"Finding the Coefficient of Determination and the Standard Error of Estimate In Exercises 11-20, use the data to (b) find the standard error of estimate s_e and interpret the result.
12. [APPLET] Median and Mean Hourly Wages The table shows the median and mean hourly wages (in dollars) in 10 states in a recent year. The equation of the regression line is y = 1.208x + 1.495. (Source: U.S. Census Bureau)
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Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the standard error of estimate (s_e) using the given data and interpreting the result. The regression equation provided is y = 1.208x + 1.495, where x represents the median hourly wage and y represents the mean hourly wage.
2. Step 2: Calculate the predicted values (ŷ) for each x value using the regression equation. Substitute each x value from the table into the equation y = 1.208x + 1.495 to compute the corresponding predicted y values.
3. Step 3: Compute the residuals for each data point. The residual for each data point is calculated as the difference between the observed y value and the predicted y value: residual = y - ŷ.
4. Step 4: Square each residual and sum them up. This gives the sum of squared residuals (SSR), which is a measure of the total deviation of the observed values from the predicted values.
5. Step 5: Use the formula for the standard error of estimate: s_e = sqrt(SSR / (n - 2)), where n is the number of data points. Divide the SSR by (n - 2) and take the square root to find s_e. Interpret the result as the average distance that the observed values fall from the regression line.

Key Concepts

Coefficient of Determination (R²)

The Coefficient of Determination, denoted as R², measures the proportion of variance in the dependent variable that can be predicted from the independent variable. It ranges from 0 to 1, where 0 indicates no explanatory power and 1 indicates perfect prediction. A higher R² value suggests a stronger relationship between the variables, making it essential for evaluating the effectiveness of a regression model.

Standard Error of Estimate (s_e)

The Standard Error of Estimate (s_e) quantifies the accuracy of predictions made by a regression model. It represents the average distance that the observed values fall from the regression line. A smaller s_e indicates that the data points are closer to the predicted values, reflecting a more reliable model. It is calculated using the residuals, which are the differences between observed and predicted values.

Regression Line

A regression line is a straight line that best fits the data points in a scatter plot, representing the relationship between the independent variable (x) and the dependent variable (y). The equation of the regression line, typically in the form y = mx + b, includes a slope (m) and y-intercept (b). In this context, the regression line helps to predict mean hourly wages based on median hourly wages, providing insights into wage trends.