Question

graduation rate is one measure used to compare colleges in national publications. one such publication compared semester tuition against graduation rate, defined as the percentage of students who graduate within four years. the value of r for the scatterplot is 0.856. how would the correlation change if the graduation rate was plotted on the x - axis and tuition plotted on the y - axis? the correlation would stay the same. the correlation would stay positive and increase. the correlation would stay positive and decrease. the correlation would be negative, rather than positive.

Explanation:

The correlation coefficient \( r \) measures the strength and direction of a linear relationship between two variables. Swapping the variables on the axes (i.e., changing which variable is \( x \) and which is \( y \)) does not change the value of \( r \). The formula for the correlation coefficient \( r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\) is symmetric with respect to \( x \) and \( y \). If we let \( x\) be the original \( y\) - variable (tuition) and \( y\) be the original \( x\) - variable (graduation rate), the numerator \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\) remains the same (since multiplication is commutative: \(ab = ba\)), and the denominator \(\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}\) also remains the same.

Answer:

The correlation would stay the same.