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


c. Determine s_{b₁}.

Verified step by step guidance

1. Identify the given data points for the variables x and y: x = {3, 4, 5, 7, 8} and y = {4, 6, 7, 12, 14}.
2. Calculate the slope \( b_1 \) of the least squares regression line using the formula: \[ b_1 = \frac{S_{xy}}{S_{xx}} \] where \[ S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}) \] and \[ S_{xx} = \sum (x_i - \bar{x})^2 \] Here, \( \bar{x} \) and \( \bar{y} \) are the means of x and y respectively.
3. Calculate the residual standard error \( s \) using the formula: \[ s = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n - 2}} \] where \( \hat{y}_i = b_0 + b_1 x_i \) are the predicted y-values from the regression line, and \( n \) is the number of data points.
4. Calculate the standard error of the slope \( s_{b_1} \) using the formula: \[ s_{b_1} = \frac{s}{\sqrt{S_{xx}}} \]
5. Summarize the steps: first find means \( \bar{x} \) and \( \bar{y} \), then compute \( S_{xx} \) and \( S_{xy} \), calculate slope \( b_1 \), find residuals and residual standard error \( s \), and finally compute \( s_{b_1} \) using the above formula.

Key Concepts

Simple Linear Regression

Simple linear regression models the relationship between a dependent variable y and an independent variable x by fitting a line y = b₀ + b₁x. The slope b₁ represents the change in y for a one-unit change in x. Understanding this relationship is essential for calculating regression parameters and interpreting their meaning.

Standard Error of the Slope (s_{b₁})

The standard error of the slope measures the variability or precision of the estimated slope coefficient b₁ in a regression model. It quantifies how much the estimated slope would vary if the sampling were repeated. Calculating s_{b₁} involves the residual standard error and the variability of the x-values.

Calculation of Regression Statistics from Data

To compute regression statistics like s_{b₁}, one must calculate sums of squares (SST, SSR, SSE), the slope b₁, residuals, and the residual standard error using the given paired data points. These calculations provide the foundation for estimating the accuracy and reliability of the regression coefficients.