Question

The following table presents the scores of 15 states on three variables. Compute \(r\) and \(r^{2}\) for each combination of variables. Assume that these 15 states are a random sample of all states, and test the correlations for their significance. Write a paragraph interpreting the relationship among these three variables. $$ \begin{array}{lccc} & & \text { Percent } & \\ & \begin{array}{c} \text { Per Capita } \\ \text { Expenditures } \\ \text { on Education, } \\ 2007 \end{array} & \begin{array}{c} \text { High } \\ \text { School } \\ \text { Graduates, } \\ \text { State } \end{array} & \begin{array}{c} \text { Rank } \\ \text { in Per } \\ \text { Capita } \\ \text { 2007 } \end{array} & \text { Income } \\ \hline \text { Arkansas } & 1546 & 80.4 & 14 \\ \text { Colorado } & 1792 & 88.9 & 6 \\ \text { Connecticut } & 2391 & 88.0 & 1 \\ \text { Florida } & 1672 & 84.9 & 8 \\ \text { Illinois } & 1887 & 85.7 & 5 \\ \text { Kansas } & 1913 & 89.1 & 9 \\ \text { Louisiana } & 1650 & 79.9 & 11 \\ \text { Maryland } & 1959 & 87.4 & 3 \\ \text { Michigan } & 1908 & 87.4 & 12 \\ \text { Mississippi } & 1313 & 78.5 & 15 \\ \text { Nebraska } & 1536 & 89.6 & 10 \\ \text { New Hampshire } & 1863 & 90.5 & 4 \\ \text { North Carolina } & 1456 & 83.0 & 13 \\ \text { Pennsylvania } & 1943 & 86.8 & 7 \\ \text { Wyoming } & 2712 & 91.2 & 2 \\ \hline \end{array} $$

Short answer

First, calculate mean values, deviations, and cross-products, then find the correlation coefficients and their significance.

Step-by-step solution

- Calculate Mean Values

Calculate the mean for each variable: Per Capita Expenditures on Education, High School Graduates Percent, and Rank in Per Capita Income.

- Calculate Deviations from the Mean

Subtract the mean from each individual value for all three variables to get the deviations.

- Compute Cross-Products

Multiply the deviations of one variable by the deviations of another variable for each combination.

- Sum the Cross-Products

Sum the cross-products for each combination of variables.

- Calculate Standard Deviations

Calculate the standard deviation for each variable.

- Compute Correlation Coefficients (r)

Divide the sum of the cross-products by the product of the standard deviations and the total number of data points minus 1.

- Compute Coefficient of Determination (r^2)

Square each correlation coefficient to find the coefficient of determination.

- Test for Significance

Use the t-test formula to determine if the correlation coefficients are significant.

- Interpret Relationships

Interpret the relationship among the variables by explaining both the magnitude and direction of the correlations.

Key concepts

Mean Calculation

To analyze the relationship among variables like per capita expenditures on education, high school graduation rates, and income ranks, we first need to calculate the mean. The mean represents the average value of each variable, and it's calculated by summing all values of a variable and dividing by the number of observations.
For example, for per capita expenditures on education, you would add up all expenditures from each state and divide by the number of states (15 in this case). It helps to get a sense of the central value around which other data points are distributed.

Standard Deviation

Standard deviation is a measure of the amount of variability or dispersion in a set of values. A low standard deviation means that the values are close to the mean, while a high standard deviation means the values are spread out over a wider range.
To calculate it:
1. Subtract the mean from each value to get the deviation.
2. Square each deviation.
3. Sum these squared deviations.
4. Divide by the number of values minus 1 (known as the degrees of freedom).
5. Take the square root of that result.
This helps to understand how spread out the data is, which is crucial before moving on to correlation analysis.

Correlation Coefficient

The correlation coefficient, represented as \(r\), quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1.
Steps to calculate \(r\):
1. Compute the deviations of each value from their respective means.
2. Find the cross-products of these deviations.
3. Sum these cross-products.
4. Divide by the product of the standard deviations of the two variables and the number of observations minus 1.
An \(r\) close to 1 implies a strong positive correlation, \(r\) close to -1 means a strong negative correlation, and \(r\) near 0 suggests no linear correlation.

Coefficient of Determination

The coefficient of determination, denoted \(r^2\), tells us the proportion of the variance in the dependent variable that is predictable from the independent variable. It is simply the square of the correlation coefficient \(r\).
If \(r = 0.8\), \(r^2 = 0.64\). This means 64% of the variance in one variable is explained by the variance in the other variable.
Understanding \(r^2\) helps in quantifying how well one variable predicts another, giving a clearer picture of the relationship's strength.

Statistical Significance

Statistical significance determines if the correlation observed is real or occurred by chance. For correlations, we typically use a t-test. The formula involves the correlation coefficient \(r\), the number of data points \(n\), and the degrees of freedom.
Steps are:
1. Calculate the t-value using the formula: \[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \]
2. Compare the t-value with critical values from the t-distribution table at a chosen significance level (typically 0.05).
If the calculated t-value exceeds the critical value, the correlation is considered statistically significant. This tells us that the observed correlation is unlikely to be due to random chance.