In Problems 5–10, use the results of Problems 7–12, respectively, from Section 4.2 to answer the following questions:


b. Compute the standard error, the point estimate for σ.

Verified step by step guidance

1. Step 1: Identify the point estimate for \( \sigma \), which in the context of regression analysis is the standard error of the estimate, often denoted as \( s_e \). This measures the typical distance that the observed values fall from the regression line.
2. Step 2: Calculate the predicted values \( \hat{y} \) for each \( x \) using the regression equation \( \hat{y} = b_0 + b_1 x \). To do this, you first need to find the slope \( b_1 \) and intercept \( b_0 \) of the regression line using the formulas:
3. \[ b_1 = \frac{n \sum xy - \sum x \sum y}{n \sum x^2 - (\sum x)^2} \]
4. \[ b_0 = \bar{y} - b_1 \bar{x} \]
5. where \( n \) is the number of data points, \( \sum xy \) is the sum of the products of \( x \) and \( y \), \( \sum x \) and \( \sum y \) are the sums of \( x \) and \( y \) values respectively, and \( \sum x^2 \) is the sum of squares of \( x \).
6. Step 3: Compute the residuals for each data point, which are the differences between the observed \( y \) values and the predicted \( \hat{y} \) values: \( e_i = y_i - \hat{y}_i \).
7. Step 4: Calculate the sum of squared residuals (SSR):
8. \[ SSR = \sum (y_i - \hat{y}_i)^2 \]
9. Step 5: Finally, compute the standard error of the estimate \( s_e \) using the formula:
10. \[ s_e = \sqrt{\frac{SSR}{n - 2}} \]
11. where \( n - 2 \) represents the degrees of freedom in simple linear regression.

Key Concepts

Point Estimate for σ (Standard Deviation of Errors)

In regression analysis, the point estimate for σ represents the estimated standard deviation of the error terms (residuals). It measures the typical distance that observed values fall from the regression line, indicating the model's accuracy. This estimate is often calculated using the residual standard error formula.

Standard Error of the Estimate

The standard error of the estimate quantifies the variability of the residuals around the regression line. It is computed as the square root of the sum of squared residuals divided by the degrees of freedom (n-2 for simple linear regression). This value helps assess the precision of predictions made by the regression model.

Simple Linear Regression and Residuals

Simple linear regression models the relationship between two variables by fitting a line that minimizes the sum of squared residuals, which are the differences between observed and predicted values. Understanding residuals is crucial for calculating both the point estimate for σ and the standard error, as they reflect the model's fit.