1. **State the problem:** We are given a regression model with sample size $n=8$, $R^2=0.937$, and an ANOVA table. We need to find the adjusted $R^2$ (denoted $\bar{R}^2$) and the F-statistic for the model. 2. **Recall formulas:** - Adjusted $R^2$ is calculated by: $$\bar{R}^2 = 1 - \frac{(1-R^2)(n-1)}{n-p-1}$$ where $p$ is the number of predictors. - The F-statistic for the model is: $$F = \frac{\text{MS}_{\text{Regression}}}{\text{MS}_{\text{Residual}}}$$ 3. **Identify values:** - $n = 8$ - $R^2 = 0.937$ - From the ANOVA table, Regression degrees of freedom $DF_{Regression} = 2$ (so $p=2$ predictors) - Mean Square Regression $MS_{Regression} = 10.5128$ - Mean Square Residual $MS_{Residual} = 0.2846$ 4. **Calculate adjusted $R^2$:** $$\bar{R}^2 = 1 - \frac{(1-0.937)(8-1)}{8-2-1} = 1 - \frac{0.063 \times 7}{5} = 1 - \frac{0.441}{5} = 1 - 0.0882 = 0.9118$$ Rounded to three decimals: $\bar{R}^2 = 0.912$ 5. **Calculate F-statistic:** $$F = \frac{10.5128}{0.2846} = 36.93$$ 6. **Interpretation:** - The adjusted $R^2$ accounts for the number of predictors and sample size, providing a more unbiased estimate of model fit. - The F-statistic tests whether the regression model explains a significant amount of variance compared to residual error. **Final answers:** - Adjusted $R^2 = 0.912$ - $F_{model} = 36.93$