"[APPLET] For Exercises 2–9, use the data in the table, which shows the average annual salaries (both in thousands of dollars) for librarians and postsecondary library science teachers in the United States for 12 years. (Source: U.S. Bureau of Labor Statistics)

7. Find the coefficient of determination r^2 and interpret the result."

Verified step by step guidance

1. Step 1: Understand that the coefficient of determination, denoted as r², measures the proportion of the variance in the dependent variable (library science teachers' salaries, y) that is predictable from the independent variable (librarians' salaries, x).
2. Step 2: Calculate the correlation coefficient r between x and y. This involves finding the covariance of x and y, and dividing it by the product of their standard deviations. The formula is: $r=\; \backslash (\backslash frac\{\backslash text\{cov\}\backslash )(x,y)\}\{s\_x\; s\_y\}$.
3. Step 3: To find covariance, calculate the mean of x and y, then compute the average product of their deviations from their means: $\backslash (\backslash text\{cov\}\backslash )(x,y)\; =\; \backslash (\backslash frac\{1\}\{n\}\backslash )\; \backslash (\backslash sum\backslash )\_\{i=1\}^n\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))(y\_i\; -\; \backslash (\backslash bar\{y\}\backslash ))$.
4. Step 4: Calculate the standard deviations of x and y using: $s\_x\; =\; \backslash (\backslash sqrt\{\backslash frac\{1\}\{n\}\backslash )\; \backslash (\backslash sum\backslash )\_\{i=1\}^n\; (x\_i\; -\; \backslash (\backslash bar\{x\}\backslash ))^2\}$ and similarly for $s\_y$.
5. Step 5: Square the correlation coefficient r to get r², the coefficient of determination. Interpret r² as the percentage of variation in y explained by x.

Key Concepts

Coefficient of Determination (r²)

The coefficient of determination, denoted as r², measures the proportion of the variance in the dependent variable that is predictable from the independent variable. It ranges from 0 to 1, where a higher value indicates a better fit of the regression model to the data. For example, an r² of 0.85 means 85% of the variation in salaries of library science teachers can be explained by the salaries of librarians.

Linear Regression and Correlation

Linear regression models the relationship between two variables by fitting a line that minimizes the sum of squared differences between observed and predicted values. Correlation measures the strength and direction of a linear relationship between variables, ranging from -1 to 1. The coefficient of determination is the square of the correlation coefficient, linking these concepts closely.

Interpreting Statistical Results in Context

Interpreting r² involves understanding what the value means in the context of the data. It helps assess how well one variable explains another, but does not imply causation. In this case, interpreting r² helps determine how well librarian salaries predict library science teacher salaries, guiding decisions or insights about their relationship.