Given the following computer output:
Dependent variable is: grade
No Selector
R -squared $=73.4 \% \quad R$-squared (adjusted) $=72.1 \%$
The value of the coefficient of determination is:
$s=4.847$ with $22-2=20$ degrees of freedom
0.734
4.847
0.721
Answer (2)
The R-squared value of 0.734 indicates that 73.4% of the variance in the 'grade' variable is explained by the model. The adjusted R-squared of 0.721 accounts for the number of predictors. The standard error of 4.847 represents the typical distance of observed values from the regression line. The degrees of freedom are 20. R 2 = 0.734 , A d j u s t e d R 2 = 0.721 , s = 4.847 , df = 20
Explanation
Understanding the Output We are given the output of a regression analysis where the dependent variable is 'grade'. We need to interpret the key statistics provided in the output.
Interpreting R-squared The R-squared value is 73.4% or 0.734. This means that 73.4% of the variance in the 'grade' variable is explained by the independent variable(s) in the regression model. In other words, the model accounts for a significant portion of the variability in grades.
Interpreting Adjusted R-squared The adjusted R-squared is 72.1% or 0.721. This is a modified version of R-squared that adjusts for the number of predictors in the model. It penalizes the inclusion of unnecessary variables that do not significantly improve the model's fit. The adjusted R-squared is slightly lower than the R-squared, which suggests that the model is not overfitted.
Interpreting Standard Error The standard error (s) is 4.847. This represents the typical distance that the observed values fall from the regression line. A smaller standard error indicates that the model's predictions are more precise.
Interpreting Degrees of Freedom The degrees of freedom are 20. This is calculated as the sample size (22) minus the number of parameters in the model (2). The degrees of freedom are used in hypothesis testing and confidence interval calculations.
Conclusion In summary, the regression model explains a substantial portion of the variance in 'grade' (73.4%), with a reasonable standard error (4.847). The adjusted R-squared (72.1%) suggests that the model is not overfitted.
Examples
In educational research, regression analysis is often used to understand the factors that influence student performance. For example, we might want to know how factors like study time, attendance, and prior grades affect a student's final grade. The R-squared value tells us how well our model explains the variation in grades, while the standard error gives us an idea of the accuracy of our predictions. This information can be used to identify areas where students need additional support and to develop interventions to improve student outcomes.
The regression output shows an R-squared of 73.4%, meaning the model explains 73.4% of the variance in grades. The adjusted R-squared of 72.1% indicates a well-fitted model without excessive predictors, and a standard error of 4.847 reflects the model's accuracy. The best answer is 0.734.
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