[DATA] Concrete As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete:

c. Determine sb_1.

Verified step by step guidance

1. Identify that \(s_{b_1}\) represents the standard error of the slope in a simple linear regression model, where the 7-day strength (\(x\)) is the independent variable and the 28-day strength (\(y\)) is the dependent variable.
2. Calculate the slope \(b_1\) of the 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 \[S_{xx} = \sum (x_i - \bar{x})^2\]
3. Compute 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 values from the regression line, and \(n\) is the number of data points.
4. Calculate \(S_{xx}\) again as the sum of squared deviations of \(x\) values from their mean, which is needed for the denominator in the formula for \(s_{b_1}\).
5. Finally, compute the standard error of the slope \(s_{b_1}\) using the formula: \[s_{b_1} = \frac{s}{\sqrt{S_{xx}}}\] This value quantifies the variability of the slope estimate.

Key Concepts

Sample Standard Deviation (sb_1)

The sample standard deviation measures the amount of variation or dispersion in a set of sample data points. Specifically, sb_1 refers to the standard deviation of the predictor variable (x-values) in regression analysis. It is calculated by taking the square root of the variance, which is the average squared deviation from the sample mean.

Regression Analysis and Predictor Variable

In regression analysis, the predictor variable (often denoted as x) is the independent variable used to predict the response variable (y). Understanding the variability of the predictor variable through its standard deviation is essential for calculating regression coefficients and assessing the strength of the relationship.

Data Organization and Interpretation

Organizing data into paired observations (7-day and 28-day strengths) allows for comparison and analysis. Properly interpreting the data table is crucial to identify which values correspond to the predictor variable (7-day strength) and to apply statistical formulas correctly for calculations like sb_1.