Regression Analysis from Computer Output

The general form of the linear regression equation is given by: $\hat{y} = a + bx$
A specific property of the least squares regression line is that it passes through the means of the data: $\bar{y} = a + b\bar{x}$
Technology is essential for finding least squares lines, and students are expected to be proficient at reading regression results from common computer statistics packages.
While output formats vary slightly across software packages, they share a standardized structure and typically provide more information than required for introductory analysis.

Dependent Variable Identification: The table explicitly identifies the response variable (y-variable). In the provided example, the output states: "Dependent variable is: Total Fat".
The S Value ( $s$ ): This represents the standard deviation of the residuals. In the example output, $s = 9.277$ .
The R-Squared Value ( $R^2$ ): This is the coefficient of determination, usually labeled as "R squared". In the example, $R^2 = 69.0\%$ .
The Coefficient Table: This is the central grid containing the values needed for the regression equation. It typically has columns for Variable, Coefficient, SE(Coeff), t-ratio, and P-value. - The y-intercept ( $a$ ): Labeled in the far-left column as "Intercept" or "Constant". Its numerical value is found in the adjacent "Coefficient" column. In the example, the intercept coefficient is $6.83077$ . - The Slope ( $b$ ): Labeled in the far-left column using the name of the independent (predictor/x) variable. In the example, the variable is "Protein", and its coefficient is $0.971381$ .

To create the equation, combine the identified intercept and slope with their respective variables.
Using the example provided: - Dependent Variable: $\text{Fat}$ - Independent Variable: $\text{Protein}$ - Intercept ( $a$ ): $6.83077$ - Slope ( $b$ ): $0.971381$
Final Equation: $\widehat{\text{Fat}} = 6.83077 + 0.971381 \times \text{Protein}$
Simplified form noted in the text: $\hat{y} = 6.83 + 0.97x$

Software Precision: Computer packages often report many more digits for the estimated slope and intercept than can be physically estimated from the original data (e.g., data measured to the nearest gram).
General Rounding Guideline: Round most reported numbers to one more digit than the precision of the original data.
Slope Rounding Guideline: Round the slope coefficient to two more digits than the precision of the original data.

Several columns in the regression table are not immediately necessary for introductory regression analysis and can be ignored for now: - SE(Coeff): Standard Error of the coefficient (Example: Intercept = $2.664$ , Protein = $0.1709$ ). - t-ratio: The test statistic for the hypothesis test on the coefficient (Example: Intercept = $2.56$ , Protein = $8.04$ ). - P-value: The significance level for the hypothesis test (Example: Intercept = $0.0158$ , Protein = $<0.0001$ ).