and intercept determined the least squares line. This line is an algebraic representation of the relationship between school funding and student performance. The positive slope from our exercise, \( 0.78 \), signifies that there is an expected increase in graduation rates of 0.78% for each additional dollar spent per pupil, according to the linear model.