[DATA] The following data represent the height (inches) of boys between the ages of 2 and 10 years.

b. Compute the standard error of the estimate, Sₑ.

Verified step by step guidance

1. Step 1: Understand that the standard error of the estimate, \(S_{e}\), measures the standard deviation of the observed values from the predicted values in a regression model. It quantifies the accuracy of predictions made by the regression line.
2. Step 2: Calculate the predicted heights (\(\hat{y}\)) for each boy's age (\(x\)) using the regression equation \(\hat{y} = b_0 + b_1 x\), where \(b_0\) is the intercept and \(b_1\) is the slope of the regression line. These values should be given or calculated from the data.
3. Step 3: Compute the residuals for each data point, which are the differences between the observed heights (\(y\)) and the predicted heights (\(\hat{y}\)): \(e_i = y_i - \hat{y}_i\).
4. Step 4: Square each residual to get \(e_i^2\), then sum all squared residuals: \(\sum e_i^2\).
5. Step 5: Use the formula for the standard error of the estimate: \[S_{e} = \sqrt{\frac{\sum e_i^2}{n - 2}}\] where \(n\) is the number of data points. This formula accounts for the degrees of freedom in simple linear regression.

Key Concepts

Standard Error of the Estimate (Sₑ)

The standard error of the estimate measures the average distance that observed values fall from the regression line. It quantifies the accuracy of predictions made by a regression model, showing how well the model fits the data. A smaller Sₑ indicates a better fit and more precise predictions.

Linear Regression and Residuals

Linear regression models the relationship between an independent variable (age) and a dependent variable (height) by fitting a line. Residuals are the differences between observed values and predicted values from this line. Calculating residuals is essential for determining the standard error of the estimate.

Sum of Squared Residuals (SSR)

The sum of squared residuals is the total of the squared differences between observed and predicted values. It is used to calculate the variance of residuals, which in turn helps compute the standard error of the estimate. SSR reflects the unexplained variation by the regression model.