Question 1202983: ***13-12
On the first statistics exam, the coefficient of determination between the hours studied and the grade earned was 76%. The standard error of estimate was 10. There were 24 students in the class. Develop an ANOVA table for the regression analysis of hours studied as a predictor of the grade earned on the first statistics exam. (Round your DF answers to nearest whole number and other answers to 2 places.)
Answer by Jason57t(3)   (Show Source):
You can put this solution on YOUR website! Sure, let's create the ANOVA (Analysis of Variance) table for a simple linear regression, using the provided information. The ANOVA table includes Source of Variation, Degrees of Freedom (DF), Sum of Squares (SS), Mean Square (MS), and the F-value.
We start by calculating the degrees of freedom:
DF_regression (or DF_model) = 1 (This is always 1 for simple linear regression because there's only 1 predictor variable)
DF_total = N - 1 = 24 - 1 = 23
DF_residual (or DF_error) = DF_total - DF_regression = 23 - 1 = 22
To calculate the Residual Sum of Squares (SS_residual or SS_error), we can use the standard error of estimate, because s = sqrt(SS_residual / DF_residual). So, SS_residual = s² * DF_residual = 10² * 22 = 2200.
The coefficient of determination (R² = 0.76) tells us the proportion of the total variability that is explained by the regression model. We can use it to estimate SS_regression and SS_total. We know that R² = SS_regression / SS_total, so SS_regression = R² * SS_total. We also know that SS_total = SS_regression + SS_residual, so SS_total = SS_residual / (1 - R²) = 2200 / (1 - 0.76) = 9166.67, and then SS_regression = 0.76 * 9166.67 = 6966.67.
The Mean Square (MS) is calculated by dividing the Sum of Squares by the corresponding degrees of freedom. Therefore, MS_regression = SS_regression / DF_regression = 6966.67 / 1 = 6966.67, and MS_residual = SS_residual / DF_residual = 2200 / 22 = 100.
Finally, the F-value can be computed by dividing MS_regression by MS_residual = 6966.67 / 100 = 69.67.
Now, we can populate the ANOVA table as follows:
Source of Variation Degrees of Freedom (DF) Sum of Squares (SS) Mean Square (MS) F-value
Regression (Model) 1 6966.67 6966.67 69.67
Residual (Error) 22 2200 100 N/A
Total 23 9166.67 N/A N/A
Please note, this analysis is based on assumptions that may or may not hold in the specific case of your data. These assumptions include linearity, independence, homoscedasticity, and normality. It is always recommended to perform residual analysis to check these assumptions before interpreting the results.
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